Analysis of Longitudinal Data with Semiparametric Estimation of Covariance Function

Abstract

Improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in analysis of longitudinal data. Both involve estimation of the covariance function. Yet, challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. A class of semiparametric models for the covariance function is proposed by imposing a parametric correlation structure while allowing a nonparametric variance function. A kernel estimator is developed for the estimation of the nonparametric variance function. Two methods, a quasi-likelihood approach and a minimum generalized variance method, are proposed for estimating parameters in the correlation structure. We introduce a semiparametric varying coefficient partially linear model for longitudinal data and propose an estimation procedure for model coefficients by using a profile weighted least squares approach. Sampling properties of the proposed estimation procedures are studied and asymptotic normality of the resulting estimators is established. Finite sample performance of the proposed procedures is assessed by Monte Carlo simulation studies. The proposed methodology is illustrated by an analysis of a real data example.

1 Introduction

Estimation of covariance functions is an important issue in the analysis of longitudinal data. It features prominently in forecasting the trajectory of an individual response over time and is closely related with improving the efficiency of estimated regression coefficients. Challenges arise in estimating the covariance function due to the fact that longitudinal data are frequently collected at irregular and possibly subject-specific time points. Interest in this kind of challenges has surged in the recent literature. Wu and Pourahmadi (2003) proposed nonparametric estimation of large covariance matrices using two-step estimation procedure (Fan and Zhang, 2000), but their method can deal with only balanced or nearly balanced longitudinal data. Recently, Huang, et al. (2006) introduced a penalized likelihood method for estimating covariance matrix when the design is balanced and Yao, Müller and Wang (2005a, b) approached the problem from the point of view of functional data analysis.

In this paper, we consider a semiparametric varying-coefficient partially linear model:

$$y(t)=\mathbf{x}{(t)}^T\mathit{\alpha }(t)+\mathbf{z}{(t)}^T\mathit{\beta }+\epsilon (t),$$ (1.1)

where α(t) consists of p unknown smooth functions, β is a q-dimensional unknown parameter vector, and E{ε(t)|x(t), z(t)} = 0. Nonparametric models for longitudinal data (Lin and Carroll, 2000; Wang, 2003) can be viewed as a special case of model (1.1). Moreover, model (1.1) is a useful extension of the partially linear model, systematically studied by Härdle, Liang and Gao (2000), and of the time-varying coefficient model (Hastie and Tibshirani, 1993). It has been considered by Zhang, Lee and Song (2002), Xia, Zhang and Tong (2004) and Fan and Huang (2005) in the case of iid observations, and by Martinussen and Scheike (1999) and Sun and Wu (2005) for longitudinal data. It is a natural extension of the models studied by Lin and Carroll (2001) (with identity link), He, Zhu and Fung (2002), He, Fung and Zhu (2005), Wang, Carroll and Lin (2005), and Huang and Zhang (2004).

We focus on parsimonious modeling of the covariance function of the random error process ε (t) for the analysis of longitudinal data, when observations are collected at irregular and possibly subject-specific time points. We approach this by assuming that var{ε (t)|x(t), z(t)} = σ²(t), which is a nonparametric smoothing function, but the correlation function between ε (s) and ε (t) has a parametric form corr{ε (s), ε (t)} = ρ (t, s, θ), where ρ (s, t, θ) is a positive definite function of s and t, and θ is an unknown parameter vector.

The covariance function is fitted by a semiparametric model, which allows the random error process ε (t) to be nonstationary as its variance function σ²(t) may be time-dependent. Compared with a fully nonparametric fit defined in (6.1) to the correlation function, our semiparametric model guarantees positive definiteness for the resulting estimate; it retains the flexibility of nonparametric modeling and parsimony and ease of interpretation of parametric modeling. To improve the efficiency of the regression coefficient, one typically takes the weight matrix in the weighted least squares method to be the inverse of estimated covariance matrix. Thus, the requirement on positive definiteness becomes necessary. Our semiparametric model allows a data analyst to easily incorporate prior information about the correlation structure. It can be used to improve the estimation efficiency of β. For example, let ρ₀(s, t) be a working correlation function (e.g. working independence) and ρ (s, t, θ) be a family of correlation functions, such as AR or ARMA correlation structure, that contains ρ₀, our method allows us to choose an appropriate θ to improve the efficiency of the estimator of β. Obviously, to improve the efficiency, the family of correlation functions {ρ (s, t, θ)} is not necessary to contain the true correlation structure.

We will also introduce an estimation procedure for the variance function, and propose two approaches to estimating the unknown vector θ, motivated from two different principles. We also propose an estimation procedure for the regression function α(t) and coefficient β using the profile least squares. Asymptotic properties of the proposed estimators are investigated, and finite sample performance is assessed via Monte Carlo simulation studies. A real data example is used to illustrate the proposed methodology.

