Structure Identification, Estimation and Variable Selection for Varying Coefficient EV Models With Longitudinal Data

ABSTRACT

In this article, we propose a bias‐corrected double penalized quadratic inference functions method to simultaneously identify model structure, estimate parameters, and perform variable selection for varying coefficient errors‐in‐variables (EV) models with longitudinal data. Unlike the linear models or the partial linear varying coefficient models, the proposed method does not assume in advance whether each regression coefficient is constant or varying. Instead, it represents each coefficient as a nonparametric function and identifies whether it is constant or varying using the proposed method. By employing a B‐spline basis to approximate the unknown coefficient functions, the proposed method integrates a bias‐corrected quadratic inference function with two penalized terms to achieve structure identification, estimation, and variable selection. Under certain regularized conditions, the consistency and sparsity properties of the estimator are established. Moreover, a three‐step iterative algorithm is developed to implement the proposed method in practice. Simulation studies and a real data analysis demonstrate the superior finite‐sample performance of the method.

1. Introduction

Varying coefficient (VC) models extend the classical linear regression framework by treating each regression coefficient as a smoothing function, thus allowing covariate effects to evolve dynamically across longitudinal measurements. Their enhanced flexibility and interpretability have spurred considerable interest in recent longitudinal studies. A wide range of estimation techniques has been developed, including kernel methods [1, 2]; locally polynomial approaches [3]; least squares estimators [4]; and spline‐based methods [5, 6, 7]. In addition, Xue and Zhu [8] proposed an empirical likelihood‐based interval estimation method; Qu and Li [9] introduced a penalized quadratic inference functions (QIF) method for model estimation. Comprehensive reviews of VC models can be found in Fan and Zhang [10] and Park et al. [11].

Suppose longitudinal data $\{(Y_{ij},X_{ij},t_{ij}):i=1,2,\dots ,n,j=1,2,\dots ,n_i\}$ satisfy the VC model

$$\begin{array}{l}\hfill Y_{ij}=X_{ij}^{\mathrm{T}}\alpha (t_{ij})+{\epsilon }_{ij},i=1,2,\dots ,n,j=1,2,\dots ,n_i,\end{array}$$ (1)

where $Y_{ij}\in ℝ$ is the response variable, $X_{ij}\in {ℝ}^q$ represents the covariates at $t_{ij}\in ℝ$, and $X_{ij}={(X_{ij}^1,X_{ij}^2,\dots ,X_{ij}^q)}^{\mathrm{T}}$. The term ${\epsilon }_{ij}\in ℝ$ denotes a zero‐mean stochastic process, and ${\epsilon }_{ij}$ is independent of $X_{ij}$. For each $t\in ℝ$, the coefficient vector $\alpha (t)={({\alpha }_1(t),{\alpha }_2(t),\dots ,{\alpha }_q(t))}^{\mathrm{T}}$ consists of some unknown smooth functions ${\alpha }_k(t)(k=1,2,\dots ,q)$ defined on the interval $t\in [0,1]$ with some scale transformation. Some moment assumptions are stated as $\mathrm{E}(Y_{ij}|X_{ij},t_{ij})={\mu }_{ij}$ and $\mathrm{var}(Y_{ij}|X_{ij},t_{ij})=\nu ({\mu }_{ij})$, where $\nu (·)$ is a known variance function.

Previous studies often assume that covariates are measured without errors. However, in practice, obtaining precise measurements for some covariates is frequently challenging or difficult to achieve, which results in inevitable measurement errors or potential unobserved covariates. Neglecting these measurement errors can result in biased parameter estimates and misleading inferences. To address this issue, we extend model (1) by incorporating additive measurement errors in the covariates, resulting in the varying coefficient errors‐in‐variables (VCEV) model

$$\begin{array}{l}\hfill \left\{\begin{array}{l}{Y}_{ij}=X_{ij}^{\mathrm{T}}\alpha (t_{ij})+{\epsilon }_{ij},\\ W_{ij}=X_{ij}+u_{ij},\end{array}\right.,i=1,2,\dots ,n,j=1,2,\dots ,n_i,\end{array}$$ (2)

where $W_{ij}={(W_{ij}^1,W_{ij}^2,\dots ,W_{ij}^q)}^{\mathrm{T}}\in {ℝ}^q$ represents the observed covariates, and $u_{ij}={(u_{ij}^1,u_{ij}^2,\dots ,u_{ij}^q)}^{\mathrm{T}}\in {ℝ}^q$ denotes the zero‐mean measurement errors with a diagonal covariance matrix ${\sum }_u$. Additionally, we assume that $\text{cov}(u_{i{j}_1},u_{i{j}_2})=0$ for $j_1\ne j_2$, the errors $u_{ij}$ are mutually independent, and all $u_{ij}$ are independent of $(X_{ij},t_{ij},{\epsilon }_{ij})$, where $\text{cov}(·)$ represents the covariance operator. To effectively account for measurement errors, supplementary information regarding ${\sum }_u$ is required in practice, and it is typically assumed that ${\sum }_u$ can be either estimated from the data or known in advance.

For model (2), Li and Greene [12] applied a locally corrected method to estimate the coefficient functions. Yang, Li and Peng [13] explored the empirical likelihood method for model (2) in the context of longitudinal data. For the VCEV models and partial linear varying coefficient EV (PLVCEV) models with longitudinal data, Zhao, Gao and Cui [14] and Zhao et al. [15] proposed a type of bias‐corrected penalized QIF method, which can handle measurement errors in covariates and within‐subject correlations simultaneously, estimate and select significant non‐zero parametric and nonparametric components. More studies about VCEV models with longitudinal data can be found in Zhao, Gao and Cui [14] and references therein, details are omitted here. Moreover, for structural change points in varying coefficient models, Zhao et al. [16] proposed the adaptive jump‐preserving (AJP) estimator. In the context of simultaneous measurement‐error correction and change‐point detection, Zhao et al. [17] developed the single‐index measurement error jump regression model. More studies about model (2) are omitted here.

Structure identification and variable selection [18, 19, 20] are of fundamental importance, as the validity of a fitted model and its subsequent inferences are critically dependent on the correctness of the specified structure. Linear models traditionally assume that all regression coefficients are constant; however, VC models generalize this concept by allowing the coefficients to be functions that vary over the domain. Building on this idea, partial linear varying coefficient (PLVC) models further distinguish between covariates by assuming that while some effects remain constant, others are set as varying functions in advance. In practice, arbitrarily designating which subset of variables should have constant versus varying effects on the response introduces a significant risk of model misspecification. Tang, Wang and Zhu [21]; Wang and Lin [22] and Xu et al. [23] have developed methodologies that consistently differentiate among varying coefficients, nonzero constant coefficients, and zero coefficients, in a manner that essentially achieves performance as if the true model structure and relevant variables were known a priori, thereby yielding robust selection outcomes in the analysis of longitudinal data. Inspired by Tang et al. [21], Xu et al. [23] and Wang and Lin [24], we propose a structure identification, estimation and variable selection approach based on the bias‐corrected double penalized quadratic inference functions (BCDPQIF), which is capable of addressing both measurement errors and within‐subject correlations, while accurately identifying the model structure. Additionally, by appropriately choosing tuning parameters, we provide a theoretical analysis of the consistency and sparsity properties of the proposed method.

The rest of this article is organized as follows. The BCDPQIF method and some theoretical results are stated in Section 2. Computational algorithm and selection of tuning parameters are presented in Section 3. Simulation studies and a real data analysis are performed to evaluate the proposed method in Section 4. Finally, we provide the conclusions and a discussion in Section 5. The derivation process of some equations, the proofs of theorems and some other numerical results are provided in Appendix A (Tables A1, A2, A3, A4).

2. Methodology and Main Results

2.1. Bias‐Corrected Double Penalized Quadratic Inference Functions Method

Following Wang and Lin [24], ${\alpha }_k(t)(k=1,2,\dots ,q)$ can be represented approximately as

$${\alpha }_k(t)={\beta }_k+{\eta }_k(t),$$ (3)

where $\mathrm{E}({\eta }_k(t))=0$. Denote a B‐spline basis $B(t)={(B_1(t),B_2(t),\dots ,B_L(t))}^{\mathrm{T}}$ with the order $d$, where $L=K+d$, $K(>0)$ is the number of interior knots. The compact support and piecewise‐polynomial structure of B‐splines markedly reduce computational complexity, facilitating rapid model fitting even in high‐dimensional or large‐scale datasets. Additionally, B‐splines deliver superior flexibility and precision in coefficient function estimation, adeptly capturing localized features and complex functional patterns. Thus, ${\eta }_k(t)$ can be approximated as

$${\eta }_k(t)\approx \sum _{l=1}^L{B}_l(t){\gamma }_{kl}=B{(t)}^{\mathrm{T}}{\gamma }_k,$$ (4)

where ${\gamma }_k={({\gamma }_{k1},{\gamma }_{k2},\dots ,{\gamma }_{kL})}^{\mathrm{T}}\in {ℝ}^L$, $k=1,2,\dots ,q$, is a regression coefficient vector of B‐spline basis. Thus we can have

$${\alpha }_k(t)\approx {\beta }_k+B{(t)}^{\mathrm{T}}{\gamma }_k,k=1,2,\dots ,q.$$ (5)

By replacing ${\alpha }_k(t)(k=1,2,\dots ,q)$ by Equation (5), model (2) can be represented as

$$\begin{array}{l}\hfill \left\{\begin{array}{cc}\hfill Y_{ij}& \approx X_{ij}^{\mathrm{T}}\beta +{\tilde{X}}_{ij}^{\mathrm{T}}\gamma +{\epsilon }_{ij}\hfill \\ \hfill W_{ij}& =X_{ij}+u_{ij}\hfill \\ \hfill {\tilde{W}}_{ij}& ={\tilde{X}}_{ij}+{ũ}_{ij}\hfill \end{array}\right.,i=1,2,\dots ,n,j=1,2,\dots ,n_i,\end{array}$$ (6)

where $\beta ={({\beta }_1,{\beta }_2,\dots ,{\beta }_q)}^{\mathrm{T}}$, $\gamma ={({\gamma }_1^{\mathrm{T}},{\gamma }_2^{\mathrm{T}},\dots ,{\gamma }_q^{\mathrm{T}})}^{\mathrm{T}}$, $B_{ij}=I_q\otimes B(t_{ij})$, ${\tilde{X}}_{ij}=B_{ij}{X}_{ij}$, ${ũ}_{ij}=B_{ij}{u}_{ij}$, ${ũ}_{ij}=B_{ij}{u}_{ij}$, $I_q$ is the $q\times q$ identity matrix. Then $u_{ij}$ are independent of $(X_{ij},t_{ij},{\epsilon }_{ij})$. $\mathrm{E}(u_{ij})=0$, $\text{cov}(u_{ij})={\sum }_u$, $\text{cov}(u_{i{j}_1},u_{i{j}_2})=0$ for $j_1\ne j_2$.

We can obtain the following generalized estimating equations (GEE) about $\theta ={({\beta }^{\mathrm{T}},{\gamma }^{\mathrm{T}})}^{\mathrm{T}}$ as

$$\begin{array}{l}\hfill \sum _{i=1}^n{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{V}_i^{-1}(Y_i-(W_i,{\tilde{W}}_i)\theta )=0.\end{array}$$ (7)

Thus, we have

$$\begin{array}{ll}& \mathrm{E}\left(\sum _{i=1}^n\left[{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{V}_i^{-1}(Y_i-(W_i,{\tilde{W}}_i)\theta )\right]\right)\\ & =\mathrm{E}\left(\sum _{i=1}^n\left[\left({(X_i,{\tilde{X}}_i)}^{\mathrm{T}}+{\left(u_i,{ũ}_i\right)}^{\mathrm{T}}\right)V_i^{-1}\left(Y_i-((X_i,{\tilde{X}}_i)+\left(u_i,{ũ}_i\right))\theta \right)\right]\right)\\ & =\mathrm{E}\left(\sum _{i=1}^n\left[{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{V}_i^{-1}(Y_i-(X_i,{\tilde{X}}_i)\theta )\right.\right.+{\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{V}_i^{-1}(Y_i-(X_i,{\tilde{X}}_i)\theta )\\ & \left.\left.-{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{V}_i^{-1}\left(u_i,{ũ}_i\right)\theta -{\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{V}_i^{-1}\left(u_i,{ũ}_i\right)\theta \right]\right)\\ & =-n\mathrm{E}\left({\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{V}_i^{-1}\left(u_i,{ũ}_i\right)\theta \right)\\ & \ne 0.\end{array}$$

This demonstrates that Equation (7) is biased when $\theta \ne 0$, which can not be used to obtain unbiased estimations. In order to overcome this drawback, we obtain an unbiased estimating equation by adding the term $\mathrm{E}\left({(u_i,{ũ}_i)}^{\mathrm{T}}{V}_i^{-1}(u_i,{ũ}_i)\theta \right)$. Accordingly, the bias‐corrected GEE for $\theta$ can be derived as

$$\begin{array}{ll}& \sum _{i=1}^n\left\{{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{V}_i^{-1}(Y_i-(W_i,{\tilde{W}}_i)\theta )\right.+\left.\mathrm{E}\left({(u_i,{ũ}_i)}^{\mathrm{T}}{V}_i^{-1}(u_i,{ũ}_i)\theta \right)\right\}=0,\end{array}$$ (8)

where $W_i={(W_{i1},W_{i2},\dots ,W_{i{n}_i})}^{\mathrm{T}}$, ${\tilde{W}}_i={({\tilde{W}}_{i1},{\tilde{W}}_{i2},\dots ,{\tilde{W}}_{i{n}_i})}^{\mathrm{T}}$, $Y_i={\left(Y_{i1},Y_{i2},\dots ,Y_{i{n}_i}\right)}^{\mathrm{T}}$, $V_i$ is the covariance matrix of $Y_i$. Then we take $V_i$ as $V_i=A_i^{1/2}{R}_i(\rho )A_i^{1/2}$, where $R_i(\rho )$ is a common working correlation matrix with a nuisance parameter $\rho$, $A_i=\text{diag}(\mathrm{var}(Y_{i1}),\mathrm{var}(Y_{i2}),\dots ,\mathrm{var}(Y_{i{n}_i}))$. Liang and Zeger [25] stated that, in some simple cases, a consistent estimator for $\rho$ may not exist, which could undermine the validity of the GEE method.

To overcome this limitation of the GEE, Qu, Lindsay and Li [26] proposed a QIF approach, assuming that $R_i^{-1}(\rho )={\sum }_{\kappa =1}^s{a}_{\kappa }{M}_{\kappa }$, where $M_{\kappa }(\kappa =1,2,\dots ,s)$ are some known simple matrices and $a_{\kappa }(\kappa =1,2,\dots ,s)$ are some unknown constants. The QIF method treats $a_{\kappa }(\kappa =1,2,\dots ,s)$ as the nuisance parameters, and approximates $R_i^{-1}(\rho )$ by a linear combination of a class of basis matrices as

$$\begin{array}{l}\hfill R_i^{-1}(\rho )=\sum _{\kappa =1}^s{a}_{\kappa }{M}_{\kappa }.\end{array}$$ (9)

One can see more details about $M_{\kappa }(\kappa =1,2,\dots ,s)$ in Qu, Lindsay, and Li [26], which are omitted here. By substituting $R_i^{-1}(\rho )$ into Equation (8), the resulting new bias‐corrected GEE is derived as

$$\begin{array}{ll}\hfill & \sum _{i=1}^n\left\{{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}\left(\sum _{\kappa =1}^s{a}_{\kappa }{M}_{\kappa }\right)A_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )\right.\\ \hfill & \left.+\mathrm{E}\left({(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}\left(\sum _{\kappa =1}^s{a}_{\kappa }{M}_{\kappa }\right)A_i^{-1/2}(u_i,{ũ}_i)\theta \right)\right\}=0.\end{array}$$ (10)

Thus, following Qu, Lindsay and Li [26], one can define a bias‐corrected extended score function ${\bar{g}}_n(\theta )$ as

$$\begin{array}{ll}{\bar{g}}_n(\theta )& =\frac{1}{n}\sum _{i=1}^n{g}_i(\theta )\\ \hfill & =\frac{1}{n}\sum _{i=1}^n\left(\begin{array}{c}{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_1{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+D_i^{(1)}\theta \\ {(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_2{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+D_i^{(2)}\theta \\ ⋮\\ {(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_s{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+D_i^{(s)}\theta \end{array}\right),\end{array}$$

where $D_i^{(\kappa )}=\mathrm{E}({(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i)),\kappa =1,2,\dots ,s.$ By some matrix calculations, we have

$$\begin{array}{ll}& D_i^{(\kappa )}=\\ & \left(\begin{array}{cc}\text{tr}\left(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\right)·{\sum }_u& {\sum }_u\otimes \left({\mathbf{1}}_{1\times n_i}\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\\ {\left({\sum }_u\otimes \left({\mathbf{1}}_{1\times n_i}\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\right)}^{\mathrm{T}}& {\sum }_u\otimes \left(B_i\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\end{array}\right),\end{array}$$ (11)

where ${\mathbf{1}}_{1\times n_i}={(1,1,\dots ,1)}_{1\times n_i}$, $B_i=(B(t_{i1}),B(t_{i2}),\dots ,B(t_{i{n}_i}))$, $\text{diag}(·)$ is a diagonal matrix operator. The detailed derivation process about Equation (11) can be found in Appendix A.

Since ${\sum }_u$ is unknown, $D_i^{(\kappa )}$ need to be estimated. Without loss of generality, we first consider a balanced longitudinal dataset, that is, $n_i=n_0$ ($i=1,2,\dots ,n$) and $n_0$ is a fixed positive integer. Suppose $W_{ij}$ ($j=1,2,\dots ,n_0$) can be observed $m_i$ times for each subject, with $W_{ij}^{(r)}=X_{ij}+u_{ij}^{(r)}$, $r=1,2,\dots ,m_i$. A consistent estimator for ${\sum }_u$ can be computed as follows

$$\begin{array}{l}\hfill {\widehat{\sum }}_u=\frac{1}{n{n}_0}\sum _{i=1}^n\sum _{j=1}^{n_0}\left(\frac{1}{m_i-1}\sum _{r=1}^{m_i}(W_{ij}^{(r)}-{\bar{W}}_{ij}){(W_{ij}^{(r)}-{\bar{W}}_{ij})}^{\mathrm{T}}\right),\end{array}$$ (12)

where ${\bar{W}}_{ij}=\frac{1}{m_i}{\sum }_{r=1}^{m_i}{W}_{ij}^{(r)}$. It should be pointed out that for unbalanced longitudinal data, following the idea from Xue, Qu and Zhou [27], one can utilize cluster‐specific transformation matrices to reformat the data with an unbalanced cluster size, details are omitted here. Then one can get a consistent estimator ${\widehat{D}}_i^{(\kappa )}$ using the plug‐in method as

$$\begin{array}{ll}& {\widehat{D}}_i^{(\kappa )}=\\ & \left(\begin{array}{cc}\text{tr}\left(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\right)·{\widehat{\sum }}_u& {\widehat{\sum }}_u\otimes \left({\mathbf{1}}_{1\times n_i}\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\\ {\left({\widehat{\sum }}_u\otimes \left({\mathbf{1}}_{1\times n_i}\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\right)}^{\mathrm{T}}& {\widehat{\sum }}_u\otimes \left(B_i\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\end{array}\right).\end{array}$$ (13)

For the sake of simplicity, for both balanced and unbalanced data, we denote the estimators of ${\sum }_u$ and $D_i^{(k)}$ as ${\widehat{\sum }}_u$ and ${\widehat{D}}_i^{(k)}$, respectively. Therefore, based on Equations (12) and (13), a consistent estimator ${\widehat{\bar{g}}}_n(\theta )$ for ${\bar{g}}_n(\theta )$ can be obtained as

$$\begin{array}{ll}{\widehat{\bar{g}}}_n(\theta )& =\frac{1}{n}\sum _{i=1}^n{ĝ}_i(\theta )\\ \hfill & =\frac{1}{n}\sum _{i=1}^n\left(\begin{array}{c}{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_1{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+{\widehat{D}}_i^{(1)}\theta \\ {(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_2{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+{\widehat{D}}_i^{(2)}\theta \\ ⋮\\ {(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_s{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+{\widehat{D}}_i^{(s)}\theta \end{array}\right).\end{array}$$

It is evident that the equation $\mathrm{E}({\widehat{\bar{g}}}_n(\theta ))=0$ involves more equations than the number of parameters to be estimated, and will result in the over‐identified problem. As a result, it can not be directly used to estimate $\theta$. Toovercome this problem, following Qu, Lindsay, and Li [26], we construct a bias‐corrected QIF (BCQIF) about $\theta$ as

$$\begin{array}{ll}\hfill Q_n(\theta )& =n{\widehat{\bar{g}}}_n^{\mathrm{T}}(\theta ){\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n(\theta ),\end{array}$$ (14)

where ${\mathrm{\Omega }}_n=\frac{1}{n}{\sum }_{i=1}^n{ĝ}_i((\theta )){ĝ}_i^{\mathrm{T}}((\theta ))$. The matrix ${\mathrm{\Omega }}_n$ is the sample covariance of the moment conditions and serves as the optimal weighting matrix in the GMM criterion. This choice guarantees the estimator's properties (Qu, Lindsay and Li [26]).