This paper is organized as follows. We propose estimation procedures for variance function and unknown parameters in the correlation matrix in Section 2. An efficient estimation procedure for α(t) and β is proposed based on the profile least squares techniques in Section 3. Sampling properties of the proposed procedures are presented in Section 4. Simulation studies and real data analysis are given in Section 5. All technical proofs are relegated to the Appendix.

2 Estimation of covariance function

Suppose that a random sample from model (1.1) consists of n subjects. For the i-th subject, i = 1,···, n, the response variable y_i(t) and the covariates {x_i(t), z_i(t)} are collected at time points t = t_ij, j = 1, ···, J_i, where J_i is the total number of observations for the i-th subject. Denote

$$r_{ij}\equiv r_{ij}(\mathit{\alpha },\mathit{\beta })=y_i(t_{ij})-{\mathbf{x}}_i{(t_{ij})}^T\mathit{\alpha }(t_{ij})-{\mathbf{z}}_i{(t_{ij})}^T\mathit{\beta },$$

and r_i(α, β) = (r_i1, ···, r_iJi)^T. Here we adopt the notation r_ij(α, β) to emphasize the parameters α and β, although for true values of α and β, r_ij(α, β) = ε_i(t_ij).

To motivate the proposed estimation procedures below, pretend for the moment that ε_i is normally distributed with zero mean and covariance matrix Σ_i. Then, the logarithm of the likelihood function for α, β, σ² and θ is

$$\ell (\mathit{\alpha },\mathit{\beta },{\sigma }^2,\mathit{\theta })=-\frac{1}{2}\sum _{i=1}^{n}log\mid {\mathrm{\sum }}_i\mid -\frac{1}{2}\sum _{i=1}^n{\mathbf{r}}_i{(\mathit{\alpha },\mathit{\beta })}^T{\mathrm{\sum }}_i^{-1}{\mathbf{r}}_i(\mathit{\alpha },\mathit{\beta })$$ (2.1)

after dropping a constant. Maximizing the log-likelihood function yields a maximum likelihood estimate (MLE) for the unknown parameters. The parameters can be estimated by iterating between estimation of (α, β) and estimation of (σ², θ). We shall discuss the estimation procedure of (α, β) for model (1.1) in details in the next section. Thus, we may substitute their estimates into r_ij(α, β), and r_ij(α̂, β̂) is computable and is denoted by r̂_ij for simplicity.

2.1 Estimation of variance function

We first propose an estimation procedure for σ²(t). Note that

$${\sigma }^2(t_{ij})=E\{{\epsilon }^2(t)\mid t=t_{ij}\}.$$

A natural estimator for σ²(t) is the kernel estimator:

$${\widehat{\sigma }}^2(t)=\frac{{\sum }_{i=1}^n{\sum }_{j=1}^{J_i}{\widehat{r}}_{ij}^2{K}_{h_1}(t-t_{ij})}{{\sum }_{i=1}^n{\sum }_{j=1}^{J_i}{K}_{h_1}(t-t_{ij})},$$

where K_h1 (x) = h₁⁻¹K(x/h₁), and K(x) is a kernel density function and h₁ is a smoothing parameter. Note that locally around a time point, there are few subjects that contribute more than one data point to the estimation of σ²(t). Thus, the estimator should behave locally as if data were independent. Ruppert et al (1997) studied local polynomial estimation of the variance function when observations are independently taken from the canonical nonparametric regression model: Y = m(X) + ε with E(ε|X) = 0 and var(ε|X) = σ²(X). Fan and Yao (1998) further showed that the local linear fit of variance function performs as well as the ideal estimator, which is a local linear fit to the true squared residuals {(Y_i − m(X_i))²}, allowing data to be taken from a stationary mixing process. A similar result was obtained by Müller and Stadtmüller (1993). The consistency and asymptotic behavior of σ̂²(t) will be studied in Theorem 4.2(B), from which we may choose an optimal bandwidth for σ̂²(t) using various existing bandwidth selectors for independent data (for example, Ruppert, Sheather and Wand, 1995).