Then a BCQIF estimator $\tilde{\theta }$ can be obtained as

$$\begin{array}{ll}\hfill \tilde{\theta }& =\underset{\theta }{\arg \min }{Q}_n(\theta ).\end{array}$$ (15)

As we know, the VCEV models assume that all the regression coefficients in the model are varying, the PLVCEV models presuppose that some of the regression coefficients in the model are constant while others are varying. All the regression coefficients in the linear EV models are set as all constants. However, these would expose us to the risk of assumptions in practice. Not only that, but the regression coefficients may also have zero coefficients. Therefore, the structure identification and variable selection of the model become very important and indispensable. To solve these problems, we propose the following BCDPQIF $Q_p(\theta )$ to do structure identification, estimation and variable selection simultaneously for model (2), defined as

(16)

where $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}={({\gamma }_k^{\mathrm{T}}\mathrm{H}{\gamma }_k)}^{1/2},$ and $\mathrm{H}={(h_{ij})}_{L\times L},h_{ij}={\int }_0^1{B}_i(t)B_j^{\mathrm{T}}(t)dt,k=1,2,\dots ,q.$ $I(·)$ is an indicator function. ${\lambda }_{1k},{\lambda }_{2k}(k=1,2,\dots ,q)$ are some tuning parameters. $p_{{\lambda }_{1k}}(·)$ and $p_{{\lambda }_{2k}}(·)$ are the SCAD [18] penalty functions defined as

$$\begin{array}{l}\hfill p_{\lambda }(|w|)=\left\{\begin{array}{ll}\lambda |w|,& |w|\le \lambda ,\\ \frac{(a^2-1){\lambda }^2-{(|w|-a\lambda )}^2}{2(a-1)},& \lambda <|w|\le a\lambda ,\\ \frac{1}{2}(a+1){\lambda }^2,& |w|>a\lambda .\end{array}\right.\end{array}$$ (17)

where $a=3.7$. It should be noted that the penalty functions $p_{{\lambda }_{1k}}(·)$ and $p_{{\lambda }_{2k}}(·)$ here do not necessarily have to be the SCAD penalty function, one can use other penalty functions that we are familiar with, such as LASSO or MCP [19] penalty functions. In our work, we employ the SCAD penalty function to evaluate the proposed method.

Then the BCDPQIF estimator of $\theta ={({\beta }^{\mathrm{T}},{\gamma }^{\mathrm{T}})}^{\mathrm{T}}$ is given by

$$\begin{array}{l}\hfill \widehat{\theta }={({\widehat{\beta }}^{\mathrm{T}},{\widehat{\gamma }}^{\mathrm{T}})}^{\mathrm{T}}=\arg \underset{\theta }{\min }{Q}_p(\theta ).\end{array}$$ (18)

Furthermore, the BCDPQIF estimators of ${\alpha }_k(t)(k=1,2,\dots ,q)$ can be obtained by

$${\widehat{\alpha }}_k(t)={\widehat{\beta }}_k+B{(t)}^{\mathrm{T}}{\widehat{\gamma }}_k.$$ (19)

Remark 1

Leveraging the SCAD penalty function's properties, the BCDPQIF method can identify, estimate and select the coefficient functions simultaneously. In Equation (16), the first penalty term, $n{\sum }_{k=1}^q{p}_{{\lambda }_{1k}}\left(\Vert {\gamma }_k{\Vert }_{\mathrm{H}}\right)$, determines whether the functional component ${\eta }_k(·)$ of ${\alpha }_k(·)$ is zero or nonzero, thereby distinguishing varying coefficients from constant ones. For coefficients identified as constant, the second penalty term, $n{\sum }_{k=1}^q{p}_{{\lambda }_{2k}}\left(|{\beta }_k|\right)I\left(\Vert {\gamma }_k{\Vert }_{\mathrm{H}}=0\right),$ further evaluates whether they are zeros or not and selects the nonzero constant coefficients. As a result, the BCDPQIF method is more versatile because it does not require pre‐specifying whether a coefficient is constant or varying. Instead, it can identify both varying and constant coefficients, and simultaneously estimate and select the varying coefficients and the nonzero constant coefficients. This generality makes it applicable to a wide range of models, including the linear EV models, VCEV models and PLVCEV models.

2.2. Asymptotic Properties

First, we give some necessary notations. Let ${\alpha }_0(·)={({\alpha }_{01}(·),{\alpha }_{02}(·),\dots ,{\alpha }_{0q}(·))}^{\mathrm{T}}$ be the real coefficients in model (2). Correspondingly, the real ${\beta }_k$ and ${\eta }_k(t)$ are denoted as ${\beta }_{0k}$ and ${\eta }_{0k}(t)$ for $k=1,2,\dots ,q$, where ${\alpha }_{0k}(t)={\beta }_{0k}+{\eta }_{0k}(t)$. Let ${\gamma }_{0k}$ be the B‐spline regression coefficient corresponding to ${\eta }_{0k}(t)$. Denote ${\beta }_0={({\beta }_{01},{\beta }_{02},\dots ,{\beta }_{0q})}^{\mathrm{T}}$, ${\gamma }_0={({\gamma }_{01}^{\mathrm{T}},{\gamma }_{02}^{\mathrm{T}},\dots ,{\gamma }_{0q}^{\mathrm{T}})}^{\mathrm{T}}$. Without loss of generality, it is assumed that ${\alpha }_{0k}(·)(k=1,2,\dots ,c)$ are nonzero constant coefficients, ${\alpha }_{0k}(·)(k=c+1,c+2,\dots ,v)$ are varying coefficients, ${\alpha }_{0k}(·)=0$ for $k=v+1,v+2,\dots ,q$.

Denote $𝒞=\{1,2,\dots ,c\},𝒱=\{c+1,c+2,\dots ,v\},𝒵=\{v+1,v+2,\dots ,q\}$. $k\in 𝒞$ implies that ${\alpha }_k(·)$ is a nonzero constant, $k\in 𝒱$ implies that ${\alpha }_k(·)$ is a varying coefficient, and $k\in 𝒵$ implies ${\alpha }_k(·)=0$. Using the BCDPQIF method, ${\alpha }_k(·)(k=1,2,\dots ,q)$ can be classified into three categories, that is, varying coefficients, nonzero constant coefficients and zero coefficients. Denote ${\alpha }^{(𝒞)}(·)={\left({\alpha }_1(·),{\alpha }_2(·),\dots ,{\alpha }_c(·)\right)}^{\mathrm{T}}$ and ${\alpha }^{(𝒱)}(·)={\left({\alpha }_{c+1}(·),{\alpha }_{c+2}(·),\dots ,{\alpha }_v(·)\right)}^{\mathrm{T}}$. The corresponding real functions of ${\alpha }^{(𝒞)}(·)$ and ${\alpha }^{(𝒱)}(·)$ are denoted as ${\alpha }_0^{(𝒞)}(·)$ and ${\alpha }_0^{(𝒱)}(·)$, respectively.

Some necessary regularity conditions for the asymptotic properties are stated as follows.

• C1: $0<n_i<\infty$ for $i=1,2,\dots ,n$.
• C2: ${\alpha }_k(t)(k=1,2,\dots ,q)$ are $r$ th continuously differentiable on $(0,1)$, and $r\ge 2$.
• C3: $\exists$ unique ${\theta }_0\in \mathrm{\Theta }$ satisfies $\mathrm{E}({\widehat{\bar{g}}}_n\left({\theta }_0\right))=o(1)$, where $\mathrm{\Theta }$ is the parameter space.
• C4:There exists an invertible matrix ${\mathrm{\Omega }}_0$ s.t. ${\mathrm{\Omega }}_n\stackrel{\text{a.s.}}{\to }{\mathrm{\Omega }}_0$.
• C5: $\sup ‖V_i‖<\infty$ and exists $\delta >0$ s.t. $\sup \mathrm{E}\{\Vert {\epsilon }_i{\Vert }^{2+\delta }\}<\infty$, $\mathrm{E}{‖u_i‖}^8<\infty$, $\mathrm{E}{‖{ũ}_i‖}^8<\infty$, where $\Vert ·\Vert$ is the modulus of the largest singular values.
• C6: $A_i\ge 0$, ${\sup }_i‖A_i‖<\infty$.
• C7: $\mathrm{E}{‖X_i‖}^4<\infty$, $\mathrm{E}\Vert {\tilde{X}}_i{\Vert }^4<\infty$, $i=1,2,\dots ,n$.
• C8:Let interior knots $\{{\omega }_i,i=1,2,\dots ,K\}$ of $B(t)$ satisfy ${\max }_{1\le i\le K}\left|\mathrm{\Delta }{\omega }_{i+1}-\mathrm{\Delta }{\omega }_i\right|=o(K^{-1})$ and $\mathrm{\Delta }{\omega }_{\max }/\mathrm{\Delta }{\omega }_{\min }\le c$, where $c\ge 0$ is a constant, $\mathrm{\Delta }{\omega }_{\max }={\max }_{1\le i\le K}{\omega }_i,\mathrm{\Delta }{\omega }_{\min }={\min }_{1\le i\le K}{\omega }_i$, $\mathrm{\Delta }{\omega }_i={\omega }_i-{\omega }_{i-1}$, ${\omega }_0=0$, ${\omega }_{K+1}=1$.
• C9:
  ${\dot{\widehat{\bar{g}}}}_n(\theta )=\frac{\partial {\widehat{\bar{g}}}_n(\theta )}{\partial \theta }$ exists and is continuous, and from the weak law of large numbers, when $\widehat{\theta }\stackrel{p}{\to }{\theta }_0$, exists $J_0$ s.t.

  $$\begin{array}{l}\hfill \underset{n\to \infty }{\lim }\frac{1}{n}\sum _{i=1}^n\mathrm{E}\left(\begin{array}{c}{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_1{A}_i^{-1/2}(X_i,{\tilde{X}}_i)\\ {(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_2{A}_i^{-1/2}(X_i,{\tilde{X}}_i)\\ ⋮\\ {(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_s{A}_i^{-1/2}(X_i,{\tilde{X}}_i)\end{array}\right)\equiv J_0.\end{array}$$
• C10:Denote $a_n=\underset{k}{\max }\left\{|p_{{\lambda }_{1k}}^{″}\left(\left|{\beta }_{k0}\right|\right)|,|p_{{\lambda }_{2k}}^{″}\left({‖{\gamma }_{0k}‖}_{\mathrm{H}}\right)|,{\beta }_{0k}\ne 0,\right.\left.{\gamma }_{0k}\ne 0\right\}$, then $a_n\to 0$ as $n\to \infty$.
• C11:
  $p_{\lambda }(t)$ satisfies

  $$\begin{array}{ll}\hfill & \underset{n\to \infty }{\lim \inf }\underset{{‖{\gamma }_k‖}_{\mathrm{H}}\to 0}{\lim \inf }{\lambda }_{1k}^{-1}{p}_{{\lambda }_{1k}}^{\prime }\left({‖{\gamma }_k‖}_{\mathrm{H}}\right)>0,\\ \hfill & \underset{n\to \infty }{\lim \inf }\underset{{\beta }_k\to 0^{+}}{\lim \inf }{\lambda }_{2k}^{-1}{p}_{{\lambda }_{2k}}^{\prime }\left(\left|{\beta }_k\right|\right)>0,\end{array}$$

  where $k=v+1,v+2,\dots ,q$.

Remark 2

These conditions are often used in the literatures for nonparametric and semi‐parametric statistical inference. C1 implies $N={\sum }_{i=1}^n{n}_i=O(n)$. C2 is the smoothness condition about ${\alpha }_k(t)(k=1,2,\dots ,q)$ and the necessary condition to study the convergence rate of B‐spline estimator. C4 and C9 can be easily obtained by the weak law of large numbers when $n\to \infty$. C3, C5‐C7, C9 can be seen in Tian, Xue and Liu [28]. C8 is necessary for knots of B‐spline basis approximations Schumaker [29]. C10 and C11 can be seen in Tian, Xue and Liu [28]; Zhao and Xue [30] and Fan and Li [18].

Theorem 1

Assuming the conditions $\mathrm{C}1\sim \mathrm{C}11$ hold and $K=O\left(N^{1/(2r+1)}\right)$, we have

$$\begin{array}{l}\hfill ‖{\widehat{\alpha }}_k(·)-{\alpha }_k(·)‖=O\left(N^{-r/(2r+1)}\right),k=1,2,\dots ,q.\end{array}$$

Theorem 2

Assuming the conditions $\mathrm{C}1\sim \mathrm{C}10$ hold, let ${\lambda }_{\max }=\underset{k}{\max }\left\{{\lambda }_{1k},{\lambda }_{2k}\right\}$ and ${\lambda }_{\min }=\underset{k}{\min }\left\{{\lambda }_{1k},{\lambda }_{2k}\right\}$ , satisfy ${\lambda }_{\max }\to 0$ and $n^{r/(2r+1)}·{\lambda }_{\min }\to \infty$ . Then with probability tending to 1, we have

• i. ${\widehat{\alpha }}_k(·)={\widehat{\beta }}_k\ne 0,k\in 𝒞$;
• ii. ${\widehat{\alpha }}_k(·)=0,k\in 𝒵$.

Theorem 3

Assuming the conditions $\mathrm{C}1\sim \mathrm{C}10$ hold, and $K=O\left(N^{1/(2r+1)}\right)$ , we have

$$\sqrt{n}({\widehat{\alpha }}^{(𝒞)}-{\alpha }_0^{(𝒞)})\stackrel{ℒ}{\to }N(0,A^{-1}B{A}^{-1}),$$

where $A$ and $B$ are denoted in proof of theorem 3 in Appendix A, “$\stackrel{ℒ}{\to }$” denotes “convergence in distribution”.

Theorem 1 shows that the BCDPQIF estimators of varying coefficients have the optimal convergence rate, while Theorem 2 states that the BCDPQIF estimators of nonzero constant coefficients and varying coefficients have the sparse property. Theorems show that the BCDPQIF method possesses the oracle property.

3. Computational Algorithm and Selection of Tuning Parameters

3.1. Computational Algorithm

According to Equation (18), $\widehat{\theta }$ does not have an explicit form, meaning that we can only obtain a numerical approximation of $\widehat{\theta }$. First, observe that the first two derivatives of $Q_n(\theta )$ are continuous. Therefore, around a given point ${\theta }^{(0)}$, $Q_n(\theta )$ can be approximated as

$$\begin{array}{ll}\hfill Q_n(\theta )& \approx Q_n({\theta }^{(0)})+{\dot{Q}}_n({\theta }^{(0)})(\theta -{\theta }^{(0)})+\frac{1}{2}{(\theta -{\theta }^{(0)})}^{\mathrm{T}}{\ddot{Q}}_n({\theta }^{(0)})(\theta -{\theta }^{(0)}).\end{array}$$

where ${\dot{Q}}_n(·)$ and ${\ddot{Q}}_n(·)$ represent the first and second derivatives of $Q_n(·)$ w.r.t. $\theta$, respectively. According to the Qu, Lindsay and Li [9] we get

$$\begin{array}{ll}\hfill n^{-1}{\dot{Q}}_n(·)& =2{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n-{\bar{g}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n^{-1}\dot{\mathrm{\Omega }}{\mathrm{\Omega }}_n^{-1}{\bar{g}}_n,\\ \hfill n^{-1}{\ddot{Q}}_n(·)& =2{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n{\dot{\widehat{\bar{g}}}}_n+R_n\end{array}$$

$$\begin{array}{ll}\hfill R_n& =2{\ddot{\widehat{\bar{g}}}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n-4{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n^{-1}{\dot{\mathrm{\Omega }}}_n{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n\\ \hfill \hspace{0.17em}& +2{\widehat{\bar{g}}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n^{-1}{\mathrm{\Omega }}_n^{-1}{\dot{\mathrm{\Omega }}}_n{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n-{\widehat{\bar{g}}}_n^{\mathrm{T}}{\mathrm{\Omega }}_n^{-1}{\ddot{\mathrm{\Omega }}}_n{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n\end{array}$$

Likewise, given an initial value $t_0$, we have

$$\begin{array}{l}\hfill p_{\lambda }(|t|)\approx p_{\lambda }(|t_0|)+\frac{1}{2}\frac{p_{\lambda }^{\prime }(|t_0|)}{|t_0|}(t^2-t_0^2),t\approx t_0.\end{array}$$

Therefore, apart from a constant, $Q_p(\theta )$ can be represented by

$$\begin{array}{ll}\hfill Q_p(\theta )& \approx Q_n({\theta }^{(0)})+{\dot{Q}}_n{({\theta }^{(0)})}^{\mathrm{T}}(\theta -{\theta }^{(0)})\\ \hfill & +\frac{1}{2}{(\theta -{\theta }^{(0)})}^{\mathrm{T}}{\ddot{Q}}_n({\theta }^{(0)})(\theta -{\theta }^{(0)})+\frac{n}{2}{\theta }^{\mathrm{T}}{\sum }_{\lambda }({\theta }^{(0)})\theta .\end{array}$$ (20)

where

$$\begin{array}{ll}\hfill {\sum }_{\lambda }({\theta }^{(0)})& =\left(\begin{array}{cc}{\sum }_{\lambda }({\beta }^{(0)})& 0\\ 0& {\sum }_{\lambda }({\gamma }^{(0)})\otimes \mathrm{H}\end{array}\right),\end{array}$$

$$\begin{array}{ll}\hfill {\sum }_{\lambda }({\gamma }^{(0)})& =\text{diag}\left\{\frac{p_{{\lambda }_{11}}^{\prime }\left(\Vert {\gamma }_1^{(0)}{\Vert }_{\mathrm{H}}\right)}{\Vert {\gamma }_1^{(0)}{\Vert }_{\mathrm{H}}},\right.\frac{p_{{\lambda }_{11}}^{\prime }\left(\Vert {\gamma }_2^{(0)}{\Vert }_{\mathrm{H}}\right)}{\Vert {\gamma }_2^{(0)}{\Vert }_{\mathrm{H}}},\dots ,\left.\frac{p_{{\lambda }_{1q}}^{\prime }\left(\Vert {\gamma }_q^{(0)}{\Vert }_{\mathrm{H}}\right)}{\Vert {\gamma }_q^{(0)}{\Vert }_{\mathrm{H}}}\right\},\end{array}$$

$$\begin{array}{ll}\hfill {\sum }_{\lambda }({\beta }^{(0)})& =\text{diag}\left\{\frac{p_{{\lambda }_{21}}^{\prime }\left(\left|{\beta }_1^{(0)}\right|\right)}{\left|{\beta }_1^{(0)}\right|}I(\Vert {\gamma }_1^{(0)}{\Vert }_{\mathrm{H}}=0),\right.\frac{p_{{\lambda }_{22}}^{\prime }\left(\left|{\beta }_2^{(0)}\right|\right)}{\left|{\beta }_2^{(0)}\right|}I(\Vert {\gamma }_2^{(0)}{\Vert }_{\mathrm{H}}=0),\dots ,\\ \hfill & \left.\frac{p_{{\lambda }_{2q}}^{\prime }\left(\left|{\beta }_q^{(0)}\right|\right)}{\left|{\beta }_q^{(0)}\right|}I(\Vert {\gamma }_q^{(0)}{\Vert }_{\mathrm{H}}=0)\right\}.\end{array}$$

To solve the problem of numerical solution, we propose a three‐step iterative computational algorithm as follows.

Step 1. This step first identifies varying and constant coefficients for each ${\alpha }_k(·)$ for $k=1,2,\dots ,q$. It provides an initial partition of the coefficient space and reduces model complexity. Specifically, we define the Step 1 estimator ${\widehat{\theta }}_{(1)}$ as

$$\begin{array}{l}\hfill {\widehat{\theta }}_{(1)}=\arg \underset{\theta }{\min }\left\{Q_n(\theta )+n\sum _{k=1}^q{p}_{{\lambda }_{1k}}\left(\Vert {\gamma }_k{\Vert }_{\mathrm{H}}\right)\right\}.\end{array}$$ (21)

Through the minimization procedure (21), nonzero values of $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}$ are identified and selected; when $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}>0$, the coefficient function ${\alpha }_k(·)$ is identified as varying, whereas $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}=0$ indicates it is constant.

Since ${\widehat{\theta }}_{(1)}$ lacks a closed‐form expression, we approximate it via the following iterative procedure

$$\begin{array}{ll}\hfill {\widehat{\theta }}_{(1)}^{(m+1)}& ={\widehat{\theta }}_{(1)}^{(m)}-{\left[{\ddot{Q}}_n\left({\widehat{\theta }}_{(1)}^{(m)}\right)+n{\sum }_{(1)}\right]}^{-1}\left[{\dot{Q}}_n\left({\widehat{\theta }}_{(1)}^{(m)}\right)+n{\sum }_{(1)}{\widehat{\theta }}_{(1)}^{(m)}\right],\end{array}$$ (22)

where ${\sum }_{(1)}=\left(\begin{array}{ll}{\sum }_0& 0\\ 0& {\sum }_{\lambda }\left({\widehat{\gamma }}^{(m)}\right)\otimes \mathrm{H}\end{array}\right),{\sum }_0=\text{diag}\{0_1,0_2,\dots ,0_q\}$ and

$$\begin{array}{ll}\hfill {\sum }_{\lambda }({\gamma }^{(m)})& =\text{diag}\left\{\frac{p_{{\lambda }_{11}}^{\prime }\left(\Vert {\gamma }_1^{(m)}{\Vert }_{\mathrm{H}}\right)}{\Vert {\gamma }_1^{(m)}{\Vert }_{\mathrm{H}}},\frac{p_{{\lambda }_{12}}^{\prime }\left(\Vert {\gamma }_2^{(m)}{\Vert }_{\mathrm{H}}\right)}{\Vert {\gamma }_2^{(m)}{\Vert }_{\mathrm{H}}},\dots ,\right.\left.\frac{p_{{\lambda }_{1q}}^{\prime }\left(\Vert {\gamma }_q^{(m)}{\Vert }_{\mathrm{H}}\right)}{\Vert {\gamma }_q^{(m)}{\Vert }_{\mathrm{H}}}\right\}.\end{array}$$ (23)

We initialize this iteration with $\tilde{\theta }$ from Equation (15) and iterate Equation (22) until convergence to obtain the approximate solution ${\widehat{\theta }}_{(1)}$. This process effectively identifies whether each ${\alpha }_k(·)$ is a varying or constant coefficient before proceeding to variable selection.

Step 2. Next, taking ${\widehat{\theta }}_{(1)}$ as an initial value, we refine the constant coefficients by selecting nonzero values. Specifically, we define

$$\begin{array}{l}\hfill {\widehat{\theta }}_{(2)}=\arg \underset{\theta }{\min }\left\{Q_n\left({\widehat{\theta }}_{(1)}\right)+n\sum _{k=1}^q{p}_{{\lambda }_{2k}}\left(|{\beta }_k|\right)\right\}.\end{array}$$ (24)

Similar to Step 1, ${\widehat{\theta }}_{(2)}$ also has no closed‐form solution, so we propose a iterative process in Step 2 as

$$\begin{array}{ll}\hfill {\widehat{\theta }}_{(2)}^{(m+1)}& ={\widehat{\theta }}_{(2)}^{(m)}-{\left[{\ddot{Q}}_n\left({\widehat{\theta }}_{(2)}^{(m)}\right)+n{\sum }_{(2)}\right]}^{-1}\left[{\dot{Q}}_n\left({\widehat{\theta }}_{(2)}^{(m)}\right)+n{\sum }_{(2)}{\widehat{\theta }}_{(2)}^{(m)}\right],\end{array}$$ (25)

where ${\sum }_{(2)}=\left(\begin{array}{ll}{\sum }_{\lambda }({\widehat{\beta }}^{(m)})& 0\\ 0& {\sum }_0\otimes \mathrm{H}\end{array}\right)$ and

$$\begin{array}{ll}\hfill {\sum }_{\lambda }({\widehat{\beta }}^{(m)})& =\text{diag}\left\{\frac{p_{{\lambda }_{21}}^{\prime }\left(|{\widehat{\beta }}_1^{(m)}|\right)}{|{\widehat{\beta }}_1^{(m)}|}I\left(\Vert {\widehat{\gamma }}_1^{(m)}{\Vert }_{\mathrm{H}}=0\right),\dots ,\right.\\ \hfill & \left.\frac{p_{{\lambda }_{2q}}^{\prime }\left(|{\widehat{\beta }}_q^{(m)}|\right)}{|{\widehat{\beta }}_q^{(m)}|}I\left(\Vert {\widehat{\gamma }}_q^{(m)}{\Vert }_{\mathrm{H}}=0\right)\right\}.\end{array}$$

Iterate Equation (25) until convergence to approximate ${\widehat{\theta }}_{(2)}$. This step focuses on eliminating uninformative constant coefficients while retaining those that are pertinent.

Step 3. Finally, we alternate between Steps 1 and 2 until overall convergence, arriving at the final estimator $\widehat{\theta }$. By iteratively identifying constant versus varying coefficients and selecting only the nonzero constants, this procedure yields a parsimonious yet flexible model that adapts to do structure identification, estimation and variable selection without requiring assumptions in advance.

3.2. Selection of Tuning Parameters

As is known to all, ${\{{\lambda }_{1k},{\lambda }_{2k}\}}_{k=1}^q$ are of vital importance for variable selection. They control the amount of penalties and determine the outcomes of structure identification and variable selection. However, the selection of ${\{{\lambda }_{1k},{\lambda }_{2k}\}}_{k=1}^q$ involves a very high computational complexity. To over this problem, in our work, we denote the adaptive tuning parameters ${\lambda }_{1k}$ and ${\lambda }_{2k}$ as

$$\begin{array}{l}\hfill {\lambda }_{1k}=\frac{{\lambda }_1}{\Vert {\tilde{\gamma }}_k{\Vert }_{\mathrm{H}}},{\lambda }_{2k}=\frac{{\lambda }_2}{|{\widehat{\beta }}_k|},\end{array}$$

where ${\tilde{\gamma }}_k$, for $k=1,2,\dots ,q$, are defined by Equation (15), and ${\widehat{\beta }}_k$ corresponds to the solution ${\widehat{\theta }}_{(1)}$ obtained in Step 1. We can see that the adaptive tuning parameters has significantly reduced the computational complexity.

To obtain the optimal tuning parameters in Steps 1 and 2, we employ a BIC‐type criterion separately for each step, thereby balancing model fit and complexity. Specifically, in Step 1, we determine the optimal ${\widehat{\lambda }}_1$ via

$$\mathrm{BI}{\mathrm{C}}_{\mathrm{1}}(\mathrm{\lambda })=Q_n\left({\widehat{\theta }}_{(1)}\right)+d{f}_{{\lambda }_{1k}}\log (n),{\widehat{\lambda }}_1=\arg \underset{\lambda }{\min }\mathrm{BI}{\mathrm{C}}_{\mathrm{1}}(\mathrm{\lambda }),$$ (26)

where $d{f}_{{\lambda }_{1k}}={\sum }_{k=1}^{q}I(\Vert {\gamma }_k{\Vert }_{\mathrm{H}}\ne 0)$. This quantity $d{f}_{{\lambda }_{1k}}$ counts the number of nonzero varying coefficients and thus penalizes more complex models. Similarly, in Step 2 we use an analogous BIC‐type criterion to obtain the optimal ${\widehat{\lambda }}_2$:

$$\mathrm{BI}{\mathrm{C}}_{\mathrm{2}}(\mathrm{\lambda })=Q_n\left({\widehat{\theta }}_{(2)}\right)+d{f}_{{\lambda }_{2k}}\log (n),{\widehat{\lambda }}_2=\arg \underset{\lambda }{\min }\mathrm{BI}{\mathrm{C}}_{\mathrm{2}}(\mathrm{\lambda }),$$ (27)

where $d{f}_{{\lambda }_{2k}}={\sum }_{k=1}^{q}I(|{\beta }_k|I(\Vert {\gamma }_k{\Vert }_{\mathrm{H}}=0)\ne 0).$ Hence, $d{f}_{{\lambda }_{2k}}$ counts only those nonzero constant coefficients that contribute to the model when the corresponding varying components are zero. By explicitly penalizing the inclusion of additional parameters, the BIC‐type criteria help select tuning parameters that yield a parsimonious yet informative model.

4. Numerical Studies

4.1. Simulations Studies

We perform some numerical simulations to evaluate the performance of the proposed method in finite samples. The performance of estimator ${\widehat{\alpha }}^{(𝒞)}$ in the simulation will be assessed by using the generalized mean square error (GMSE) [28], which is defined as

$$\begin{array}{ll}\hfill \text{GMSE}& ={\left({\widehat{\alpha }}^{(𝒞)}-{\alpha }_0^{(𝒞)}\right)}^{\mathrm{T}}\mathrm{E}\left(X^{(𝒞)}{(X^{(𝒞)})}^{\mathrm{T}}\right)\left({\widehat{\alpha }}^{(𝒞)}-{\alpha }_0^{(𝒞)}\right),\end{array}$$ (28)

where ${\widehat{\alpha }}^{(𝒞)}={({\widehat{\beta }}_1,{\widehat{\beta }}_2,\dots ,{\widehat{\beta }}_c)}^{\mathrm{T}}$. The performance of estimator ${\widehat{\alpha }}^{(𝒱)}$ in the simulation will be assessed by using the square root of average square errors (RASE) [28]

$$\begin{array}{ll}\hfill \text{RASE}& ={\left\{\frac{1}{M}\sum _{m=1}^M\sum _{k=c+1}^v{\left[{\widehat{\alpha }}_k\left(t_m\right)-{\alpha }_{0k}\left(t_m\right)\right]}^2\right\}}^{1/2}.\end{array}$$ (29)

where $t_m(m=1,2,\dots ,M)$ are grid points at which ${\widehat{\alpha }}_k\left(t_m\right)$ is evaluated. A smaller RASE or GMSE signifies higher estimation accuracy, indicating that $\widehat{\alpha }(·)$ is closer to the true value $\alpha (·)$. In our simulations, grid points were equally spaced on $[0,1]$ and $M=200$.

In order to assess the performance of structure identification, estimation and variable selection, we give some denotations. Let “CZ” denote the average number of correctly identified zero coefficients; “IZ” represent the average number of nonzero coefficients incorrectly identified as zero; “CV” denote the average number of correctly identified varying coefficients; “IV” represent the average number of non‐varying coefficients incorrectly identified as varying; “CC” denote the average number of correctly identified constant coefficients; and “IC” represent the average number of non‐constant coefficients incorrectly identified as constant. “CF” represents the percentage of simulations in which the true model structure was correctly identified. Smaller values of IZ, IV, and IC, along with values of CZ, CV, and CC closer to the true model, indicate better performance in structure identification and selection. A lower GMSE or RASE indicates better estimation accuracy, implying that $\widehat{\alpha }(·)$ is closer to the true parameter function ${\alpha }_0(·)$ on average.

Suppose that the real model (2) satisfies $𝒞=\{1,2\},𝒱=\{3,4,5\},𝒵=\{6,7,8\}$ and

$$\begin{array}{ll}\hfill {\alpha }^{(𝒞)}(t)& =({\alpha }_1(t),{\alpha }_2(t))=(5,6),\\ \hfill {\alpha }^{(𝒱)}(t)& =({\alpha }_3(t),{\alpha }_4(t),{\alpha }_5(t))=(0.4·e^{2t-1},\sin (\pi t),0.1·{(2-2t)}^3),\\ \hfill {\alpha }^{(𝒵)}(t)& =({\alpha }_6(t),{\alpha }_7(t),{\alpha }_8(t))=(0,0,0).\end{array}$$

We took $X_{ij}\sim N(8,{\sigma }_X^2{I}_8),u_{ij}\sim N(0,{\sigma }_u^2{I}_8)$, where $j=$ $1,2,\dots ,8,{\sigma }_X=8,I_8$ is $8\times 8$ identifymatrix. We set ${\sigma }_u$ as $0.2,0.4,0.6$. $t_{ij}\sim U[0,1]$. ${\epsilon }_i={\left({\epsilon }_{i1},{\epsilon }_{i2},\dots ,{\epsilon }_{i{n}_i}\right)}^{\mathrm{T}}\sim N\left(0,{\sigma }^2\text{Corr}\left({\epsilon }_{\mathrm{i}},\rho \right)\right)$, where ${\sigma }^2=1$ and $\left.\text{Corr}\left({\epsilon }_{\mathrm{i}},\rho \right)\right)$ is a known correlation matrix with parameter $\rho$. Thus, we can get $A_i=\text{diag}(1,1,\dots ,1)$. In our work, we set $n=300,350,400$, $n_i=10,20$ and ${\epsilon }_i$ has the first‐order autoregressive $(\text{AR}(1))$ and exchangeable (EX) correlation structures with $\rho =0.3,0.7$. The cubic B‐spline basis was applied with the knots being equally spaced in $[0,1],K=⌊c_0\times N^{1/5}⌋$, where $\lfloor c_0\rfloor$ denotes the largest integer less than $c_0$. Following Tian, Xue and Liu [28], we choose $c_0=0.4$.

For each simulated longitudinal data, we compared the BCDPQIF method with the LASSO, MCP and the SCAD penalty functions and the one neglecting measurement errors with SCAD penalty function, denoted as BCDPQIF‐LASSO, BCDPQIF‐MCP, BCDPQIF‐SCAD and BCDPQIF‐nSCAD, respectively. For the sake of simplicity, BCDPQIF‐LASSO, BCDPQIF‐MCP, BCDPQIF‐SCAD and BCDPQIF‐nSCAD are denoted by “LASSO”, “MCP”, “SCAD” and “nSCAD” in the following tables respectively. ${\widehat{\lambda }}_{1k},{\widehat{\lambda }}_{2k}(k=1,2,\dots ,q)$ were selected by Equations (26) and (27).

Among them, Tables 1 and 3 show the model estimation results of longitudinal data with EX correlation structures under different $n_i$; while Tables 2 and 4 show the structure identification and variable selection results of longitudinal data with AR(1) correlation structures under different $n_i$. Tables 5 and 7 continue this analysis for the EX correlation structures with under different $n_i$; Tables 6 and 8 present the corresponding results for the AR(1) correlation structures, following the same organization but focusing on different and parameter combinations.

TABLE 1.

Model estimation with the EX correlation structure ($n_i=10$).

			$n=300$		$n=350$		$n=400$
$\rho$	${\sigma }_u$	Method	GMSE	RASE	GMSE	RASE	GMSE	RASE
0.3	0.2	LASSO	0.006310	0.020265	0.005318	0.018661	0.004500	0.017346
		MCP	0.005170	0.020117	0.004126	0.018650	0.003502	0.017217
		SCAD	0.005350	0.020085	0.004272	0.018533	0.003607	0.017198
		nSCAD	0.007214	0.020461	0.006025	0.018826	0.005227	0.017122
	0.4	LASSO	0.021447	0.034504	0.022946	0.031387	0.022236	0.029445
		MCP	0.013685	0.033768	0.013808	0.030912	0.011439	0.028973
		SCAD	0.014983	0.033763	0.014196	0.031039	0.012539	0.028929
		nSCAD	0.036712	0.035110	0.038602	0.031825	0.034361	0.030109
	0.6	LASSO	0.067761	0.050695	0.061829	0.046198	0.062955	0.043836
		MCP	0.033222	0.049347	0.025128	0.044601	0.022594	0.041510
		SCAD	0.034123	0.049820	0.027653	0.044584	0.023667	0.041760
		nSCAD	0.143232	0.053116	0.120532	0.047648	0.129252	0.044608
0.7	0.2	LASSO	0.005144	0.018004	0.004381	0.017552	0.003754	0.017102
		MCP	0.004269	0.017916	0.003456	0.017511	0.002752	0.017053
		SCAD	0.004457	0.017971	0.003609	0.017524	0.002794	0.017049
		nSCAD	0.006139	0.017992	0.004973	0.017827	0.004717	0.017225
	0.4	LASSO	0.022754	0.030121	0.024283	0.031665	0.020305	0.028867
		MCP	0.014025	0.029910	0.014666	0.031067	0.011411	0.028234
		SCAD	0.015138	0.029797	0.014507	0.030956	0.011690	0.028112
		nSCAD	0.036937	0.030407	0.037578	0.031802	0.033961	0.028730
	0.6	LASSO	0.079412	0.045530	0.067183	0.046290	0.063914	0.042990
		MCP	0.033737	0.044692	0.028633	0.044774	0.025319	0.041379
		SCAD	0.034388	0.044384	0.030047	0.044629	0.025696	0.041235
		nSCAD	0.144105	0.046027	0.130574	0.047405	0.126245	0.043912

TABLE 3.

Model estimation with the EX correlation structure ($n_i=20$).

			$n=300$		$n=350$		$n=400$
$\rho$	${\sigma }_u$	Method	GMSE	RASE	GMSE	RASE	GMSE	RASE
0.3	0.2	LASSO	0.002549	0.014618	0.002126	0.012966	0.001890	0.012367
		MCP	0.002203	0.014552	0.002022	0.012872	0.001506	0.012342
		SCAD	0.002390	0.014589	0.002042	0.012890	0.001584	0.012278
		nSCAD	0.004089	0.014739	0.003274	0.013051	0.003253	0.012508
	0.4	LASSO	0.010233	0.024553	0.009077	0.022487	0.007610	0.021105
		MCP	0.007098	0.024214	0.006204	0.022208	0.005311	0.020843
		SCAD	0.007654	0.024516	0.006359	0.022315	0.005350	0.020757
		nSCAD	0.031571	0.025823	0.029056	0.023655	0.027567	0.022327
	0.6	LASSO	0.026446	0.034791	0.025621	0.032215	0.022216	0.030112
		MCP	0.015586	0.033946	0.013372	0.031520	0.011238	0.029165
		SCAD	0.016384	0.034301	0.014309	0.031568	0.011855	0.029246
		nSCAD	0.127600	0.038364	0.124034	0.036174	0.115740	0.033771
0.7	0.2	LASSO	0.011945	0.021913	0.002154	0.012708	0.001604	0.011912
		MCP	0.009801	0.021855	0.001838	0.012687	0.001377	0.011901
		SCAD	0.009734	0.021733	0.001872	0.012659	0.001403	0.011876
		nSCAD	0.031744	0.022608	0.004244	0.012763	0.003275	0.012015
	0.4	LASSO	0.010903	0.024006	0.008541	0.021701	0.008329	0.020267
		MCP	0.008149	0.024126	0.005660	0.021320	0.005311	0.020045
		SCAD	0.008548	0.024027	0.005942	0.021531	0.005358	0.020022
		nSCAD	0.032437	0.024605	0.030968	0.022312	0.029802	0.021041
	0.6	LASSO	0.022386	0.035063	0.024778	0.031569	0.021510	0.030168
		MCP	0.015425	0.034393	0.014333	0.031084	0.010042	0.029233
		SCAD	0.017002	0.034705	0.014282	0.030983	0.010343	0.029355
		nSCAD	0.114663	0.037800	0.124042	0.034664	0.120750	0.034191

TABLE 2.

Model estimation with the AR(1) correlation structure ($n_i=10$).

			$n=300$		$n=350$		$n=400$
$\rho$	${\sigma }_u$	Method	GMSE	RASE	GMSE	RASE	GMSE	RASE
0.3	0.2	LASSO	0.005067	0.020788	0.004916	0.018783	0.004804	0.017505
		MCP	0.004491	0.020572	0.004093	0.018768	0.003825	0.017414
		SCAD	0.004634	0.020750	0.004196	0.018769	0.003833	0.017394
		nSCAD	0.005291	0.020818	0.005076	0.018990	0.004946	0.017501
	0.4	LASSO	0.024966	0.034666	0.020821	0.032083	0.020043	0.030934
		MCP	0.015624	0.034262	0.012613	0.031646	0.011056	0.030374
		SCAD	0.015726	0.034356	0.013997	0.031824	0.011561	0.030416
		nSCAD	0.037888	0.035410	0.031702	0.031750	0.030249	0.031064
	0.6	LASSO	0.076168	0.051224	0.071597	0.046558	0.064740	0.043943
		MCP	0.032306	0.049338	0.026233	0.044643	0.023371	0.042344
		SCAD	0.032083	0.049692	0.027803	0.044845	0.024317	0.042389
		nSCAD	0.142457	0.052199	0.133610	0.048358	0.112066	0.045093
0.7	0.2	LASSO	0.006348	0.020563	0.004656	0.018560	0.003374	0.017292
		MCP	0.005043	0.020318	0.003572	0.018437	0.002682	0.017168
		SCAD	0.005124	0.020370	0.003710	0.018446	0.002878	0.017186
		nSCAD	0.007161	0.020453	0.004955	0.018410	0.003912	0.017337
	0.4	LASSO	0.022236	0.034752	0.021721	0.031719	0.021380	0.030122
		MCP	0.015580	0.034360	0.012842	0.031043	0.011394	0.029633
		SCAD	0.017398	0.034046	0.013235	0.030967	0.011727	0.029443
		nSCAD	0.035691	0.035031	0.032769	0.032387	0.033609	0.030349
	0.6	LASSO	0.062320	0.050684	0.073482	0.045768	0.065321	0.045198
		MCP	0.029349	0.049476	0.029730	0.043799	0.020605	0.043522
		SCAD	0.030288	0.050020	0.031292	0.043488	0.020856	0.043477
		nSCAD	0.133572	0.050931	0.124449	0.046314	0.120049	0.045432

TABLE 4.

Model estimation with the AR(1) correlation structure ($n_i=20$).

			$n=300$		$n=350$		$n=400$
$\rho$	${\sigma }_u$	Method	GMSE	RASE	GMSE	RASE	GMSE	RASE
0.3	0.2	LASSO	0.003031	0.014615	0.002251	0.013376	0.002001	0.012555
		MCP	0.002757	0.014560	0.002151	0.013237	0.001696	0.012479
		SCAD	0.002769	0.014623	0.002149	0.013293	0.001736	0.012396
		nSCAD	0.004133	0.014716	0.003351	0.013513	0.003169	0.012532
	0.4	LASSO	0.008171	0.023962	0.008643	0.023011	0.007514	0.021520
		MCP	0.006882	0.023678	0.006250	0.022543	0.005167	0.021042
		SCAD	0.007213	0.023850	0.006275	0.022768	0.005229	0.021094
		nSCAD	0.025199	0.025266	0.026554	0.024029	0.025662	0.022530
	0.6	LASSO	0.026435	0.035735	0.024354	0.032726	0.022199	0.029838
		MCP	0.017692	0.035287	0.013702	0.032172	0.012334	0.029107
		SCAD	0.018006	0.035391	0.013737	0.031987	0.012320	0.029036
		nSCAD	0.118056	0.038565	0.112571	0.036187	0.111787	0.033875
0.7	0.2	LASSO	0.002589	0.014614	0.002075	0.013225	0.001798	0.012293
		MCP	0.002230	0.014582	0.001883	0.013104	0.001592	0.012196
		SCAD	0.002253	0.014590	0.001872	0.013095	0.001635	0.012196
		nSCAD	0.003952	0.014572	0.003295	0.013263	0.003026	0.012375
	0.4	LASSO	0.009626	0.024461	0.008094	0.022350	0.006898	0.020712
		MCP	0.007010	0.024280	0.006497	0.022211	0.005155	0.020495
		SCAD	0.007257	0.024291	0.006661	0.022158	0.005165	0.020371
		nSCAD	0.029485	0.025269	0.024169	0.023269	0.024452	0.021766
	0.6	LASSO	0.026455	0.035658	0.023275	0.032126	0.022329	0.029903
		MCP	0.016085	0.034704	0.012669	0.031518	0.011182	0.028931
		SCAD	0.017075	0.034907	0.013049	0.031266	0.010934	0.028944
		nSCAD	0.117351	0.039046	0.117707	0.035933	0.117497	0.033745

TABLE 5.