2.2 Estimation of θ

Decompose the covariance matrix Σ_i into variance-correlation form, that is,

$${\mathrm{\sum }}_i=V_i{C}_i(\mathit{\theta })V_i,$$

where V_i = diag{σ (t_i1), ···, σ (t_iJi)} and C_i(θ) is the correlation matrix of ε_i, whose (k, l)-element equals ρ (t_ik, t_il, θ). To construct an estimator for θ, we maximize ℓ (α̂, β̂, σ̂², θ) with respect to θ. In other words,

$$\widehat{\mathit{\theta }}={\text{argmax}}_{\mathit{\theta }}-\frac{1}{2}\sum _{i=1}^n\left\{log\mid C_i(\mathit{\theta })\mid +{\widehat{\mathbf{r}}}_i^T{\widehat{V}}_i^{-1}{C}_i^{-1}(\mathit{\theta }){\widehat{V}}_i^{-1}{\widehat{\mathbf{r}}}_i\right\},$$ (2.2)

where V̂_i = diag{σ̂(t_i1), ···, σ̂(t_iJi)}, and r̂_i = (r̂_i1, ···, r̂_iJi)^T. The estimator in (2.2) is referred to as a quasi-likelihood (QL) estimator.

Optimizing QL may provide us a good estimate for θ when the correlation structure is correctly specified, but when it is misspecified, the QL might not be the best criterion to optimize. We may, for example, be interested in improving the efficiency for β, treating α, σ² and θ as nuisance parameters. In such a case, we are interested in choosing θ to minimize the estimated variance of β̂. For example, for a given working correlation function ρ₀(s, t) (e.g. working independence), we can embed this matrix into a family of parametric models ρ (s, t, θ) (e.g. autocovariance function of the ARMA(1, 1) model). Even though ρ (s, t, θ) might not be the true correlation function, we can always find a θ to improve the efficiency of β. More generally, suppose that the current working correlation function is ρ₀(s, t; θ₀). Let ρ₁(s, t), ···, ρ_m(s, t) be given family of correlation functions. We can always embed the current working correlation function ρ₀(s, t; θ₀) into the family of the correlation functions

$$\rho (s,t;\mathit{\theta })={\tau }_0{\rho }_0(s,t;{\mathit{\theta }}_0)+{\tau }_1{\rho }_1(s,t)+\cdots +{\tau }_m{\rho }_m(s,t).$$

where θ = (θ₀, τ₀, ···, τ_m), and τ₀ + ··· + τ_m = 1 with all τ_i ≥ 0. Thus, by optimizing the parameters θ₀, τ₀, ···, τ_m, the efficiency of the resulting estimator β̂ can be improved.

To fix the idea, let Γ(σ̂², θ) be the estimated covariance matrix of β̂ derived in (3.7) for a given working correlation function ρ (s, t, θ). Define the generalized variance of β̂ as the determinant of Γ(σ̂², θ). Minimizing the volume of the confidence ellipsoid of (β̂ − β)^TΓ⁻¹(σ̂², θ)(β̂ − β) < c for any positive constant c is equivalent to minimizing the generalized variance. Thus, we may choose θ to minimize the volume of the confidence ellipsoid:

$$\widehat{\mathit{\theta }}={\text{argmin}}_{\mathit{\theta }}\mid \mathrm{\Gamma }({\widehat{\sigma }}^2,\mathit{\theta })\mid .$$ (2.3)

We refer to this approach as the minimum generalized variance (MGV) method.

3 Estimation of regression coefficients

As mentioned in Section 2, the estimation of σ² and θ depends on the estimation of α(t) and β. On the other hand, improving the efficiency of the estimate for (α, β) relies on the estimation of σ² and θ. In practice, therefore, estimation needs to be done in steps: the initial estimates of (α(t), β) are constructed by ignoring within subject correlation. With this initial estimate, one can further estimate σ²(t) and θ. Finally, we can now estimate α(t) and β more efficiently by using the estimate of σ²(t) and θ. In this section, we propose efficient estimates for α(t) and β using profile least squares techniques.

For a given β, let y*(t) = y(t) − z(t)^Tβ. Then model (1.1) can be written as

$$y^{\ast }(t)=\mathbf{x}{(t)}^T\mathit{\alpha }(t)+\epsilon .$$ (3.1)

This is a varying coefficient model, studied by Fan and Zhang (2000) in the context of longitudinal data and by Hastie and Tibshirani (1993) for the case of iid observations. Thus, α(t) can be easily estimated by using any linear smoother. Here we employ local linear regression (Fan and Gijbels, 1996). For any t in a neighborhood of t₀, it follows from Taylor’s expansion that

$${\alpha }_l(t)\approx {\alpha }_l(t_0)+{\alpha }_l^{\prime }(t_0)(t-t_0)\equiv a_l+b_l(t-t_0),\text{for}l=1,\cdots ,q.$$

Let K(·) be a kernel function and h be a bandwidth. Thus, we can find local parameters (a₁, ···, a_q, b₁, ···, b_q) that minimize

$$\sum _{i=1}^n\sum _{j=1}^{J_i}{[y_i^{\ast }(t_{ij})-\sum _{l=1}^q\{a_l+b_l(t_{ij}-t_0)\}{x}_{il}(t_{ij})]}^2{K}_h(t_{ij}-t_0),$$ (3.2)

where K_h(·) = h ⁻¹K(·/h). The local linear estimate for α(t₀) is then simply α̂ (t₀, β̂) = (a₁, ···, a_q)^T. Note that since the data are localized in time, the covariance structure does not greatly affect the local linear estimator.