Structure identification and variable selection with the EX correlation structure ($n_i=10$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.3	0.2	300	LASSO	2.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8650
			MCP	2.8100	0.0000	3.0000	0.0000	2.0000	0.1900	0.8450
			SCAD	2.8850	0.0000	3.0000	0.0000	2.0000	0.1150	0.9100
			nSCAD	2.7850	0.0000	3.0000	0.0000	2.0000	0.2150	0.8200
		350	LASSO	2.8800	0.0000	3.0000	0.0000	2.0000	0.1200	0.9000
			MCP	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8750
			SCAD	2.9050	0.0000	3.0000	0.0000	2.0000	0.0950	0.9250
			nSCAD	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8700
		400	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8900
			MCP	2.8800	0.0000	3.0000	0.0000	2.0000	0.1200	0.8900
			SCAD	2.9300	0.0000	3.0000	0.0000	2.0000	0.0700	0.9400
			nSCAD	2.8350	0.0000	3.0000	0.0000	2.0000	0.1650	0.8600
	0.4	300	LASSO	2.5700	0.0000	3.0000	0.0000	2.0000	0.4300	0.6400
			MCP	2.5750	0.0000	3.0000	0.0000	2.0000	0.4250	0.6400
			SCAD	2.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8700
			nSCAD	2.5150	0.0000	3.0000	0.0000	2.0000	0.4850	0.5900
		350	LASSO	2.8000	0.0000	3.0000	0.0000	2.0000	0.2000	0.8400
			MCP	2.7450	0.0000	3.0000	0.0000	2.0000	0.2550	0.7850
			SCAD	2.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.8950
			nSCAD	2.7550	0.0000	3.0000	0.0000	2.0000	0.2450	0.7850
		400	LASSO	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.8950
			MCP	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8550
			SCAD	2.9250	0.0000	3.0000	0.0000	2.0000	0.0750	0.9300
			nSCAD	2.7650	0.0000	3.0000	0.0000	2.0000	0.2350	0.7950
	0.6	300	LASSO	2.4500	0.0000	3.0000	0.0000	2.0000	0.5500	0.5350
			MCP	2.3850	0.0000	3.0000	0.0000	2.0000	0.6150	0.5150
			SCAD	2.7950	0.0000	3.0000	0.0000	2.0000	0.2050	0.8250
			nSCAD	2.3150	0.0000	3.0000	0.0000	2.0000	0.6850	0.4750
		350	LASSO	2.6100	0.0000	3.0000	0.0000	2.0000	0.3900	0.6650
			MCP	2.6300	0.0000	3.0000	0.0000	2.0000	0.3700	0.6750
			SCAD	2.8150	0.0000	3.0000	0.0000	2.0000	0.1850	0.8400
			nSCAD	2.4250	0.0000	3.0000	0.0000	2.0000	0.5750	0.5550
		400	LASSO	2.6800	0.0000	3.0000	0.0000	2.0000	0.3200	0.7150
			MCP	2.6450	0.0000	3.0000	0.0000	2.0000	0.3550	0.7000
			SCAD	2.8500	0.0000	3.0000	0.0000	2.0000	0.1500	0.8700
			nSCAD	2.5350	0.0000	3.0000	0.0000	2.0000	0.4650	0.6350

TABLE 7.

Structure identification and variable selection with the EX correlation structure ($n_i=20$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.3	0.2	300	LASSO	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9150
			MCP	2.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.8900
			SCAD	2.9350	0.0000	3.0000	0.0000	2.0000	0.0650	0.9450
			nSCAD	2.9050	0.0000	3.0000	0.0000	2.0000	0.0950	0.9250
		350	LASSO	2.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9250
			MCP	2.8900	0.0000	3.0000	0.0000	2.0000	0.1100	0.9000
			SCAD	2.9800	0.0000	3.0000	0.0000	2.0000	0.0200	0.9850
			nSCAD	2.8500	0.0000	3.0000	0.0000	2.0000	0.1500	0.8850
		400	LASSO	2.9450	0.0000	3.0000	0.0000	2.0000	0.0550	0.9500
			MCP	2.9400	0.0000	3.0000	0.0000	2.0000	0.0600	0.9450
			SCAD	2.9950	0.0000	3.0000	0.0000	2.0000	0.0050	0.9950
			nSCAD	2.9250	0.0000	3.0000	0.0000	2.0000	0.0750	0.9350
	0.4	300	LASSO	2.8100	0.0000	3.0000	0.0000	2.0000	0.1900	0.8450
			MCP	2.8100	0.0000	3.0000	0.0000	2.0000	0.1900	0.8500
			SCAD	2.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9200
			nSCAD	2.6250	0.0000	3.0000	0.0000	2.0000	0.3750	0.6600
		350	LASSO	2.8700	0.0000	3.0000	0.0000	2.0000	0.1300	0.8850
			MCP	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8750
			SCAD	2.9200	0.0000	3.0000	0.0000	2.0000	0.0800	0.9300
			nSCAD	2.7300	0.0000	3.0000	0.0000	2.0000	0.2700	0.7800
		400	LASSO	2.9200	0.0000	3.0000	0.0000	2.0000	0.0800	0.9250
			MCP	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.8850
			SCAD	2.9300	0.0000	3.0000	0.0000	2.0000	0.0700	0.9350
			nSCAD	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8350
	0.6	300	LASSO	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8150
			MCP	2.7850	0.0000	3.0000	0.0000	2.0000	0.2150	0.7950
			SCAD	2.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9100
			nSCAD	2.5550	0.0000	3.0000	0.0000	2.0000	0.4450	0.6350
		350	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8700
			MCP	2.8550	0.0000	3.0000	0.0000	2.0000	0.1450	0.8650
			SCAD	2.9150	0.0000	3.0000	0.0000	2.0000	0.0850	0.9250
			nSCAD	2.5750	0.0000	3.0000	0.0000	2.0000	0.4250	0.6550
		400	LASSO	2.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9050
			MCP	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8300
			SCAD	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9050
			nSCAD	2.6350	0.0000	3.0000	0.0000	2.0000	0.3650	0.6800

TABLE 6.

Structure identification and variable selection with the AR(1) correlation structure ($n_i=10$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.3	0.2	300	LASSO	2.6800	0.0000	3.0000	0.0000	2.0000	0.3200	0.7350
			MCP	2.6600	0.0000	3.0000	0.0000	2.0000	0.3400	0.7150
			SCAD	2.6600	0.0000	3.0000	0.0000	2.0000	0.3400	0.7000
			nSCAD	2.6550	0.0000	3.0000	0.0000	2.0000	0.3450	0.6950
		350	LASSO	2.7400	0.0000	3.0000	0.0000	2.0000	0.2600	0.7650
			MCP	2.7150	0.0000	3.0000	0.0000	2.0000	0.2850	0.7550
			SCAD	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9050
			nSCAD	2.6950	0.0000	3.0000	0.0000	2.0000	0.3050	0.7350
		400	LASSO	2.8700	0.0000	3.0000	0.0000	2.0000	0.1300	0.8850
			MCP	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8800
			SCAD	2.9150	0.0000	3.0000	0.0000	2.0000	0.0850	0.9150
			nSCAD	2.8900	0.0000	3.0000	0.0000	2.0000	0.1100	0.9050
	0.4	300	LASSO	2.6050	0.0000	3.0000	0.0000	2.0000	0.3950	0.6650
			MCP	2.5800	0.0000	3.0000	0.0000	2.0000	0.4200	0.6350
			SCAD	2.6250	0.0000	3.0000	0.0000	2.0000	0.3750	0.6700
			nSCAD	2.4950	0.0000	3.0000	0.0000	2.0000	0.5050	0.5750
		350	LASSO	2.6850	0.0000	3.0000	0.0000	2.0000	0.3150	0.7000
			MCP	2.6800	0.0000	3.0000	0.0000	2.0000	0.3200	0.7150
			SCAD	2.8100	0.0000	3.0000	0.0000	2.0000	0.1900	0.8300
			nSCAD	2.5850	0.0000	3.0000	0.0000	2.0000	0.4150	0.6400
		400	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8800
			MCP	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8400
			SCAD	2.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8600
			nSCAD	2.7100	0.0000	3.0000	0.0000	2.0000	0.2900	0.7650
	0.6	300	LASSO	2.5600	0.0000	3.0000	0.0000	2.0000	0.4400	0.6150
			MCP	2.4950	0.0000	3.0000	0.0000	2.0000	0.5050	0.5700
			SCAD	2.5200	0.0000	3.0000	0.0000	2.0000	0.4800	0.5950
			nSCAD	2.4050	0.0000	3.0000	0.0000	2.0000	0.5950	0.5100
		350	LASSO	2.6250	0.0000	3.0000	0.0000	2.0000	0.3750	0.6750
			MCP	2.5550	0.0000	3.0000	0.0000	2.0000	0.4450	0.6100
			SCAD	2.6150	0.0000	3.0000	0.0000	2.0000	0.3850	0.6650
			nSCAD	2.4750	0.0000	3.0000	0.0000	2.0000	0.5250	0.5500
		400	LASSO	2.7450	0.0000	3.0000	0.0000	2.0000	0.2550	0.7900
			MCP	2.7700	0.0000	3.0000	0.0000	2.0000	0.2300	0.7850
			SCAD	2.7800	0.0000	3.0000	0.0000	2.0000	0.2200	0.8100
			nSCAD	2.6000	0.0000	3.0000	0.0000	2.0000	0.4000	0.6650

TABLE 8.

Structure identification and variable selection with the AR(1) correlation structure ($n_i=20$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.3	0.2	300	LASSO	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8450
			MCP	2.8150	0.0000	3.0000	0.0000	2.0000	0.1850	0.8500
			SCAD	2.8500	0.0000	3.0000	0.0000	2.0000	0.1500	0.8700
			nSCAD	2.7700	0.0000	3.0000	0.0000	2.0000	0.2300	0.8100
		350	LASSO	2.9150	0.0000	3.0000	0.0000	2.0000	0.0850	0.9250
			MCP	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.9100
			SCAD	2.9500	0.0000	3.0000	0.0000	2.0000	0.0500	0.9500
			nSCAD	2.7600	0.0000	3.0000	0.0000	2.0000	0.2400	0.8250
		400	LASSO	2.9250	0.0000	3.0000	0.0000	2.0000	0.0750	0.9300
			MCP	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9100
			SCAD	2.9950	0.0000	3.0000	0.0000	2.0000	0.0050	0.9950
			nSCAD	2.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9150
	0.4	300	LASSO	2.8150	0.0000	3.0000	0.0000	2.0000	0.1850	0.8350
			MCP	2.7750	0.0000	3.0000	0.0000	2.0000	0.2250	0.8050
			SCAD	2.8300	0.0000	3.0000	0.0000	2.0000	0.1700	0.8500
			nSCAD	2.6550	0.0000	3.0000	0.0000	2.0000	0.3450	0.6900
		350	LASSO	2.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8550
			MCP	2.7850	0.0000	3.0000	0.0000	2.0000	0.2150	0.8300
			SCAD	2.8900	0.0000	3.0000	0.0000	2.0000	0.1100	0.9050
			nSCAD	2.6900	0.0000	3.0000	0.0000	2.0000	0.3100	0.7350
		400	LASSO	2.8700	0.0000	3.0000	0.0000	2.0000	0.1300	0.8800
			MCP	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8800
			SCAD	2.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9200
			nSCAD	2.7500	0.0000	3.0000	0.0000	2.0000	0.2500	0.7950
	0.6	300	LASSO	2.7550	0.0000	3.0000	0.0000	2.0000	0.2450	0.7850
			MCP	2.7300	0.0000	3.0000	0.0000	2.0000	0.2700	0.7600
			SCAD	2.7850	0.0000	3.0000	0.0000	2.0000	0.2150	0.8050
			nSCAD	2.5600	0.0000	3.0000	0.0000	2.0000	0.4400	0.6650
		350	LASSO	2.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8400
			MCP	2.7900	0.0000	3.0000	0.0000	2.0000	0.2100	0.8050
			SCAD	2.7950	0.0000	3.0000	0.0000	2.0000	0.2050	0.8100
			nSCAD	2.6150	0.0000	3.0000	0.0000	2.0000	0.3850	0.6600
		400	LASSO	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8700
			MCP	2.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8350
			SCAD	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.8950
			nSCAD	2.6350	0.0000	3.0000	0.0000	2.0000	0.3650	0.6900

For completeness, any additional or more detailed tables not included in the main text are provided in Appendix A, ensuring that all results are fully documented and reproducible. In our work, we have included simulation results for higher dimensional settings ($q=20,50$) in Appendix A, and the conclusions in these higher dimensional scenarios coincide with those presented in the main text. In summary, from Tables 1, 2, 3, 4, 5, 6, 7, 8, we can draw the following conclusions.

• 1.In the vast majority of experimental scenarios, the estimation accuracies of BCDPQIF‐LASSO, BCDPQIF‐SCAD, and BCDPQIF‐MCP consistently surpasses that of BCDPQIF‐nSCAD, thereby demonstrating the effectiveness of the proposed bias‐corrected strategy. Overall, the four methods exhibit comparable good performances in structure identification. Moreover, neglecting measurement errors results in biased estimation for model (2).
• 2.Under same conditions, as both the sample size and the number of observations increase, the performances of BCDPQIF‐SCAD, BCDPQIF‐MCP, and BCDPQIF‐LASSO improve. Notably, BCDPQIF‐SCAD and BCDPQIF‐MCP generally outperform BCDPQIF‐LASSO when estimating varying coefficients.
• 3.Similarly, as the magnitude of measurement errors increases, the performances of BCDPQIF‐SCAD, BCDPQIF‐MCP, and BCDPQIF‐LASSO deteriorate under the same conditions. When measurement errors are small, the performance differences among these methods are minimal; however, when measurement errors become substantial, BCDPQIF‐SCAD and BCDPQIF‐MCP significantly outperform BCDPQIF‐LASSO, indicating that BCDPQIF‐LASSO is less robust than BCDPQIF‐SCAD and BCDPQIF‐MCP.

Overall, these numerical study results have confirmed that the proposed method make sense, which is manifested in the dealing with measurement errors and within‐subject correlations, structural identification, estimation and variable selection.

4.2. Real Data Analysis

We now apply the proposed BCDPQIF method to data from the Multicenter AIDS Cohort Study, comprising 283 homosexual men infected with HIV between 1984 and 1991. This dataset has been widely used to illustrate VC models [5]; VCEV models [31] and PLVCEVM [15]. Because CD4 cells are crucial for immune function, the study focused on how risk factors—such as cigarette smoking, drug use, and pre‐infection CD4 cell levels—influence the post‐infection depletion of CD4 percentages. Previous analyses aimed to describe the trend of mean CD4 depletion over time and to evaluate the effects of pre‐infection CD4 percentage and age at HIV infection. In our application, we account for measurement errors in the covariates and demonstrate the utility of the BCDPQIF method on this dataset.

Let $Y$ be the individual's CD4 percentage, $X_1$ be the centered preCD4 percentage, $X_2$ be the centered age at HIV infection, $X_3=X_1·X_2$, $X_4=X_1^2$ and $X_5=X_2^2$. Then we consider the following model

$$Y={\alpha }_0(t)+X_1{\alpha }_1(t)+X_2{\alpha }_2(t)+X_3{\alpha }_3(t)+X_4{\alpha }_4(t)+X_5{\alpha }_5(t)+\epsilon$$ (30)

where ${\alpha }_0(t)$ is the baseline of CD4 percentage; ${\alpha }_1(t)$ and ${\alpha }_2(t)$ describe the effects of preCD4 percentage and age at HIV infection, two covariates that, in clinical practice, are particularly prone to measurement error (due to laboratory assay variability and patient recall), ${\alpha }_3(t)$ describes the interaction effect between the preCD4 percentage and age at HIV infection, ${\alpha }_4(t)$ and ${\alpha }_5(t)$ correspond to the $X_1^2$ and $X_2^2$ terms, respectively, capturing the quadratic effects of pre‐infection CD4 percentage and age at HIV infection. $t$ is the visiting time for each patient.

In this application, we considered observations of the pre‐infection CD4 percentage and age may contain measurement errors. The validity of the BCDPQIF method was verified by adding some measurement errors to the covariates, that is,

$$W_1=X_1+u_1,W_2=X_2+u_2$$

where ${\left(u_1,u_2\right)}^{\mathrm{T}}\sim N\left(0,{\sum }_u\right),{\sum }_u={\sigma }_u^2{I}_2$. We took ${\sigma }_u=0$, which assumes no measurement error. 0.4 and 0.6 represent different levels of measurement errors.

The BCDPQIF identified one varying coefficient ${\alpha }_0(t)$. Figure 1 shows the curve of ${\widehat{\alpha }}_0(t)$ over time under different measurement errors. It shows that ${\alpha }_0(t)$ decreases quickly at the beginning of HIV infection, and the rate of decrease slows down, which is similar to Zhao and Xue [31]. Furthermore, we found that the estimated functional curve ${\widehat{\alpha }}_1(t)$ under different measurement errors are very close to each other, which means that our bias‐corrected model selection scheme works well. This further demonstrates that the proposed model structure identification, estimation and variable selection method is valuable practically.

FIGURE 1.

The fitted plot of the BCDPQIF estimation ${\widehat{\alpha }}_0(t)$.

5. Conclusion and Discussion

In this article, combining the merits of Xu et al. [23] and Wang and Lin [24], we proposed a BCDPQIF for varying coefficient EV models with longitudinal data. Xu et al. [23] focused on a unified variable selection for longitudinal varying coefficient models, and Wang and Lin [24] conducted research on generalized partial linear varying coefficient models with longitudinal data. Notably, their approaches can do structure identification and variable selection simultaneously. However, they do not take into account the situation where the model contains measurement errors. It is worth noting that measurement errors are inevitable in practice. Especially for longitudinal data, both measurement errors and unknown working correlation matrices need to be handled appropriately. And precisely for this reason, we aim to study the structure identification, estimation and variable selection of the VCEV models with longitudinal data.

It is important to highlight that the VCEV models discussed here fall under a broad category of models, which includes both the linear EV models and the PLVCEV models. The proposed BCDPQIF method can identify the model structure, estimation and variable selection simultaneously for these models. To be precise, the BCDPQIF method can not onlyhandle measurement errors and unknown within subject correlations, but also identify whether the regression coefficients in the model are constant or varying coefficients, and select out the nonzero constant coefficients. This means that the BCDPQIF method avoids the assumption risks of the linear EV models, VCEV models and PLVCEV models. Theoretical and numerical results confirm that this method makes sense.

Furthermore, the BCDPQIF method is versatile and can be extended to structure identification, estimation and variable selection in a variety of models, including the additive models and the single‐index varying coefficient models, among others. Additionally, the BCDPQIF method is applicable to other forms of correlated data analysis, such as panel data and clustered data. In future work, we plan to use this method to investigate more complex modeling frameworks.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A.

Derivation Process of $D_i^{(\kappa )}$

$$\begin{array}{ll}& D_i^{(\kappa )}=\\ & \left(\begin{array}{cc}\text{tr}\left(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\right)·{\sum }_u& {\sum }_u\otimes \left({\mathbf{1}}_{1\times n_i}\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\\ {\left({\sum }_u\otimes \left({\mathbf{1}}_{1\times n_i}\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\right)}^{\mathrm{T}}& {\sum }_u\otimes \left(B_i\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right)\end{array}\right).\end{array}$$

First, we know that

$$\begin{array}{ll}\hfill & B_{ij}=I_q\otimes B(t_{ij})={\left(\begin{array}{cccc}B(t_{ij})& 0& \cdots & 0\\ 0& B(t_{ij})& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & B(t_{ij})\end{array}\right)}_{qL\times q},{ũ}_{ij}={\left(\begin{array}{c}B(t_{ij})u_{ij}^1\\ B(t_{ij})u_{ij}^2\\ ⋮\\ B(t_{ij})u_{ij}^q\end{array}\right)}_{qL\times 1},\end{array}$$

and

$$\begin{array}{ll}\hfill & {ũ}_i={({ũ}_{i1},{ũ}_{i2},\dots ,{ũ}_{i{n}_i})}^{\mathrm{T}}={\left(\begin{array}{cccc}{u}_{i1}^1{B}^{\mathrm{T}}(t_{i1})& u_{i1}^2{B}^{\mathrm{T}}(t_{i1})& \cdots & u_{i1}^q{B}^{\mathrm{T}}(t_{i1})\\ u_{i2}^1{B}^{\mathrm{T}}(t_{i2})& u_{i2}^2{B}^{\mathrm{T}}(t_{i2})& \cdots & u_{i2}^q{B}^{\mathrm{T}}(t_{i2})\\ ⋮& ⋮& \ddots & ⋮\\ u_{i{n}_i}^1{B}^{\mathrm{T}}(t_{i{n}_i})& u_{i{n}_i}^2{B}^{\mathrm{T}}(t_{i{n}_i})& \cdots & u_{i{n}_i}^q{B}^{\mathrm{T}}(t_{i{n}_i})\end{array}\right)}_{n_i\times qL}.\end{array}$$

For simplicity, we define the matrix ${\zeta }^{\kappa }$ as

$${\zeta }^{\kappa }=A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}=\left[\begin{array}{cccc}{\iota }_{11}^{\kappa }& {\iota }_{12}^{\kappa }& \cdots & {\iota }_{1n_i}^{\kappa }\\ {\iota }_{21}^{\kappa }& {\iota }_{22}^{\kappa }& \cdots & {\iota }_{2n_i}^{\kappa }\\ ⋮& ⋮& \ddots & ⋮\\ {\iota }_{n_{i}1}^{\kappa }& {\iota }_{n_{i}2}^{\kappa }& \cdots & {\iota }_{n_i{n}_i}^{\kappa }\end{array}\right].$$

Then, $D_i^{(\kappa )}$ can be reexpressed as follows

$$\begin{array}{cc}\hfill D_i^{(\kappa )}& =\mathrm{E}({(u_i,{ũ}_i)}^{\mathrm{T}}{\zeta }^{\kappa }(u_i,{ũ}_i))=\left(\begin{array}{cc}\mathrm{E}(u_i^{\mathrm{T}}{\zeta }^{\kappa }{u}_i)& \mathrm{E}(u_i^{\mathrm{T}}{\zeta }^{\kappa }{ũ}_i)\\ \mathrm{E}({ũ}_i^{\mathrm{T}}{\zeta }^{\kappa }{u}_i)& \mathrm{E}({ũ}_i^{\mathrm{T}}{\zeta }^{\kappa }{ũ}_i)\end{array}\right).\hfill \end{array}$$

After some matrix calculations, we can have

$$\begin{array}{cc}\mathrm{E}(u_i{\zeta }^{\kappa }{u}_i^{\mathrm{T}})\hfill & =\mathrm{E}\left(\left(u_{i1},u_{i2},\dots ,u_{i{n}_i}\right)\left[\begin{array}{cccc}{\iota }_{11}^{\kappa }& {\iota }_{12}^{\kappa }& \cdots & {\iota }_{1n_i}^{\kappa }\\ {\iota }_{21}^{\kappa }& {\iota }_{22}^{\kappa }& \cdots & {\iota }_{2n_i}^{\kappa }\\ ⋮& ⋮& \ddots & ⋮\\ {\iota }_{n_{i}1}^{\kappa }& {\iota }_{n_{i}2}^{\kappa }& \cdots & {\iota }_{n_i{n}_i}^{\kappa }\end{array}\right]\left(\begin{array}{c}{u}_{i1}^{\mathrm{T}}\\ u_{i2}^{\mathrm{T}}\\ ⋮\\ u_{i{n}_i}^{\mathrm{T}}\end{array}\right)\right)\\ \hfill & =\mathrm{E}\left(\sum _{j=1}^{n_i}{u}_{ij}{\iota }_{j1}^{\kappa }{u}_{i1}^{\mathrm{T}}+\sum _{j=1}^{n_i}{u}_{ij}{\iota }_{j2}^{\kappa }{u}_{i2}^{\mathrm{T}}+\cdots +\sum _{j=1}^{n_i}{u}_{ij}{\iota }_{j{n}_i}^{\kappa }{u}_{i{n}_i}^{\mathrm{T}}\right)\hfill \\ \hfill & =\left[{\iota }_{11}^{\kappa }\mathrm{E}\left(u_{i1}{u}_{i1}^{\mathrm{T}}\right)+{\iota }_{22}^{\kappa }\mathrm{E}\left(u_{i2}{u}_{i2}^{\mathrm{T}}\right)+\cdots +{\iota }_{n_i{n}_i}^{\kappa }\mathrm{E}\left(u_{i{n}_i}{u}_{i{n}_i}^{\mathrm{T}}\right)\right]\hfill \\ \hfill & =\text{tr}({\zeta }^{\kappa })·{\sum }_u.\hfill \end{array}$$

$$\begin{array}{cc}\mathrm{E}(u_i{\zeta }^{\kappa }{ũ}_i^{\mathrm{T}})\hfill & =\mathrm{E}\left({\left(u_{i1},u_{i2},\dots ,u_{i{n}_i}\right)}_{q\times n_i}\left[\begin{array}{cccc}{\iota }_{11}^{\kappa }& {\iota }_{12}^{\kappa }& \cdots & {\iota }_{1n_i}^{\kappa }\\ {\iota }_{21}^{\kappa }& {\iota }_{22}^{\kappa }& \cdots & {\iota }_{2n_i}^{\kappa }\\ ⋮& ⋮& \ddots & ⋮\\ {\iota }_{n_{i}1}^{\kappa }& {\iota }_{n_{i}2}^{\kappa }& \cdots & {\iota }_{n_i{n}_i}^{\kappa }\end{array}\right]{\left(\begin{array}{c}{ũ}_{i1}^{\mathrm{T}}\\ {ũ}_{i2}^{\mathrm{T}}\\ ⋮\\ {ũ}_{i{n}_i}^{\mathrm{T}}\end{array}\right)}_{n_i\times Lq}\right)\hfill \\ \hfill & =\mathrm{E}\left(\sum _{j=1}^{n_i}{u}_{ij}{\iota }_{j1}^{\kappa }{ũ}_{i1}^{\mathrm{T}}+\sum _{j=1}^{n_i}{u}_{ij}{\iota }_{j2}^{\kappa }{ũ}_{i2}^{\mathrm{T}}+\cdots +\sum _{j=1}^{n_i}{u}_{ij}{\iota }_{j{n}_i}^{\kappa }{ũ}_{i{n}_i}^{\mathrm{T}}\right)\hfill \\ \hfill & ={\left[{\iota }_{11}^{\kappa }\mathrm{E}(u_{i1}{ũ}_{i1}^{\mathrm{T}})+{\iota }_{22}^{\kappa }\mathrm{E}(u_{i2}{ũ}_{i2}^{\mathrm{T}})+\cdots +{\iota }_{n_i{n}_i}^{\kappa }\mathrm{E}(u_{i{n}_i}{ũ}_{i{n}_i}^{\mathrm{T}})\right]}_{q\times Lq}\hfill \\ \hfill & ={\sum }_u\otimes \left({(1,1,\dots 1)}_{1\times n_i}\text{diag}(\zeta ){B_i}^{\mathrm{T}}\right).\hfill \end{array}$$

$$\begin{array}{cc}\mathrm{E}({ũ}_i{\zeta }^{\kappa }{ũ}_i^{\mathrm{T}})\hfill & =\mathrm{E}\left(\left({ũ}_{i1},{ũ}_{i2},\dots ,{ũ}_{i{n}_i}\right)\left[\begin{array}{cccc}{\iota }_{11}^{\kappa }& {\iota }_{12}^{\kappa }& \cdots & {\iota }_{1n_i}^{\kappa }\\ {\iota }_{21}^{\kappa }& {\iota }_{22}^{\kappa }& \cdots & {\iota }_{2n_i}^{\kappa }\\ ⋮& ⋮& \ddots & ⋮\\ {\iota }_{n_{i}1}^{\kappa }& {\iota }_{n_{i}2}^{\kappa }& \cdots & {\iota }_{n_i{n}_i}^{\kappa }\end{array}\right]{\left(\begin{array}{c}{ũ}_{i1}^{\mathrm{T}}\\ {ũ}_{i2}^{\mathrm{T}}\\ ⋮\\ {ũ}_{i{n}_i}^{\mathrm{T}}\end{array}\right)}_{n_i\times Lq}\right)\hfill \\ \hfill & =\mathrm{E}{\left(\sum _{j=1}^{n_i}{ũ}_{ij}{\iota }_{j1}^{\kappa }{ũ}_{i1}^{\mathrm{T}}+\sum _{j=1}^{n_i}{ũ}_{ij}{\iota }_{j2}^{\kappa }{ũ}_{i2}^{\mathrm{T}}+\cdots +\sum _{j=1}^{n_i}{ũ}_{ij}{\iota }_{j{n}_i}^{\kappa }{ũ}_{i{n}_i}^{\mathrm{T}}\right)}_{Lq\times Lq}\hfill \\ \hfill & ={\left[{\iota }_{11}^{\kappa }\mathrm{E}({ũ}_{i1}{ũ}_{i1}^{\mathrm{T}})+{\iota }_{22}^{\kappa }\mathrm{E}({ũ}_{i2}{ũ}_{i2}^{\mathrm{T}})+\cdots +{\iota }_{n_i{n}_i}^{\kappa }\mathrm{E}({ũ}_{i{n}_i}{ũ}_{i{n}_i}^{\mathrm{T}})\right]}_{Lq\times Lq}\hfill \\ \hfill & ={\sum }_u\otimes \left(B_i\text{diag}(A_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2})B_i^{\mathrm{T}}\right).\hfill \end{array}$$

Therefore, we can obtain $D_i^{(\kappa )}$ defined as equation (11).

Proof of Theorems

Firstly, we present two necessary lemmas.

Lemma 1

If C1‐C11 hold, and $K=O\left(N^{1/(2r+1)}\right)$ , then we have

$$\begin{array}{l}\hfill {\dot{\widehat{\bar{g}}}}_n(\theta )\stackrel{p}{\to }-J_0,\sqrt{n}{\widehat{\bar{g}}}_n\left({\theta }_0\right)\stackrel{ℒ}{\to }N\left(0,{\mathrm{\Omega }}_0\right).\end{array}$$

According to Equation (14), we have

$$\begin{array}{ll}\hfill & {\widehat{\bar{g}}}_n(\theta )=\frac{1}{n}\sum _{i=1}^n{ĝ}_i(\theta )=\frac{1}{n}\sum _{i=1}^n\left(\begin{array}{c}{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_1{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+{\widehat{D}}_i^{(1)}\theta \\ {(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_2{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+{\widehat{D}}_i^{(2)}\theta \\ ⋮\\ {(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_s{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\theta )+{\widehat{D}}_i^{(s)}\theta \end{array}\right).\end{array}$$

Denote the $\kappa \text{th}$ block matrix of ${\dot{\widehat{\bar{g}}}}_n(\theta )$ as ${\dot{\widehat{\bar{g}}}}_{n\kappa }(\theta )$, $\kappa =1,2,\dots ,s$,

$$\begin{array}{ll}& {\dot{\widehat{\bar{g}}}}_{n\kappa }(\theta )\\ & =-\frac{1}{n}\sum _{i=1}^n\left({(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(W_i,{\tilde{W}}_i)-{\widehat{D}}_i^{(\kappa )}\right)\\ & =-\frac{1}{n}\sum _{i=1}^n\left({(X_i+u_i,{\tilde{X}}_i+{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\right.\left.(X_i+u_i,{\tilde{X}}_i+{ũ}_i)-{\widehat{D}}_i^{(\kappa )}\right)\\ & =-\frac{1}{n}\sum _{i=1}^n\left(\underset{{\mathrm{\Delta }}_{1i}}{\underbrace{{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(X_i,{\tilde{X}}_i)}}\right.+\underset{{\mathrm{\Delta }}_{2i}}{\underbrace{{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i)}}\\ & +\underset{{\mathrm{\Delta }}_{3i}}{\underbrace{{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(X_i,{\tilde{X}}_i)}}+\left.\underset{{\mathrm{\Delta }}_{4i}}{\underbrace{{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i)}}-{\widehat{D}}_i^{(\kappa )}\right)\\ & =-\left({\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2+{\mathrm{\Delta }}_3+{\mathrm{\Delta }}_4-\frac{1}{n}\sum _{i=1}^n{\widehat{D}}_i^{(\kappa )}\right).\end{array}$$

Then we have

$$\begin{array}{ll}\hfill {\mathrm{\Delta }}_4-\frac{1}{n}\sum _{i=1}^n{\widehat{D}}_i^{(\kappa )}& =\frac{1}{n}\sum _{i=1}^n{\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\left(u_i,{ũ}_i\right)-D_i^{(\kappa )}+D_i^{(\kappa )}-\frac{1}{n}\sum _{i=1}^n{\widehat{D}}_i^{(\kappa )}.\end{array}$$

Clearly, according to the law of large numbers, we have $\frac{1}{n}{\sum }_{i=1}^n{\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\left(u_i,{ũ}_i\right)-D_i^{(\kappa )}\stackrel{p}{\to }0$ and $D_i^{(\kappa )}-\frac{1}{n}{\sum }_{i=1}^n{\widehat{D}}_i^{(\kappa )}\stackrel{p}{\to }0$ as the $n\to \infty$. So we get ${\mathrm{\Delta }}_4-\frac{1}{n}{\sum }_{i=1}^n{\widehat{D}}_i^{(\kappa )}\stackrel{p}{\to }0$. Under C9, we can get ${\mathrm{\Delta }}_1\stackrel{p}{\to }{J}_0^{(\kappa )}$. Now, let's prove that ${\mathrm{\Delta }}_2\stackrel{p}{\to }0$ and ${\mathrm{\Delta }}_3\stackrel{p}{\to }0$.

Denote ${\mathrm{\Delta }}_2=\frac{1}{n}{\sum }_{i=1}^n{\xi }_{i\kappa }$, where ${\xi }_{i\kappa }={(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i)$. Obviously, we can get $\mathrm{E}\left({\xi }_{i\kappa }\right)=0$. From C4‐C7, we see that $\text{cov}\left({\xi }_{i\kappa }\right)$ are bounded. By the law of large numbers, we can get ${\mathrm{\Delta }}_3^{\mathrm{T}}={\mathrm{\Delta }}_2\stackrel{p}{\to }0$. Thus, we have ${\dot{\widehat{\bar{g}}}}_{n\kappa }(\theta )\stackrel{p}{\to }-J_0^{(\kappa )}$ and ${\dot{\widehat{\bar{g}}}}_n(\theta )\stackrel{p}{\to }-J_0$ where $J_0={\left(J_0^{(1)},J_0^{(2)},\dots ,J_0^{(s)}\right)}^{\mathrm{T}}$. According to the Taylor expansion to ${\widehat{\bar{g}}}_n(\theta )$ at ${\theta }_0$, we have

$$\begin{array}{l}\hfill {\widehat{\bar{g}}}_n(\theta )={\widehat{\bar{g}}}_n\left({\theta }_0\right)+{\dot{\widehat{\bar{g}}}}_n\left({\theta }_0\right)\left(\theta -{\theta }_0\right)+o\left(\theta -{\theta }_0\right).\end{array}$$

Denote the $\kappa \text{th}$ block matrix of ${\widehat{\bar{g}}}_n\left({\theta }_0\right)$ as ${\widehat{\bar{g}}}_{n\kappa }\left({\theta }_0\right)$, $\kappa =1,2,\dots ,s$,

$$\begin{array}{ll}\hfill {\widehat{\bar{g}}}_{n\kappa }({\theta }_0)& =\frac{1}{n}\sum _{i=1}^n\left[{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\right.\left.\left(Y_i-(W_i,{\tilde{W}}_i){\theta }_0\right)+{\widehat{D}}_i^{(\kappa )}{\theta }_0\right]\\ \hfill & =\frac{1}{n}\sum _{i=1}^n\left[{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i\right]\\ \hfill & -\frac{1}{n}\sum _{i=1}^n\left[{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i){\theta }_0\right]\\ \hfill & +\frac{1}{n}\sum _{i=1}^n\left[{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{J}_{XR}\right]\\ \hfill & +\frac{1}{n}\sum _{i=1}^n\left[{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i\right]\\ \hfill & +\frac{1}{n}\sum _{i=1}^n\left[{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{J}_{XR}\right]\\ \hfill & -\frac{1}{n}\sum _{i=1}^n\left[{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i){\theta }_0\right]+\frac{1}{n}\sum _{i=1}^n{D}_i^{(\kappa )}{\theta }_0\\ \hfill & =J_1-J_2+J_3+J_4+J_5-J_6+\frac{1}{n}\sum _{i=1}^n{D}_i^{(\kappa )}{\theta }_0.\end{array}$$

where $R(t_{ij})={\left(R_1(t_{ij}),R_2(t_{ij}),\dots ,R_q(t_{ij})\right)}^{\mathrm{T}}$, $R_k(t_{ij})={\alpha }_k(t_{ij})-B{(t_{ij})}^{\mathrm{T}}{\gamma }_k-{\beta }_k,k=1,2,\dots ,q$, and $J_{XR}=\text{diag}(X_i{R}_i^{\mathrm{T}}){(1,1,\dots ,1)}_{n_i\times 1}$.

$$\begin{array}{ll}\hfill & R_i^{\mathrm{T}}=\left(R\left(t_{i1}\right),R\left(t_{i2}\right),\dots ,R\left(t_{i{n}_i}\right)\right)=\left(\begin{array}{cccc}{R}_1\left(t_{i1}\right)& R_1\left(t_{i2}\right)& \cdots & R_1\left(t_{i{n}_i}\right)\\ R_2\left(t_{i1}\right)& R_2\left(t_{i2}\right)& \cdots & R_2\left(t_{i{n}_i}\right)\\ ⋮& ⋮& \ddots & ⋮\\ R_q\left(t_{i1}\right)& R_q\left(t_{i2}\right)& \cdots & R_q\left(t_{i{n}_i}\right)\end{array}\right).\end{array}$$

Denote $J_1=\frac{1}{n}{\sum }_{i=1}^n{\phi }_i$, where ${\phi }_i={(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i$. According to C5‐C7, we have $\mathrm{E}\left({\phi }_i\right)=0$ and

$$\text{cov}\left({\phi }_i\right)={(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{V}_i{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(X_i,{\tilde{X}}_i)<\infty .$$

By the law of the large numbers, we get $J_1\stackrel{p}{\to }0$. Similarly, we have $J_2\stackrel{p}{\to }0$ and $J_3\stackrel{p}{\to }0$.

Denote $J_4=\frac{1}{n}{\sum }_{i=1}^n{\varphi }_i$, ${\varphi }_i={\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i$. And since ${\epsilon }_i,u_i$ are independent of each other, we have $E\left({\varphi }_i\right)=0$. According to the Cauchy‐Schwarz inequality and C5‐C7 we have

$$\begin{array}{ll}\hfill {\left(\text{cov}\left({\phi }_i\right)\right)}^2& =\mathrm{E}\left({\left(u_i,{ũ}_i\right)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\left(u_i,{ũ}_i\right)\right)\mathrm{E}\left({\epsilon }_i^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i\right)<\infty .\end{array}$$

Thus, $J_4\stackrel{p}{\to }0$. By the law of large numbers, from the definition of ${\widehat{D}}_i^{(\kappa )}$, we have $J_6-\frac{1}{n}{\sum }_{i=1}^n{\widehat{D}}_i^{(\kappa )}{\theta }_0\stackrel{p}{\to }0$. From C8, we have $J_5=O_p(n^{-1/2}{K}^{-r})=o_p(n^{-1/2})$ and $J_3=o_p(n^{-1/2})$. So, we have ${\widehat{\bar{g}}}_n(\theta )\stackrel{p}{\to }{J}_0\left({\theta }_0-\theta \right),\forall \theta \in \mathrm{\Theta }.$

Following Tian, Xue and Liu [28], according to the results above, we have

$$\begin{array}{ll}\hfill {\widehat{\bar{g}}}_{n\kappa }({\theta }_0)& =\frac{1}{n}\sum _{i=1}^n\left[{((X_i,{\tilde{X}}_i)+(u_i,{ũ}_i))}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}\right.\\ \hfill & \left.({\epsilon }_i-(u_i,{ũ}_i){\theta }_0)+{\widehat{D}}_i^{(\kappa )}{\theta }_0\right]+o_p(n^{-1/2})\\ \hfill & =\frac{1}{n}\sum _{i=1}^n\left[{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i\right.-{(X_i,{\tilde{X}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i){\theta }_0\\ \hfill & +{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}{\epsilon }_i\\ \hfill & -\left.{(u_i,{ũ}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{\kappa }{A}_i^{-1/2}(u_i,{ũ}_i){\theta }_0+{\widehat{D}}_i^{(\kappa )}{\theta }_0\right]+o_p(n^{-1/2})\\ \hfill & =\frac{1}{n}\sum _{i=1}^n\left({\psi }_{i\kappa 1}+{\psi }_{i\kappa 2}+{\psi }_{i\kappa 3}+{\psi }_{i\kappa 4}\right)+o_p(n^{-1/2})\\ \hfill & =\frac{1}{n}\sum _{i=1}^n{\psi }_{i\kappa }+o_p(n^{-1/2}).\end{array}$$

where ${\psi }_i={\left({\psi }_{i1},{\psi }_{i2},\dots ,{\psi }_{is}\right)}^{\mathrm{T}},{\psi }_{i\kappa }={\psi }_{i\kappa 1}+{\psi }_{i\kappa 2}+{\psi }_{i\kappa 3}+{\psi }_{i\kappa 4}$. So we have ${\widehat{\bar{g}}}_n\left({\theta }_0\right)=\frac{1}{n}{\sum }_{i=1}^n{\psi }_i+o_p(n^{-1/2}),\text{and}{\mathrm{\Omega }}_n\left({\theta }_0\right)=\frac{1}{n}{\sum }_{i=1}^n{\psi }_i{\psi }_i^{\mathrm{T}}+o(1).$ From C5‐C7, we get $\mathrm{E}\left({\psi }_{ikm}\right)=0,\text{cov}\left({\psi }_{i\kappa m}\right)<\infty ,m=1,2,3,4.$Following the properties of covariance matrix, we have

$$\begin{array}{ll}\hfill & \text{cov}\left({\psi }_{i\kappa }\right)\le \sum _{m=1}^4\text{cov}\left({\psi }_{i\kappa m}\right)+\sum _{m\ne l}\sqrt{\text{cov}\left({\psi }_{i\kappa m}\right)\text{cov}\left({\psi }_{i\kappa l}\right)}<\infty ,\\ \hfill & \forall a\in {ℝ}^{s(q+qL)},a^{\mathrm{T}}a=1,\mathrm{E}(a^{\mathrm{T}}{\psi }_i)=0,\underset{i}{\sup }\mathrm{E}\Vert a^{\mathrm{T}}{\psi }_i{\Vert }^3\le \Vert a^{\mathrm{T}}\Vert \underset{i}{\sup }\mathrm{E}\Vert {\psi }_i{\Vert }^3.\end{array}$$

According to the Slutsky Theorem, we have $\sqrt{n}{\widehat{\bar{g}}}_n\left({\theta }_0\right)\stackrel{ℒ}{\to }N\left(0,{\mathrm{\Omega }}_0\right),{\widehat{\bar{g}}}_n\left({\theta }_0\right)=O_p(n^{-1/2})$. The proof of Lemma 1 is completed.

Lemma 2

If C1‐C11 hold, we get

$$\Vert n^{-1}{\dot{Q}}_n\left({\theta }_0\right)-2{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}\left({\theta }_0\right){\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n\left({\theta }_0\right)\Vert =O_p(n^{-1}),$$ (A1)

$$\Vert n^{-1}{\ddot{Q}}_n\left({\theta }_0\right)-2{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}\left({\theta }_0\right){\mathrm{\Omega }}_n^{-1}{\dot{\widehat{\bar{g}}}}_n\left({\theta }_0\right)\Vert =o_p(1).$$ (A2)

The proof of Lemma 2 is similar as Lemma 2 in Tian, Xue and Liu [28] and details are omitted here.

Proof of Theorem 1

Let $\delta =n^{-r/(2r+1)},\beta ={\beta }_0+\delta C_1,\gamma ={\gamma }_0+\delta C_2$ and $C={(C_1^{\mathrm{T}},C_2^{\mathrm{T}})}^{\mathrm{T}}$. To prove Theorem 1, it is sufficient to show that $\forall \epsilon >0,\exists$ a large constant $C_0$ satisfies

$$P\left\{\underset{\Vert C\Vert =C_0}{\inf }{Q}_p(\theta )⩾Q_p\left({\theta }_0\right)\right\}⩾1-\epsilon .$$ (A3)

Obviously, when $\epsilon \ge 1$, Equation (A3) is always true. Therefore, we consider the case that $\epsilon \in (0,1)$. Assume ${\alpha }_k(·)=0(k=q_1+1,q_1+2,\dots ,q)$, $p_{\lambda }(0)=0$ and let $\mathrm{\Delta }(\beta ,\gamma )=\frac{1}{K}[Q_p(\theta )-Q_p\left({\theta }_0\right)],{\theta }_0={({\beta }_0^{\mathrm{T}},{\gamma }_0^{\mathrm{T}})}^{\mathrm{T}}$, we have

$$\begin{array}{ll}\hfill \mathrm{\Delta }(\beta ,\gamma )& \ge \frac{1}{K}\left[Q_n(\theta )-Q_n\left({\theta }_0\right)\right]+\frac{n}{K}\sum _{l=1}^{q_1}\left[p_{{\lambda }_{1l}}\left({‖{\gamma }_l‖}_H\right)-p_{{\lambda }_{1l}}\left({‖{\gamma }_{l0}‖}_H\right)\right]\\ \hfill & +\frac{n}{K}\sum _{k=1}^{q_1}\left[p_{{\lambda }_{2k}}\left(\left|{\beta }_k\right|\right)-p_{{\lambda }_{2k}}\left(\left|{\beta }_{k0}\right|\right)\right]\\ \hfill & ={\tilde{\mathrm{\Delta }}}_1+{\tilde{\mathrm{\Delta }}}_2+{\tilde{\mathrm{\Delta }}}_3.\end{array}$$

Apply Taylor expansion to $Q_n(\theta )$ at ${\theta }_0$, we have $Q_n(\theta )=Q_n\left({\theta }_0+\delta C\right)=Q_n\left({\theta }_0\right)+\delta C^{\mathrm{T}}{\dot{Q}}_n\left({\theta }_0\right)+\frac{1}{2}{\delta }^2{C}^{\mathrm{T}}{\ddot{Q}}_n(\tilde{\theta })C,$ where $\tilde{\theta }$ lies between $\theta$ and ${\theta }_0$. According to Lemmas 1 and 2, we can get

$$\begin{array}{cc}\hfill \delta C^{\mathrm{T}}{\dot{Q}}_n\left({\theta }_0\right)& =\delta C^{\mathrm{T}}\left\{2n{\dot{\widehat{\bar{g}}}}_n\left({\theta }_0\right){\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n\left({\theta }_0\right)+n{O}_p\left(n^{-1}\right)\right\}\hfill \\ \hfill & =\Vert C\Vert O_p(\sqrt{n}\delta )+\Vert C\Vert O_p(\delta ),\hfill \\ \hfill \frac{1}{2}{\delta }^2{C}^{\mathrm{T}}{\ddot{Q}}_n\left({\theta }_0\right)C& ={\delta }^2{C}^{\mathrm{T}}\left\{2n{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}\left({\theta }_0\right){\mathrm{\Omega }}_n^{-1}{\dot{\widehat{\bar{g}}}}_n\left({\theta }_0\right)+n{o}_p(1)\right\}C\hfill \\ \hfill & =n{\delta }^2{C}^{\mathrm{T}}{\dot{\widehat{\bar{g}}}}_n^{\mathrm{T}}\left({\theta }_0\right){\mathrm{\Omega }}_n^{-1}{\dot{\widehat{\bar{g}}}}_n\left({\theta }_0\right)C+n{\delta }^2\Vert C{\Vert }^2{o}_p(1).\hfill \end{array}$$

Therefore, we have

$$\begin{array}{ll}\hfill {\tilde{\mathrm{\Delta }}}_1& =\frac{1}{K}\left\{n{\delta }^2\Vert C{\Vert }^2{J}_0^{\mathrm{T}}{\mathrm{\Omega }}_0^{-1}{J}_0\right.+\left.\Vert C\Vert O_p(\sqrt{n}\delta )+\Vert C\Vert O_p(\delta )+n{\delta }^2\Vert C{\Vert }^2{o}_p(1)\right\}.\end{array}$$

Obviously, $n{\delta }^2\Vert C{\Vert }^2{J}_0^{\mathrm{T}}{\mathrm{\Omega }}_0^{-1}{J}_0\ge 0$. When $C$ is large enough,

$$\begin{array}{ll}\hfill n{\delta }^2\Vert C{\Vert }^2{J}_0^{\mathrm{T}}{\mathrm{\Omega }}_0^{-1}{J}_0& \ge \Vert C\Vert O_p(\sqrt{n}\delta ),n{\delta }^2\Vert C{\Vert }^2{J}_0^{\mathrm{T}}{\mathrm{\Omega }}_0^{-1}{J}_0\ge n{\delta }^2\Vert C{\Vert }^2{o}_p(1).\end{array}$$

So when $C$ is large enough, ${\mathrm{\Delta }}_1>0$. Next, by Taylor expansion, we get that

$$\begin{array}{ll}\hfill {\tilde{\mathrm{\Delta }}}_2& =\frac{n}{K}\sum _{k=1}^{p_1}\left[p_{{\lambda }_{2k}}\left(\left|{\beta }_k\right|\right)-p_{{\lambda }_{2k}}\left(\left|{\beta }_{k0}\right|\right)\right]\\ \hfill & =\frac{1}{K}\sum _{k=1}^{p_1}\left[n\delta p_{{\lambda }_{2k}}^{\prime }\left(\left|{\beta }_{k0}\right|\right)\text{sgn}\left({\beta }_{k0}\right)\left|C_2\right|\right.+\left.n{\delta }^2{p}_{{\lambda }_{2k}}^{″}\left({\beta }_{k0}\right){\left|C_2\right|}^2(1+o(1))\right]\\ \hfill & \le \frac{1}{K}\left\{\sqrt{p_1}n\delta a_n\Vert C\Vert +n{\delta }^2{a}_n\Vert C{\Vert }^2\right\},\\ \hfill {\tilde{\mathrm{\Delta }}}_3& =\frac{n}{K}\sum _{k=1}^{p_1}\left[p_{{\lambda }_{2k}}\left(\left|{\beta }_k\right|\right)-p_{{\lambda }_{2k}}\left(\left|{\beta }_{k0}\right|\right)\right]\\ \hfill & =\frac{1}{K}\sum _{k=1}^{p_1}\left[n\delta p_{{\lambda }_{2k}}^{\prime }\left(\left|{\beta }_{k0}\right|\right)\text{sgn}\left({\beta }_{k0}\right)\left|C_1\right|\right.+\left.n{\delta }^2{p}_{{\lambda }_{2k}}^{″}\left({\beta }_{k0}\right){\left|C_1\right|}^2(1+o(1))\right]\\ \hfill & \le \frac{1}{K}\left\{\sqrt{p_1}n\delta a_n\Vert C\Vert +n{\delta }^2{a}_n\Vert C{\Vert }^2\right\}.\end{array}$$

We can see that for sufficiently large $C$, ${\tilde{\mathrm{\Delta }}}_1\ge {\mathrm{\Delta }}_2$ and ${\tilde{\mathrm{\Delta }}}_1\ge {\tilde{\mathrm{\Delta }}}_3$ uniformly in $\Vert {\mathrm{\Delta }}_{\beta }\Vert =C$. Thus, inequality (A3) holds. According to Schumaker [29], we get

$$\Vert {\alpha }_k(t)-{\beta }_{0k}-B^{\mathrm{T}}(t){\gamma }_{0k}\Vert =O_p\left(n^{-r/(2r+1)}\right),k=1,2,\dots ,q.$$

And then we have

$$\begin{array}{ll}\hfill & {‖{\widehat{\alpha }}_k(t)-{\alpha }_k(t)‖}^2\\ \hfill & ={\int }_0^1\left\{B^{\mathrm{T}}(t){\widehat{\gamma }}_k+{\widehat{\beta }}_k-B^{\mathrm{T}}(t){\gamma }_{0k}-{\beta }_{0k}-{\alpha }_k(t)+{\beta }_{0k}\right.+{\left.B^{\mathrm{T}}(t){\gamma }_{0k}\right\}}^{2}dt\\ \hfill & \le 2{\int }_0^1{\left\{B^{\mathrm{T}}(t){\widehat{\gamma }}_k-B^{\mathrm{T}}(t){\gamma }_{0k}+{\widehat{\beta }}_k-{\beta }_{0k}\right\}}^{2}dt+2{\int }_0^1{\left\{{\alpha }_k(t)-{\beta }_{0k}-B^{\mathrm{T}}(t){\gamma }_{0k}\right\}}^{2}dt\\ \hfill & =O_p\left(n^{-2r/(2r+1)}\right).\end{array}$$

The proof of Theorem 1 is finished. See the reference Fan and Li [18].

Proof of Theorem 2

To prove part (i), we just need to prove ${‖{\gamma }_k‖}_{\mathrm{H}}=0$ for $k\in 𝒞\cup 𝒵$. According to Theorem 1, it is sufficient to show that, for any $\theta$ that satisfies $\Vert {\widehat{\theta }}^{(𝒞)}-{\theta }_0^{(𝒞)}\Vert =O_p\left(N^{-1/(2r+1)}\right)$ and $\Vert {\widehat{\theta }}^{(𝒱)}-{\theta }_0^{(𝒱)}\Vert =O_p\left(N^{-1/(2r+1)}\right)$, and $\exists$ a small $e=C{n}^{-1/(2r+1)}$, when $n\to \infty$, with probability tending to 1, we have

$$\frac{\partial Q_p(\beta )}{\partial {\gamma }_{kl}}<0,-e<{\gamma }_{kl}<0,l=1,2,\dots ,L,k\in 𝒞\cup 𝒵,$$ (A4)

$$\frac{\partial Q_p(\beta )}{\partial {\gamma }_{kl}}>0,e>{\gamma }_{kl}>0,l=1,2,\dots ,L,k\in 𝒞\cup 𝒵.$$ (A5)

According to Equations (A1) and (A2), we have

$$\begin{array}{cc}\hfill \frac{\partial Q_p(\theta )}{\partial {\gamma }_{kl}}& =2n\frac{\partial {\widehat{\bar{g}}}_n(\theta )}{\partial {\gamma }_{kl}}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n(\theta )+o_p(n^{-1})+n{p}_{{\lambda }_{1k}}^{\prime }\left({\Vert \gamma \Vert }_{\mathrm{H}}\right)\frac{{\sum }_{j=1}^L{h}_{lj}{\gamma }_{kj}}{\Vert {\gamma }_k{\Vert }_{\mathrm{H}}}\hfill \\ & =2n\frac{\partial {\widehat{\bar{g}}}_n(\theta )}{\partial {\gamma }_{kl}}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n(\theta )+o_p(n^{-1})+n{p}_{{\lambda }_{1k}}^{\prime }(\Vert {\gamma }_k{\Vert }_{\mathrm{H}})\frac{{(\mathrm{H}{\gamma }_k)}_l}{\Vert {\gamma }_k{\Vert }_{\mathrm{H}}}\hfill \\ & =n{\lambda }_{1k}\left\{2\frac{\partial {\widehat{\bar{g}}}_n(\theta )}{\partial {\gamma }_{kl}}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n(\theta )\right.+\left.{\lambda }_{1k}^{-1}{p}_{{\lambda }_{1k}}^{\prime }(\Vert {\gamma }_k{\Vert }_{\mathrm{H}})\frac{{(\mathrm{H}{\gamma }_k)}_l}{\Vert {\gamma }_k{\Vert }_{\mathrm{H}}}\right\}+o_p(n^{-1}).\hfill \end{array}$$

According to the condition C10 and $n^{r/(2r+1)}{\lambda }_{\min }\to \infty$, it is clear that the $\text{sgn}$ of $\frac{\partial Q_p(\theta )}{\partial {\gamma }_{kl}}$ is completely determined by that of ${\gamma }_{kl}$, then Equations (A4) and (A5) hold. This completes the proof of part (i).

Similarly, to prove part (ii), we need to prove ${‖{\gamma }_k‖}_{\mathrm{H}}=0$ for $k\in 𝒞\cup 𝒵$ holds with probability tending to one. It is clear that ${\widehat{\gamma }}_k=0$ for $k=1,2,\dots ,c$, and then ${\widehat{\alpha }}_k(·)$ has been reduced to a constant; it remains to prove that ${\widehat{\beta }}_k=0$ for $k\in 𝒵$. It is sufficient to show that, for any $\beta$ that satisfies $\Vert {\widehat{\theta }}^{(𝒞)}-{\theta }_0^{(𝒞)}\Vert =O_p\left(N^{-1/(2r+1)}\right)$ and $\Vert {\widehat{\theta }}^{(𝒱)}-{\theta }_0^{(𝒱)}\Vert =O_p\left(N^{-1/(2r+1)}\right)$, and for some given small $e=C{n}^{-1/(2r+1)}$, when $n\to \infty$, with probability tending to 1, we have

$$\begin{array}{ll}\hfill \frac{\partial Q_p(\theta )}{\partial {\beta }_k}& <0,-e<{\beta }_k<0,\text{and}\\ \hfill \frac{\partial Q_p(\theta )}{\partial {\beta }_k}& >0,e>{\beta }_k>0,k=v+1,v+2,\dots ,q.\end{array}$$

Applying similar techniques as in the analysis of part (i), we have

$$\begin{array}{ll}\hfill \frac{\partial Q_p(\theta )}{\partial {\beta }_k}& =2n\frac{\partial {\widehat{\bar{g}}}_n^{\mathrm{T}}(\theta )}{\partial {\beta }_k}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n(\theta )+o_p\left(n^{-1}\right)+n{p}_{2k}^{\prime }\left(\left|{\beta }_k\right|\right)\text{sgn}\left({\beta }_k\right)\\ \hfill & =n{\lambda }_{2k}\left\{2{\lambda }_{2k}^{-1}\frac{\partial {\widehat{\bar{g}}}_n^{\mathrm{T}}(\theta )}{\partial {\beta }_k}{\mathrm{\Omega }}_n^{-1}{\widehat{\bar{g}}}_n(\theta )+{\lambda }_{2k}^{-1}{p}_{2k}^{\prime }\left(\left|{\beta }_k\right|\right)\text{sgn}\left({\beta }_k\right)\right\}+o_p(n^{-1}).\end{array}$$

It is clear that the sign of $\frac{\partial Q_p(\theta )}{\partial {\beta }_k}$ is completely determined by the sign of ${\beta }_k$. Then, with probability tending to 1, ${\widehat{\beta }}_k=0$ for $k\in 𝒵$. The proof of part (ii) is finished. See the Xu et al. [23] and reference therein.

Proof of Theorem 3

Let ${\alpha }_0(·)={({\alpha }_{01}(·),{\alpha }_{02}(·),\dots ,{\alpha }_{0q}(·))}^{\mathrm{T}}$ be the real coefficients in model (2).

$$\begin{array}{ll}\hfill {\theta }_0& ={\left({\beta }_0^{\mathrm{T}},{\gamma }_0^{\mathrm{T}}\right)}^{\mathrm{T}}=\left({({\beta }_0^{(𝒞)})}^{\mathrm{T}},{({\beta }_0^{(𝒱)})}^{\mathrm{T}},{({\beta }_0^{(𝒵)})}^{\mathrm{T}},\right.{\left.{({\gamma }_0^{(𝒞)})}^{\mathrm{T}},{({\gamma }_0^{(𝒱)})}^{\mathrm{T}},{({\gamma }_0^{(𝒵)})}^{\mathrm{T}}\right)}^{\mathrm{T}}.\end{array}$$

where ${\beta }_0^{(𝒞)}={({\beta }_{01},\dots ,{\beta }_{0c})}^{\mathrm{T}},{\beta }_0^{(𝒱)}={({\beta }_{0(c+1)},\dots ,{\beta }_{0v})}^{\mathrm{T}},{\beta }_0^{(𝒵)}={({\beta }_{0(v+1)},\dots ,{\beta }_{0q})}^{\mathrm{T}}$, and ${\gamma }_0^{(𝒞)}={({\gamma }_{01}^{\mathrm{T}},\dots ,{\gamma }_{0c}^{\mathrm{T}})}^{\mathrm{T}},{\gamma }_0^{(𝒱)}={({\gamma }_{0(c+1)}^{\mathrm{T}},\dots ,{\gamma }_{0v}^{\mathrm{T}})}^{\mathrm{T}},{\gamma }_0^{(𝒵)}={({\gamma }_{0(v+1)}^{\mathrm{T}},\dots ,{\gamma }_{0q}^{\mathrm{T}})}^{\mathrm{T}}.$

Thus,

$$\begin{array}{ll}\hfill (W_i,{\tilde{W}}_i)\theta & =W_i^{(𝒞)}{\widehat{\beta }}^{(𝒞)}+W_i^{(𝒱)}{\widehat{\beta }}^{(𝒱)}+{\tilde{W}}_i^{(𝒱)}{\widehat{\gamma }}^{(𝒱)}.\end{array}$$

Then, Theorems 1 and 2 imply that, as $n\to \infty$, with probability tending to 1, the objective function $Q_p(\theta )$ attains its minimum at $\widehat{\theta }={\left({({\widehat{\beta }}^{(𝒞)})}^{\mathrm{T}},{({\widehat{\beta }}^{(𝒱)})}^{\mathrm{T}},{(0)}^{\mathrm{T}},{(0)}^{\mathrm{T}},{({\widehat{\gamma }}^{(𝒱)})}^{\mathrm{T}},{(0)}^{\mathrm{T}}\right)}^{\mathrm{T}}.$ Denote $Q_{1p}(\theta )=\frac{\partial Q_p(\theta )}{\partial {\beta }_1^{(𝒞)}},Q_{2p}(\theta )=\frac{\partial Q_p(\theta )}{\partial {\gamma }^{(𝒱)}}$. We know that

$$\begin{array}{ll}& Q_{1p}\left({\left({({\widehat{\beta }}^{(𝒞)})}^{\mathrm{T}},{({\widehat{\beta }}^{(𝒱)})}^{\mathrm{T}},{(0)}^{\mathrm{T}},{(0)}^{\mathrm{T}},{({\widehat{\gamma }}^{(𝒱)})}^{\mathrm{T}},{(0)}^{\mathrm{T}}\right)}^{\mathrm{T}}\right)\\ & =-\frac{2}{n}\sum _{i=1}^n\sum _{r=1}^s\sum _{r^{\prime }=1}^s\left[{(X_i^{(𝒞)})}^{\mathrm{T}}{A}_i^{-1/2}{M}_r{A}_i^{-1/2}(W_i,{\tilde{W}}_i){\mathrm{\Omega }}_{r,r^{\prime }}^{-1}\right.\\ & \left.\{{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{r^{\prime }}{A}_i^{-1/2}(Y_i-(W_i,{\tilde{W}}_i)\widehat{\theta })+{\widehat{D}}_i^{(r^{\prime })}\widehat{\theta }\}\right]\\ & +{\mathrm{o}}_p(n^{-1})+\sum _{k=1}^c{p}_{2k}^{\prime }(|{\widehat{\beta }}_k|)\text{sgn}({\widehat{\beta }}_k)\\ & =-\frac{2}{n}\sum _{i=1}^n{(X_i^{(𝒞)})}^{\mathrm{T}}{\tau }_i\left\{W_i^{(𝒞)}{\beta }^{(𝒞)}+W_i^{(𝒱)}{\beta }^{(𝒱)}+{\tilde{W}}_i^{(𝒱)}{\gamma }^{(𝒱)}\right.\\ & -\left.[W_i^{(𝒞)}{\widehat{\beta }}^{(𝒞)}+W_i^{(𝒱)}{\widehat{\beta }}^{(𝒱)}+{\tilde{W}}_i^{(𝒱)}{\widehat{\gamma }}^{(𝒱)}]+{\epsilon }_i+J_{XR}\right\}\\ & +{\mathrm{o}}_p(n^{-1})+\sum _{k=1}^c{p}_{2k}^{\prime }(|{\widehat{\beta }}_k|)\text{sgn}({\widehat{\beta }}_k)\\ & =0.\end{array}$$ (A6)

and

$$\begin{array}{ll}\hfill & Q_{2p}\left({\left({({\widehat{\beta }}^{(𝒞)})}^{\mathrm{T}},{({\widehat{\beta }}^{(𝒱)})}^{\mathrm{T}},{(0)}^{\mathrm{T}},{(0)}^{\mathrm{T}},{({\widehat{\gamma }}^{(𝒱)})}^{\mathrm{T}},{(0)}^{\mathrm{T}}\right)}^{\mathrm{T}}\right)\\ \hfill & =-\frac{2}{n}\sum _{i=1}^n\sum _{r=1}^s\sum _{r^{\prime }=1}^s\left[{({\tilde{X}}_i^{(𝒱)})}^{\mathrm{T}}{A}_i^{-1/2}{M}_r{A}_i^{-1/2}(W_i,{\tilde{W}}_i){\mathrm{\Omega }}_{r,r^{\prime }}^{-1}\right.\\ \hfill & \left\{{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{r^{\prime }}{A}_i^{-1/2}(Y_i-[W_i^{(𝒞)}{\widehat{\beta }}^{(𝒞)}\right.+W_i^{(𝒱)}{\widehat{\beta }}^{(𝒱)}\\ \hfill & +\left.\left.{\tilde{W}}_i^{(𝒱)}{\widehat{\gamma }}^{(𝒱)}])+{\widehat{D}}_i^{(1)}\widehat{\theta }\right\}\right]+{\mathrm{o}}_p(n^{-1})+\sum _{k=c+1}^v{p}_{1k}^{\prime }\left({‖{\widehat{\gamma }}_k‖}_{\mathrm{H}}\right)\frac{\mathrm{H}{\gamma }_k}{{‖{\widehat{\gamma }}_k‖}_{\mathrm{H}}}\\ \hfill & =-\frac{2}{n}\sum _{i=1}^n{({\tilde{X}}_i^{(𝒱)})}^{\mathrm{T}}{\tau }_i\left\{W_i^{(𝒞)}{\beta }^{(𝒞)}+W_i^{(𝒱)}{\beta }^{(𝒱)}+{\tilde{W}}_i^{(𝒱)}{\gamma }^{(𝒱)}\right.\\ \hfill & -\left.[W_i^{(𝒞)}{\widehat{\beta }}^{(𝒞)}+W_i^{(𝒱)}{\widehat{\beta }}^{(𝒱)}+{\tilde{W}}_i^{(𝒱)}{\widehat{\gamma }}^{(𝒱)}]+{\epsilon }_i+J_{XR}\right\}\\ \hfill & +{\mathrm{o}}_p(n^{-1})+\sum _{k=c+1}^v{p}_{1k}^{\prime }\left({‖{\widehat{\gamma }}_k‖}_{\mathrm{H}}\right)\frac{\mathrm{H}}{{\gamma }_k}{‖{\widehat{\gamma }}_k‖}_{\mathrm{H}}\\ \hfill & =0.\end{array}$$ (A7)

where ${\mathrm{\Omega }}_{\kappa {\kappa }^{\prime }}^{-1}$ is the $\left(\kappa ,{\kappa }^{\prime }\right)$ block of ${\mathrm{\Omega }}_0^{-1}$ ${\tau }_i={\sum }_{r=1}^s{\sum }_{r^{\prime }=1}^s{A}_i^{-1/2}{M}_r{A}_i^{-1/2}(W_i,{\tilde{W}}_i){\mathrm{\Omega }}_{r{r}^{\prime }}^{-1}{(W_i,{\tilde{W}}_i)}^{\mathrm{T}}{A}_i^{-1/2}{M}_{r^{\prime }}{A}_i^{-1/2}$.

Apply the Taylor expansion to $p_{{\lambda }_{2k}}^{\prime }(|{\widehat{\beta }}_k|)$, we have

$$p_{{\lambda }_{2k}}^{\prime }(|{\widehat{\beta }}_k|)=p_{{\lambda }_{2k}}^{\prime }\left(\left|{\beta }_{0k}\right|\right)+\left\{p_{{\lambda }_{2k}}^{″}\left(\left|{\beta }_{0k}\right|\right)+o_p(1)\right\}({\widehat{\beta }}_k-{\beta }_{0k}).$$

Condition C10 implies that $p_{{\lambda }_{2k}}^{″}\left(\left|{\beta }_{0k}\right|\right)=o_p(1)$, and note that $p_{{\lambda }_{2k}}^{\prime }\left(\left|{\beta }_{0k}\right|\right)=0$ as ${\lambda }_{\max }\to o_p(\widehat{\beta }-{\beta }_0)$. Thus, $p_{{\lambda }_{2k}}^{\prime }(|{\widehat{\beta }}_k|)=0$. By the same argument, we know that ${‖{\widehat{\gamma }}_k‖}_{\mathrm{H}}\ge a{\lambda }_{1k}$ for $n$ large enough. Thus, $p_{{\lambda }_{1k}}^{\prime }\left({‖{\gamma }_{0k}‖}_{\mathrm{H}}\right)=0$ and $p_{{\lambda }_{1k}}^{″}\left({‖{\gamma }_{0k}‖}_{\mathrm{H}}\right)=0$, which imply that $p_{1k}^{\prime }\left(\Vert {\widehat{\gamma }}_k{\Vert }_{\mathrm{H}}\right)=0$.

Hence, according to equations (A6) and (A7), we have

$$\begin{array}{ll}\hfill & -\frac{2}{n}\sum _{i=1}^n{(X_i^{(𝒞)})}^{\mathrm{T}}{\tau }_i\left(W_i^{(𝒞)}\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)+W_i^{(𝒱)}\left({\beta }^{(𝒱)}-{\widehat{\beta }}^{(𝒱)}\right)\right.\\ \hfill & +\left.{\tilde{W}}_i^{(𝒱)}\left({\gamma }^{(𝒱)}-{\widehat{\gamma }}^{(𝒱)}\right)+{\epsilon }_i+J_{XR}\right)+o_p\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)=0.\end{array}$$ (A8)

$$\begin{array}{ll}\hfill & -\frac{2}{n}\sum _{i=1}^n{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i\left(W_i^{(𝒞)}\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)+W_i^{(𝒱)}\left({\beta }^{(𝒱)}-{\widehat{\beta }}^{(𝒱)}\right)\right.\\ \hfill & +\left.{\tilde{W}}_i^{(𝒱)}\left({\gamma }^{(𝒱)}-{\widehat{\gamma }}^{(𝒱)}\right)+{\epsilon }_i+J_{XR}\right)+o_p\left(\mathrm{\Delta }{\theta }^{(𝒱)}\right)=0.\end{array}$$ (A9)

To clearly present the combined term, we have

$${\mathbf{W}}_i^{(𝒱)}\mathrm{\Delta }{\theta }^{(𝒱)}=W_i^{(𝒱)}\left({\beta }^{(𝒱)}-{\widehat{\beta }}^{(𝒱)}\right)+{\tilde{W}}_i^{(𝒱)}\left({\gamma }^{(𝒱)}-{\widehat{\gamma }}^{(𝒱)}\right),$$

where $\mathrm{\Delta }{\theta }^{(𝒱)}=\left(\begin{array}{l}{\beta }^{(𝒱)}-{\widehat{\beta }}^{(𝒱)}\\ {\gamma }^{(𝒱)}-{\widehat{\gamma }}^{(𝒱)}\end{array}\right)$ and ${\mathbf{W}}_i^{(𝒱)}=\left(\begin{array}{l}{W}_i^{(𝒱)},{\tilde{W}}_i^{(𝒱)}\end{array}\right).$ Denote ${\mathrm{\Phi }}_n\equiv \frac{1}{n}{\sum }_{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i{W}_i^{(𝒞)}$, ${\mathrm{\Psi }}_n\equiv \frac{1}{n}{\sum }_{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i{\mathbf{W}}_i^{(𝒱)}$, and $\varrho \equiv \frac{1}{n}{\sum }_{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i({\epsilon }_i+J_{XR})$. Then Equation (A8) can be rewritten as

$${\mathrm{\Phi }}_n\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)+{\mathrm{\Psi }}_n\mathrm{\Delta }{\theta }^{(𝒱)}+\varrho +o_p\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)=0.$$

Similarly, denote $Ã\equiv \frac{1}{n}{\sum }_{i=1}^n{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i{W}_i^{(𝒞)},\tilde{B}\equiv \frac{1}{n}{\sum }_{i=1}^n{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i{\mathbf{W}}_i^{(𝒱)},$and $\tilde{R}\equiv \frac{1}{n}{\sum }_{i=1}^n{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i({\epsilon }_i+J_{XR}).$ Thus, Equation (A9) becomes $Ã\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)+\tilde{B}\mathrm{\Delta }{\theta }^{(𝒱)}+\tilde{R}+o_p\left(\mathrm{\Delta }{\theta }^{(𝒱)}\right)=0$.Thus, we have $\mathrm{\Delta }{\theta }^{(𝒱)}=-{\tilde{B}}^{-1}Ã\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)-{\tilde{B}}^{-1}\tilde{R}+o_p\left(\mathrm{\Delta }{\theta }^{(𝒱)}\right)$. Substitute $\mathrm{\Delta }{\theta }^{(𝒱)}$ into Equation (A8), then we can get

$$\begin{array}{ll}\hfill & {\mathrm{\Phi }}_n\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)-{\mathrm{\Psi }}_n{\tilde{B}}^{-1}Ã\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)-{\mathrm{\Psi }}_n{\tilde{B}}^{-1}\tilde{R}+\varrho +o_p\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)=0.\end{array}$$ (A10)

The foregoing formula is equivalent to

$$\begin{array}{cc}\hfill & \frac{1}{n}\sum _{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i{W}_i^{(𝒞)}\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)\hfill \\ \hfill & -\left[\frac{1}{n}\sum _{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i{\mathbf{W}}_i^{(𝒱)}\right]{\tilde{B}}^{-1}\left[\frac{1}{n}\sum _{i=1}^n{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i{W}_i^{(𝒞)}\right]\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)\hfill \\ \hfill & -\left[\frac{1}{n}\sum _{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i{\mathbf{W}}_i^{(𝒱)}\right]{\tilde{B}}^{-1}\left[\frac{1}{n}\sum _{i=1}^n{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i\left({\epsilon }_i+J_{XR}\right)\right]\hfill \\ \hfill & +\frac{1}{n}\sum _{i=1}^n{\left(X_i^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i\left({\epsilon }_i+J_{XR}\right)+o_p\left({\beta }^{(𝒞)}-{\widehat{\beta }}^{(𝒞)}\right)=0.\hfill \end{array}$$

According Equation (A10), we have

$$\frac{1}{n}\sum _{i=1}^nÃ{\tilde{B}}^{-1}{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i\left\{W_i^{(𝒞)}-{\mathbf{W}}_i^{(𝒱)}{\tilde{B}}^{-1}Ã\right\}=0,$$

$$\frac{1}{n}\sum _{i=1}^nÃ{\tilde{B}}^{-1}{\left({\tilde{X}}_i^{(𝒱)}\right)}^{\mathrm{T}}{\tau }_i\left[{\epsilon }_i+J_{XR}-{\mathbf{W}}_i^{(𝒱)}{\tilde{B}}^{-1}\tilde{R}\right]=0.$$

and

$$\begin{array}{cc}\hfill & \left\{\frac{1}{n}\sum _{i=1}^n{{\breve{\mathrm{W}}}_i^{(𝒞)}{\tau }_i\left({\breve{\mathrm{W}}}_i^{(𝒞)}\right)}^{\mathrm{T}}+o_p(1)\right\}\sqrt{n}\left({\widehat{\beta }}_1^{(𝒞)}-{\beta }_{01}^{(𝒞)}\right)\hfill \\ \hfill & =\frac{1}{\sqrt{n}}\sum _{i=1}^n{\breve{\mathrm{W}}}_i^{(𝒞)}{\tau }_i{\epsilon }_i-\frac{1}{\sqrt{n}}\sum _{i=1}^n{\breve{\mathrm{W}}}_i^{(𝒞)}{\tau }_i{\tilde{X}}_i^{(𝒱)}\left[{\tilde{B}}^{-1}+o_p(1)\right]\tilde{R}+\frac{1}{\sqrt{n}}\sum _{i=1}^n{\breve{\mathrm{W}}}_i^{(𝒞)}{\tau }_i{J}_{XR}\hfill \\ \hfill & =G_1+G_2+G_3,\hfill \end{array}$$

where ${\breve{W}}_i^{(C)}={\left(W_i^{(C)}\right)}^{\mathrm{T}}-Ã{\tilde{B}}^{-1}{\left({\mathbf{W}}_i^{(𝒱)}\right)}^{\mathrm{T}}$.

It is clear that $\frac{1}{\sqrt{n}}{\sum }_{i=1}^n{\breve{W}}_i^{(C)}{\tau }_i{\tilde{X}}_i^{(𝒱)}=0$ implies $G_2=0$. Using the law of large numbers, we can obtain $\frac{1}{n}{\sum }_{i=1}^n{\breve{W}}_i^{(C)}{\tau }_i{\tau }_i{({\breve{W}}_i^{(C)})}^{\mathrm{T}}\stackrel{P}{\to }A$. According to the central limit theorem, we have

$$G_1=\frac{1}{\sqrt{n}}\sum _{i=1}^n{\breve{\mathrm{W}}}_i^{(𝒞)}{\tau }_i{\epsilon }_i\stackrel{ℒ}{\to }N(0,B),$$

$$\begin{array}{cc}\hfill A& =\mathrm{E}\left\{{\left(W^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i\right.-\mathrm{E}\left({\left({\tilde{X}}^{(𝒱)}\right)}^{\mathrm{T}}\text{diag}(\tau )W^{(𝒞)}|t\right).\\ & {\left.{\left[\mathrm{E}\left({\left({\tilde{X}}^{(𝒱)}\right)}^{\mathrm{T}}\text{diag}(\tau ){\mathbf{\text{W}}}^{(𝒱)}|t\right)\right]}^{-1}\mathrm{E}\left({\left({\mathbf{\text{W}}}^{(𝒱)}\right)}^{\mathrm{T}}\tau |t\right)\right\}}^{\otimes 2},\end{array}$$

$$\begin{array}{ll}\hfill B& =\mathrm{E}\left\{{\left(W^{(𝒞)}\right)}^{\mathrm{T}}{\tau }_i-\mathrm{E}\left({\left({\tilde{X}}^{(𝒱)}\right)}^{\mathrm{T}}\text{diag}(\tau ){\mathbf{\text{W}}}^{(𝒱)}|t\right)\right.\\ \hfill & {\left.{\left[\mathrm{E}\left({\left({\tilde{X}}^{(𝒱)}\right)}^{\mathrm{T}}\text{diag}(\tau ){\mathbf{\text{W}}}^{(𝒱)}|t\right)\right]}^{-1}\mathrm{E}\left({\left({\mathbf{\text{W}}}^{(𝒱)}\right)}^{\mathrm{T}}\tau |t\right)\epsilon \right\}}^{\otimes 2},\end{array}$$

where the superscript symbol “” is defined as a matrix operator for a matrix $\mathrm{M}$ such as ${\mathrm{M}}^{\otimes 2}=\mathrm{M}{\mathrm{M}}^{\mathrm{T}}$.

$$\begin{array}{ll}{W}^{(𝒞)}& ={\left({\left(W_1^{(𝒞)}\right)}^{\mathrm{T}},{\left(W_2^{(𝒞)}\right)}^{\mathrm{T}},\dots ,{\left(W_n^{(𝒞)}\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},\\ {\mathbf{\text{W}}}^{(𝒱)}& ={\left({\left({\mathbf{\text{W}}}_1^{(𝒱)}\right)}^{\mathrm{T}},{\left({\mathbf{\text{W}}}_2^{(𝒱)}\right)}^{\mathrm{T}},\dots ,{\left({\mathbf{\text{W}}}_{\mathrm{n}}^{(𝒱)}\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},\\ {\tilde{X}}^{(𝒱)}& ={\left({\left({\tilde{X}}_1^{(𝒱)}\right)}^{\mathrm{T}},{\left({\tilde{X}}_2^{(𝒱)}\right)}^{\mathrm{T}},\dots ,{\left({\tilde{X}}_n^{(𝒱)}\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},\tau ={({\tau }_1,{\tau }_n,\dots ,{\tau }_n)}^{\mathrm{T}}.\end{array}$$

Following Tian, Xue and Liu [28], we know that $G_3=o_p(1)$. According to the Slutsky Theorem, Theorem 3 is proved.

Some Additional Numerical Results

TABLE A1.

Structure identification and variable selection with the EX correlation structure ($n_i=10$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.7	0.2	300	LASSO	2.7200	0.0000	3.0000	0.0000	2.0000	0.2800	0.7400
			MCP	2.6800	0.0000	3.0000	0.0000	2.0000	0.3200	0.7200
			SCAD	2.7150	0.0000	3.0000	0.0000	2.0000	0.2850	0.7500
			nSCAD	2.6750	0.0000	3.0000	0.0000	2.0000	0.3250	0.7450
		350	LASSO	2.8200	0.0000	3.0000	0.0000	2.0000	0.1800	0.8650
			MCP	2.8000	0.0000	3.0000	0.0000	2.0000	0.2000	0.8450
			SCAD	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8900
			nSCAD	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8850
		400	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.9000
			MCP	2.8350	0.0000	3.0000	0.0000	2.0000	0.1650	0.8750
			SCAD	2.9200	0.0000	3.0000	0.0000	2.0000	0.0800	0.9300
			nSCAD	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9100
	0.4	300	LASSO	2.6500	0.0000	3.0000	0.0000	2.0000	0.3500	0.6800
			MCP	2.6450	0.0000	3.0000	0.0000	2.0000	0.3550	0.6800
			SCAD	2.7200	0.0000	3.0000	0.0000	2.0000	0.2800	0.7350
			nSCAD	2.5700	0.0000	3.0000	0.0000	2.0000	0.4300	0.6200
		350	LASSO	2.6900	0.0000	3.0000	0.0000	2.0000	0.3100	0.7200
			MCP	2.6850	0.0000	3.0000	0.0000	2.0000	0.3150	0.7250
			SCAD	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8800
			nSCAD	2.6000	0.0000	3.0000	0.0000	2.0000	0.4000	0.6550
		400	LASSO	2.8100	0.0000	3.0000	0.0000	2.0000	0.1900	0.8500
			MCP	2.7800	0.0000	3.0000	0.0000	2.0000	0.2200	0.8250
			SCAD	2.8700	0.0000	3.0000	0.0000	2.0000	0.1300	0.9000
			nSCAD	2.7450	0.0000	3.0000	0.0000	2.0000	0.2550	0.8000
	0.6	300	LASSO	2.3500	0.0000	3.0000	0.0000	2.0000	0.6500	0.5100
			MCP	2.2350	0.0000	3.0000	0.0000	2.0000	0.7650	0.4800
			SCAD	2.6600	0.0000	3.0000	0.0000	2.0000	0.3400	0.7300
			nSCAD	2.1750	0.0000	3.0000	0.0000	2.0000	0.8250	0.4150
		350	LASSO	2.4450	0.0000	3.0000	0.0000	2.0000	0.5550	0.5450
			MCP	2.4450	0.0000	3.0000	0.0000	2.0000	0.5550	0.5500
			SCAD	2.7800	0.0000	3.0000	0.0000	2.0000	0.2200	0.8300
			nSCAD	2.3150	0.0000	3.0000	0.0000	2.0000	0.6850	0.4600
		400	LASSO	2.7100	0.0000	3.0000	0.0000	2.0000	0.2900	0.7400
			MCP	2.6500	0.0000	3.0000	0.0000	2.0000	0.3500	0.6950
			SCAD	2.8500	0.0000	3.0000	0.0000	2.0000	0.1500	0.8550
			nSCAD	2.5000	0.0000	3.0000	0.0000	2.0000	0.5000	0.5800

TABLE A2.

Structure identification and variable selection with the EX correlation structure ($n_i=20$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.7	0.2	300	LASSO	2.8550	0.0000	3.0000	0.0000	2.0000	0.1450	0.8700
			MCP	2.8200	0.0000	3.0000	0.0000	2.0000	0.1800	0.8500
			SCAD	2.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9150
			nSCAD	2.7700	0.0000	3.0000	0.0000	2.0000	0.2300	0.8200
		350	LASSO	2.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.9000
			MCP	2.8500	0.0000	3.0000	0.0000	2.0000	0.1500	0.8850
			SCAD	2.9550	0.0000	3.0000	0.0000	2.0000	0.0450	0.9650
			nSCAD	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.9000
		400	LASSO	2.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9050
			MCP	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9000
			SCAD	2.9850	0.0000	3.0000	0.0000	2.0000	0.0150	0.9850
			nSCAD	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9250
	0.4	300	LASSO	2.7800	0.0000	3.0000	0.0000	2.0000	0.2200	0.8200
			MCP	2.7300	0.0000	3.0000	0.0000	2.0000	0.2700	0.7900
			SCAD	2.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.8850
			nSCAD	2.7200	0.0000	3.0000	0.0000	2.0000	0.2800	0.7450
		350	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8650
			MCP	2.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8450
			SCAD	2.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9150
			nSCAD	2.7200	0.0000	3.0000	0.0000	2.0000	0.2800	0.7650
		400	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8950
			MCP	2.8350	0.0000	3.0000	0.0000	2.0000	0.1650	0.8750
			SCAD	2.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9250
			nSCAD	2.8000	0.0000	3.0000	0.0000	2.0000	0.2000	0.8200
	0.6	300	LASSO	2.6950	0.0000	3.0000	0.0000	2.0000	0.3050	0.7350
			MCP	2.6750	0.0000	3.0000	0.0000	2.0000	0.3250	0.7300
			SCAD	2.8200	0.0000	3.0000	0.0000	2.0000	0.1800	0.8550
			nSCAD	2.5300	0.0000	3.0000	0.0000	2.0000	0.4700	0.6150
		350	LASSO	2.8100	0.0000	3.0000	0.0000	2.0000	0.1900	0.8400
			MCP	2.7650	0.0000	3.0000	0.0000	2.0000	0.2350	0.8150
			SCAD	2.8550	0.0000	3.0000	0.0000	2.0000	0.1450	0.8800
			nSCAD	2.5650	0.0000	3.0000	0.0000	2.0000	0.4350	0.6300
		400	LASSO	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.8900
			MCP	2.8550	0.0000	3.0000	0.0000	2.0000	0.1450	0.8750
			SCAD	2.9050	0.0000	3.0000	0.0000	2.0000	0.0950	0.9150
			nSCAD	2.6500	0.0000	3.0000	0.0000	2.0000	0.3500	0.7050

TABLE A3.

Structure identification and variable selection with the AR(1) correlation structure ($n_i=10$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.7	0.2	300	LASSO	2.6450	0.0000	3.0000	0.0000	2.0000	0.3550	0.6900
			MCP	2.6100	0.0000	3.0000	0.0000	2.0000	0.3900	0.6700
			SCAD	2.6400	0.0000	3.0000	0.0000	2.0000	0.3600	0.6800
			nSCAD	2.6050	0.0000	3.0000	0.0000	2.0000	0.3950	0.6650
		350	LASSO	2.6700	0.0000	3.0000	0.0000	2.0000	0.3300	0.7150
			MCP	2.7150	0.0000	3.0000	0.0000	2.0000	0.2850	0.7450
			SCAD	2.7300	0.0000	3.0000	0.0000	2.0000	0.2700	0.7500
			nSCAD	2.6950	0.0000	3.0000	0.0000	2.0000	0.3050	0.7150
		400	LASSO	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8750
			MCP	2.8300	0.0000	3.0000	0.0000	2.0000	0.1700	0.8600
			SCAD	2.8700	0.0000	3.0000	0.0000	2.0000	0.1300	0.8850
			nSCAD	2.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8550
	0.4	300	LASSO	2.5950	0.0000	3.0000	0.0000	2.0000	0.4050	0.6500
			MCP	2.6000	0.0000	3.0000	0.0000	2.0000	0.4000	0.6350
			SCAD	2.5950	0.0000	3.0000	0.0000	2.0000	0.4050	0.6200
			nSCAD	2.5300	0.0000	3.0000	0.0000	2.0000	0.4700	0.6100
		350	LASSO	2.6000	0.0000	3.0000	0.0000	2.0000	0.4000	0.6500
			MCP	2.5950	0.0000	3.0000	0.0000	2.0000	0.4050	0.6400
			SCAD	2.6100	0.0000	3.0000	0.0000	2.0000	0.3900	0.6600
			nSCAD	2.5100	0.0000	3.0000	0.0000	2.0000	0.4900	0.6100
		400	LASSO	2.6800	0.0000	3.0000	0.0000	2.0000	0.3200	0.7250
			MCP	2.7050	0.0000	3.0000	0.0000	2.0000	0.2950	0.7350
			SCAD	2.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.8800
			nSCAD	2.5750	0.0000	3.0000	0.0000	2.0000	0.4250	0.6450
	0.6	300	LASSO	2.5450	0.0000	3.0000	0.0000	2.0000	0.4550	0.6100
			MCP	2.4450	0.0000	3.0000	0.0000	2.0000	0.5550	0.5400
			SCAD	2.5150	0.0000	3.0000	0.0000	2.0000	0.4850	0.5850
			nSCAD	2.3050	0.0000	3.0000	0.0000	2.0000	0.6950	0.4450
		350	LASSO	2.6600	0.0000	3.0000	0.0000	2.0000	0.3400	0.7100
			MCP	2.5700	0.0000	3.0000	0.0000	2.0000	0.4300	0.6300
			SCAD	2.6000	0.0000	3.0000	0.0000	2.0000	0.4000	0.6500
			nSCAD	2.4950	0.0000	3.0000	0.0000	2.0000	0.5050	0.5800
		400	LASSO	2.6450	0.0000	3.0000	0.0000	2.0000	0.3550	0.6750
			MCP	2.6450	0.0000	3.0000	0.0000	2.0000	0.3550	0.6800
			SCAD	2.6550	0.0000	3.0000	0.0000	2.0000	0.3450	0.6750
			nSCAD	2.5500	0.0000	3.0000	0.0000	2.0000	0.4500	0.6150

TABLE A4.

Structure identification and variable selection with the AR(1) correlation structure ($n_i=20$).

				Structure identification and variable selection
$\rho$	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
0.7	0.2	300	LASSO	2.8200	0.0000	3.0000	0.0000	2.0000	0.1800	0.8400
			MCP	2.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8400
			SCAD	2.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8500
			nSCAD	2.7750	0.0000	3.0000	0.0000	2.0000	0.2250	0.8000
		350	LASSO	2.8300	0.0000	3.0000	0.0000	2.0000	0.1700	0.8750
			MCP	2.8550	0.0000	3.0000	0.0000	2.0000	0.1450	0.8700
			SCAD	2.8750	0.0000	3.0000	0.0000	2.0000	0.1250	0.8900
			nSCAD	2.8150	0.0000	3.0000	0.0000	2.0000	0.1850	0.8550
		400	LASSO	2.9300	0.0000	3.0000	0.0000	2.0000	0.0700	0.9350
			MCP	2.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9200
			SCAD	2.9600	0.0000	3.0000	0.0000	2.0000	0.0400	0.9600
			nSCAD	2.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8700
	0.4	300	LASSO	2.7350	0.0000	3.0000	0.0000	2.0000	0.2650	0.7600
			MCP	2.7350	0.0000	3.0000	0.0000	2.0000	0.2650	0.7550
			SCAD	2.8000	0.0000	3.0000	0.0000	2.0000	0.2000	0.8100
			nSCAD	2.6200	0.0000	3.0000	0.0000	2.0000	0.3800	0.6650
		350	LASSO	2.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.8650
			MCP	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8300
			SCAD	2.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.8750
			nSCAD	2.7100	0.0000	3.0000	0.0000	2.0000	0.2900	0.7450
		400	LASSO	2.8550	0.0000	3.0000	0.0000	2.0000	0.1450	0.8650
			MCP	2.8450	0.0000	3.0000	0.0000	2.0000	0.1550	0.8550
			SCAD	2.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9000
			nSCAD	2.7200	0.0000	3.0000	0.0000	2.0000	0.2800	0.7650
	0.6	300	LASSO	2.7000	0.0000	3.0000	0.0000	2.0000	0.3000	0.7350
			MCP	2.7050	0.0000	3.0000	0.0000	2.0000	0.2950	0.7400
			SCAD	2.7750	0.0000	3.0000	0.0000	2.0000	0.2250	0.8050
			nSCAD	2.5650	0.0000	3.0000	0.0000	2.0000	0.4350	0.6250
		350	LASSO	2.7500	0.0000	3.0000	0.0000	2.0000	0.2500	0.7800
			MCP	2.7250	0.0000	3.0000	0.0000	2.0000	0.2750	0.7700
			SCAD	2.8300	0.0000	3.0000	0.0000	2.0000	0.1700	0.8450
			nSCAD	2.5700	0.0000	3.0000	0.0000	2.0000	0.4300	0.6400
		400	LASSO	2.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8650
			MCP	2.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8450
			SCAD	2.8700	0.0000	3.0000	0.0000	2.0000	0.1300	0.8900
			nSCAD	2.6350	0.0000	3.0000	0.0000	2.0000	0.3650	0.6850

TABLE A5.

Model estimation results ($q=20,n_i=20,\rho =0.7$).

				$n=400$		$n=500$		$n=600$
Corstr	$\rho$	${\sigma }_u$	Method	GMSE	RASE	GMSE	RASE	GMSE	RASE
AR(1)	0.7	0.4	LASSO	0.016985	0.062900	0.016123	0.056167	0.015715	0.052109
			MCP	0.008156	0.061890	0.005986	0.053690	0.003870	0.049172
			SCAD	0.008249	0.061449	0.006156	0.053667	0.004017	0.049403
			nSCAD	0.144295	0.078095	0.144290	0.071223	0.142096	0.067723
		0.6	LASSO	0.066033	0.099954	0.065660	0.087637	0.063251	0.080042
			MCP	0.017984	0.091795	0.011278	0.079057	0.008998	0.070233
			SCAD	0.019279	0.093662	0.012399	0.080058	0.009650	0.071932
			nSCAD	0.692059	0.141239	0.663470	0.133779	0.657981	0.125970
EX	0.7	0.4	LASSO	0.019243	0.063297	0.016841	0.055279	0.016527	0.050517
			MCP	0.008648	0.061288	0.005200	0.052335	0.004022	0.048058
			SCAD	0.008824	0.061437	0.005602	0.052955	0.004182	0.048277
			nSCAD	0.175114	0.072692	0.166807	0.066381	0.158860	0.061465
		0.6	LASSO	0.072644	0.098746	0.071361	0.086338	0.067447	0.079995
			MCP	0.017900	0.092662	0.011884	0.077784	0.008770	0.070767
			SCAD	0.018550	0.093136	0.011706	0.078541	0.008805	0.071786
			nSCAD	0.760916	0.131829	0.735102	0.126376	0.715669	0.120093

TABLE A6.

Structure identification and variable selection results ($q=20,n_i=20,\rho =0.7$).

				Structure identification and variable selection
Corstr	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
EX	0.4	400	LASSO	14.8950	0.0000	3.0000	0.0000	2.0000	0.1050	0.9200
			MCP	14.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.8900
			SCAD	14.9450	0.0000	3.0000	0.0000	2.0000	0.0550	0.9450
			nSCAD	14.5550	0.0000	3.0000	0.0000	2.0000	0.4450	0.7150
		500	LASSO	14.9050	0.0000	3.0000	0.0000	2.0000	0.0950	0.9150
			MCP	14.8850	0.0000	3.0000	0.0000	2.0000	0.1150	0.9050
			SCAD	14.9150	0.0000	3.0000	0.0000	2.0000	0.0850	0.9700
			nSCAD	14.5550	0.0000	3.0000	0.0000	2.0000	0.4450	0.7600
		600	LASSO	14.8800	0.0000	3.0000	0.0000	2.0000	0.1200	0.9550
			MCP	14.9300	0.0000	3.0000	0.0000	2.0000	0.0700	0.9500
			SCAD	14.9300	0.0000	3.0000	0.0000	2.0000	0.0700	0.9900
			nSCAD	14.6550	0.0000	3.0000	0.0000	2.0000	0.3450	0.7850
	0.6	400	LASSO	13.2200	0.0000	3.0000	0.0000	2.0000	1.7800	0.5750
			MCP	12.4450	0.0000	3.0000	0.0000	2.0000	2.5550	0.4600
			SCAD	14.5550	0.0000	3.0000	0.0000	2.0000	0.4450	0.8550
			nSCAD	13.1600	0.0000	3.0000	0.0000	2.0000	1.8400	0.6700
		500	LASSO	14.4250	0.0000	3.0000	0.0000	2.0000	0.5750	0.6850
			MCP	14.0250	0.0000	3.0000	0.0000	2.0000	0.9750	0.4950
			SCAD	14.8000	0.0000	3.0000	0.0000	2.0000	0.2000	0.9150
			nSCAD	13.7800	0.0000	3.0000	0.0000	2.0000	1.2200	0.7300
		600	LASSO	14.8250	0.0000	3.0000	0.0000	2.0000	0.1750	0.8500
			MCP	14.7400	0.0000	3.0000	0.0000	2.0000	0.2600	0.8200
			SCAD	14.9050	0.0000	3.0000	0.0000	2.0000	0.0950	0.9300
			nSCAD	14.1350	0.0000	3.0000	0.0000	2.0000	0.8650	0.7600
AR(1)	0.4	400	LASSO	14.7650	0.0000	3.0000	0.0000	2.0000	0.2350	0.8100
			MCP	14.7500	0.0000	3.0000	0.0000	2.0000	0.2500	0.7800
			SCAD	14.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9350
			nSCAD	14.2950	0.0000	3.0000	0.0000	2.0000	0.7050	0.6500
		500	LASSO	14.9650	0.0000	3.0000	0.0000	2.0000	0.0350	0.9700
			MCP	14.9550	0.0000	3.0000	0.0000	2.0000	0.0450	0.9800
			SCAD	14.9300	0.0000	3.0000	0.0000	2.0000	0.0700	0.9750
			nSCAD	14.6600	0.0000	3.0000	0.0000	2.0000	0.3400	0.8200
		600	LASSO	14.9900	0.0000	3.0000	0.0000	2.0000	0.0100	0.9900
			MCP	14.9000	0.0000	3.0000	0.0000	2.0000	0.1000	0.9900
			SCAD	14.9950	0.0000	3.0000	0.0000	2.0000	0.0050	0.9950
			nSCAD	14.8200	0.0000	3.0000	0.0000	2.0000	0.1800	0.8600
	0.6	400	LASSO	13.1750	0.0000	3.0000	0.0000	2.0000	1.8250	0.5900
			MCP	12.1650	0.0000	3.0000	0.0000	2.0000	2.8350	0.4500
			SCAD	14.1700	0.0000	3.0000	0.0000	2.0000	0.8300	0.6700
			nSCAD	12.7650	0.0000	3.0000	0.0000	2.0000	2.2350	0.5100
		500	LASSO	14.7050	0.0000	3.0000	0.0000	2.0000	0.2950	0.7850
			MCP	14.5800	0.0000	3.0000	0.0000	2.0000	0.4200	0.6900
			SCAD	14.9200	0.0000	3.0000	0.0000	2.0000	0.0800	0.9500
			nSCAD	13.9900	0.0000	3.0000	0.0000	2.0000	1.0100	0.8000
		600	LASSO	14.9050	0.0000	3.0000	0.0000	2.0000	0.0950	0.9250
			MCP	14.8400	0.0000	3.0000	0.0000	2.0000	0.1600	0.8800
			SCAD	14.9700	0.0000	3.0000	0.0000	2.0000	0.0300	0.9750
			nSCAD	14.2900	0.0000	3.0000	0.0000	2.0000	0.7100	0.8050

TABLE A7.

Model estimation results $\left(q=50,n_i=20\right)$.

				$n=400$		$n=700$		$n=1000$
Corstr	$\rho$	${\sigma }_u$	Method	GMSE	RASE	GMSE	RASE	GMSE	RASE
AR(1)	0.3	0.6	LASSO	0.064341	0.114592	0.031191	0.067650	0.042419	0.059851
			MCP	0.029075	0.106077	0.015399	0.066843	0.010787	0.056794
			SCAD	0.027497	0.108374	0.015930	0.067008	0.011287	0.056623
			nSCAD	0.146192	0.137163	0.128575	0.082160	0.125350	0.074936
		0.7	LASSO	0.112530	0.138968	0.051961	0.079297	0.043889	0.070980
			MCP	0.043292	0.128857	0.022971	0.075008	0.020832	0.069268
			SCAD	0.046243	0.129316	0.025064	0.075690	0.021372	0.069486
			nSCAD	0.253093	0.170923	0.222431	0.103973	0.184898	0.084143
EX	0.7	0.6	LASSO	0.089834	0.131975	0.033852	0.069576	0.039466	0.057969
			MCP	0.042206	0.124017	0.018991	0.067334	0.011058	0.055223
			SCAD	0.037738	0.122862	0.018790	0.067459	0.011709	0.055316
			nSCAD	0.211071	0.150111	0.190835	0.084163	0.155614	0.072097
		0.7	LASSO	0.147694	0.157707	0.059348	0.076340	0.049364	0.071095
			MCP	0.048806	0.145195	0.021851	0.072973	0.020772	0.068934
			SCAD	0.048354	0.147517	0.023644	0.073777	0.021018	0.068360
			nSCAD	0.316021	0.190023	0.352679	0.097988	0.191410	0.083184

TABLE A8.

Structure identification and variable selection results $\left(q=50,n_i=20\right)$.

				Structure identification and variable selection
Corstr	${\sigma }_u$	$n$	Method	CZ	IZ	CV	IV	CC	IC	CF
EX	0.6	400	LASSO	43.7350	0.0000	3.0000	0.0000	2.0000	1.2650	0.6300
			MCP	44.3550	0.0000	3.0000	0.0000	2.0000	0.6450	0.6600
			SCAD	44.5850	0.0000	3.0000	0.0000	2.0000	0.4150	0.7850
			nSCAD	43.5950	0.0000	3.0000	0.0000	2.0000	1.4050	0.4250
		700	LASSO	44.5250	0.0000	3.0000	0.0000	2.0000	0.4750	0.7950
			MCP	44.6300	0.0000	3.0000	0.0000	2.0000	0.3700	0.8100
			SCAD	44.6950	0.0000	3.0000	0.0000	2.0000	0.3050	0.8550
			nSCAD	43.4700	0.0000	3.0000	0.0000	2.0000	1.5300	0.4700
		1000	LASSO	44.8850	0.0000	3.0000	0.0000	2.0000	0.1150	0.9800
			MCP	44.7750	0.0000	3.0000	0.0000	2.0000	0.2250	0.9750
			SCAD	44.7150	0.0000	3.0000	0.0000	2.0000	0.2850	0.9650
			nSCAD	44.3200	0.0000	3.0000	0.0000	2.0000	0.6800	0.7150
	0.7	400	LASSO	43.9000	0.0000	3.0000	0.0000	2.0000	1.1000	0.5000
			MCP	44.2400	0.0000	3.0000	0.0000	2.0000	0.7600	0.5800
			SCAD	44.2700	0.0000	3.0000	0.0000	2.0000	0.7300	0.7500
			nSCAD	42.5300	0.0000	3.0000	0.0000	2.0000	2.4700	0.2500
		700	LASSO	44.3050	0.0000	3.0000	0.0000	2.0000	0.6950	0.7650
			MCP	44.5450	0.0000	3.0000	0.0000	2.0000	0.4550	0.7500
			SCAD	44.7350	0.0000	3.0000	0.0000	2.0000	0.2650	0.8200
			nSCAD	43.0150	0.0000	3.0000	0.0000	2.0000	1.9850	0.4100
		1000	LASSO	44.8650	0.0000	3.0000	0.0000	2.0000	0.1350	0.9050
			MCP	44.8200	0.0000	3.0000	0.0000	2.0000	0.1800	0.8800
			SCAD	44.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8650
			nSCAD	43.8600	0.0000	3.0000	0.0000	2.0000	1.1400	0.6250
AR(1)	0.6	400	LASSO	44.3300	0.0000	3.0000	0.0000	2.0000	0.6700	0.6700
			MCP	44.4300	0.0000	3.0000	0.0000	2.0000	0.5700	0.6800
			SCAD	44.6150	0.0000	3.0000	0.0000	2.0000	0.3850	0.8150
			nSCAD	43.1700	0.0000	3.0000	0.0000	2.0000	1.8300	0.3450
		700	LASSO	44.7150	0.0000	3.0000	0.0000	2.0000	0.2850	0.9400
			MCP	44.7300	0.0000	3.0000	0.0000	2.0000	0.2700	0.8900
			SCAD	44.7950	0.0000	3.0000	0.0000	2.0000	0.2050	0.9050
			nSCAD	43.7650	0.0000	3.0000	0.0000	2.0000	1.2350	0.5450
		1000	LASSO	44.9100	0.0000	3.0000	0.0000	2.0000	0.0900	0.9900
			MCP	44.8600	0.0000	3.0000	0.0000	2.0000	0.1400	0.9800
			SCAD	44.7650	0.0000	3.0000	0.0000	2.0000	0.2350	0.9900
			nSCAD	44.6150	0.0000	3.0000	0.0000	2.0000	0.3850	0.9050
	0.7	400	LASSO	44.2400	0.0000	3.0000	0.0000	2.0000	0.7600	0.6000
			MCP	43.9600	0.0000	3.0000	0.0000	2.0000	1.0400	0.5700
			SCAD	44.5100	0.0000	3.0000	0.0000	2.0000	0.4900	0.7650
			nSCAD	42.5300	0.0000	3.0000	0.0000	2.0000	2.4700	0.3000
		700	LASSO	44.7250	0.0000	3.0000	0.0000	2.0000	0.2750	0.8300
			MCP	44.4550	0.0000	3.0000	0.0000	2.0000	0.5450	0.8000
			SCAD	44.7500	0.0000	3.0000	0.0000	2.0000	0.2500	0.8250
			nSCAD	43.2500	0.0000	3.0000	0.0000	2.0000	1.7500	0.3850
		1000	LASSO	44.8050	0.0000	3.0000	0.0000	2.0000	0.1950	0.8900
			MCP	44.6400	0.0000	3.0000	0.0000	2.0000	0.3600	0.8850
			SCAD	44.2850	0.0000	3.0000	0.0000	2.0000	0.7150	0.8800
			nSCAD	43.8650	0.0000	3.0000	0.0000	2.0000	1.1350	0.6950

Higher Dimensional Simulation Results

When considering the higher dimensional case, under reasonable conditions, we have relaxed the criteria in our code for determining whether the varying coefficient is zero. According to Equation (21), nonzero values of $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}$ are identified and selected. In our simulation with eight varying coefficients, we adopt the threshold $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}>10^{-7}$ to declare the coefficient function ${\alpha }_k(·)$ as varying, whereas $\Vert {\gamma }_k{\Vert }_{\mathrm{H}}<10^{-7}$ indicates it is constant. In the higher dimensional setting, that is, $q=20,50$, we replace the threshold $10^{-7}$ by $10^{-3}$. We reached the same conclusion in higher dimensional setting. The detailed results are as follows.

• 1.
  Suppose that the real model (2) satisfies $𝒞=\{1,2\},𝒱=\{3,4,5\},𝒵=\{6,7,\dots ,20\}$ and

  $$\begin{array}{ll}\hfill {\alpha }^{(𝒞)}(t)& =({\alpha }_1(t),{\alpha }_2(t))=(5,6),\\ \hfill {\alpha }^{(𝒱)}(t)& =({\alpha }_3(t),{\alpha }_4(t),{\alpha }_5(t))\\ \hfill & =(0.7·e^{2t-1},1.5·sin(\pi t),0.2·{(2-2t)}^3),\\ \hfill {\alpha }^{(𝒵)}(t)& =({\alpha }_6(t),{\alpha }_7(t),\dots ,{\alpha }_{20}(t))=(0,0,\dots ,0).\end{array}$$

  We took $X_{ij}\sim N(3,{\sigma }_X^2{I}_{20}),u_{ij}\sim N(0,{\sigma }_u^2{I}_{20})$, where $j=$ $1,2,\dots ,20,{\sigma }_X=3,I_{20}$ is $20\times 20$ identify matrix. We set ${\sigma }_u$ as $0.4,0.6$. $t_{ij}\sim U[0,1]$. ${\epsilon }_i={\left({\epsilon }_{i1},{\epsilon }_{i2},\dots ,{\epsilon }_{i{n}_i}\right)}^{\mathrm{T}}\sim N\left(0,{\sigma }^2\text{Corr}\left({\epsilon }_{\mathrm{i}},\rho \right)\right)$, where ${\sigma }^2=1$ and $\left.\text{Corr}\left({\epsilon }_{\mathrm{i}},\rho \right)\right)$ is a known correlation matrix with parameter $\rho$. Thus, we can get $A_i=\text{diag}(1,1,\dots ,1)$. In our work, we set $n=400,500,600$, $n_i=20$ and ${\epsilon }_i$ has the first‐order autoregressive $(\text{AR}(1))$ and exchangeable (EX) correlation structures with $\rho =0.7$. The cubic B‐spline basis was applied with the knots being equally spaced in $[0,1],K=⌊c_0\times N^{1/5}⌋$, where $\lfloor c_0\rfloor$ denotes the largest integer less than $c_0$ [28]. Please see Tables A5 and A6.

• 2.
  Suppose that the real model (2) satisfies $𝒞=\{1,2\},𝒱=\{3,4,5\},𝒵=\{6,7,\dots ,50\}$ and

  $$\begin{array}{ll}\hfill {\alpha }^{(𝒞)}(t)& =({\alpha }_1(t),{\alpha }_2(t))=(5,6),\\ \hfill {\alpha }^{(𝒱)}(t)& =({\alpha }_3(t),{\alpha }_4(t),{\alpha }_5(t))=(e^{2t},6·sin(\pi t),{(2-2t)}^3),\\ \hfill {\alpha }^{(𝒵)}(t)& =({\alpha }_6(t),{\alpha }_7(t),\dots ,{\alpha }_{50}(t))=(0,0,\dots ,0).\end{array}$$

  We took $X_{ij}\sim N(5,{\sigma }_X^2{I}_{50}),u_{ij}\sim N(0,{\sigma }_u^2{I}_{50})$, where $j=$ $1,2,\dots ,50,{\sigma }_X=5,I_{50}$ is $50\times 50$ identify matrix. We set ${\sigma }_u$ as $0.6,0.7$. $t_{ij}\sim U[0,1]$. ${\epsilon }_i={\left({\epsilon }_{i1},{\epsilon }_{i2},\dots ,{\epsilon }_{i{n}_i}\right)}^{\mathrm{T}}\sim N\left(0,{\sigma }^2\text{Corr}\left({\epsilon }_{\mathrm{i}},\rho \right)\right)$, where ${\sigma }^2=1$ and $\left.\text{Corr}\left({\epsilon }_{\mathrm{i}},\rho \right)\right)$ is a known correlation matrix with parameter $\rho$. Thus, we can get $A_i=\text{diag}(1,1,\dots ,1)$. In our work, we set $n=400,700,1000$, $n_i=20$ and ${\epsilon }_i$ has the first‐order autoregressive $(\text{AR}(1))$ with $\rho =0.3$ and exchangeable (EX) correlation structures with $\rho =0.7$. The cubic B‐spline basis was applied with the knots being equally spaced in $[0,1],K=⌊c_0\times N^{1/5}⌋$, where $\lfloor c_0\rfloor$ denotes the largest integer less than $c_0$ [28]. Please see Tables A7 and A8.

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.