The profile least-squares estimator of (α, β) has a closed form using the following matrix notation. Let y_i = (y_i(t_i1), ···, y_i(t_iJi))^T, X_i = (x_i(t_i1), ···, x_i(t _iJi))^T, Z_i = (z_i(t_i1), ···, z_i(t_iJi))^T, and m_i = (x_i(t_i1)^Tα(t_i1), ···, x_i(t_iJi)^Tα(t_iJi))^T. Denote by $\mathbf{y}={({\mathbf{y}}_1^T,\cdots ,{\mathbf{y}}_n^T)}^T,\mathbf{X}={({\mathbf{X}}_1^T,\cdots ,{\mathbf{X}}_n^T)}^T,\mathbf{Z}={({\mathbf{Z}}_1^T,\cdots ,{\mathbf{Z}}_n^T)}^T$, and $\mathbf{m}={({\mathbf{m}}_1^T,\cdots ,{\mathbf{m}}_n^T)}^T$. Then, model (3.1) can be written as

$$\mathbf{y}-\mathbf{Z}\mathit{\beta }=\mathbf{m}+\mathit{\epsilon },$$ (3.3)

where ε = (ε₁(t₁₁), ···, ε _n(t _nJn))^T. It is known that the local linear regression results in a linear estimate in y*(t_ij) for α(·) (Fan and Gijbels, 1996). Thus, the estimate of α(·) is linear in y − Zβ, and the estimate of m is of the form m̂ = S(y − Zβ). The matrix S is usually called a smoothing matrix of the local linear smoother, and depends only on the observations {t_ij, x_i(t_ij), j = 1, ···, J_i, i = 1, ···, n}. Substituting m̂ into (3.3) results in the synthetic linear model

$$(I-\mathbf{S})\mathbf{y}=(I-\mathbf{S})\mathbf{Z}\mathit{\beta }+\mathit{\epsilon },$$ (3.4)

where I is the identity matrix of order $n^{\ast }={\sum }_{i=1}^n{J}_i$.

To improve efficiency for estimating β, we minimize the weighted least squares

$${(\mathbf{y}-\mathbf{Z}\mathit{\beta })}^T{(\mathbf{I}-\mathbf{S})}^T\mathbf{W}(\mathbf{I}-\mathbf{S})(\mathbf{y}-\mathbf{Z}\mathit{\beta }),$$ (3.5)

where W is a weight matrix, called a working covariance matrix. As usual, misspecification of the working covariance matrix does not affect the consistency of the resulting estimate, but it does affect the efficiency. The weighted least squares estimator for β is

$$\widehat{\mathit{\beta }}={\{{\mathbf{Z}}^T{(I-\mathbf{S})}^T\mathbf{W}(I-\mathbf{S})\mathbf{Z}\}}^{-1}{\mathbf{Z}}^T{(I-\mathbf{S})}^T\mathbf{W}(I-\mathbf{S})\mathbf{y}.$$ (3.6)

This estimator is called the profile weighted least squares estimator. The profile least squares estimator for the nonparametric component is simply α̂ (·; β̂). Using (3.4), it follows that when the weight matrix does not depend on y,

$$cov\{\widehat{\mathit{\beta }}\mid t_{ij},{\mathbf{x}}_i(t_{ij}),{\mathbf{z}}_i(t_{ij})\}={\mathbf{D}}^{-1}\mathbf{V}{\mathbf{D}}^{-1}\widehat{=}\mathrm{\Gamma }({\sigma }^2,\mathit{\theta }),$$ (3.7)

where D = Z^T(I − S)^TW(I − S)Z and V = cov{Z^T(I − S)^TWε}. In practice, Γ̂ (σ̂², θ) is estimated by a sandwich formula by taking V̂ = Z^T(I − S)^TWRW^T (I − S)Z, where $\mathbf{R}=\text{diag}\{{\mathbf{r}}_1{\mathbf{r}}_1^T,\cdots ,{\mathbf{r}}_n{\mathbf{r}}_n^T\}$ with r_i = y_i − ŷ_i. Speckman (1988) derived a partial residual estimator of β for partially linear models with independent and identically distributed data; the form of this estimator is the same as that in (3.5) with W set to be an identity matrix. However, the idea of partial residual approach is difficult to implement for model (1.1).

4 Sampling properties

In this section, we investigate sampling properties of the profile weighted least squares estimator. The proposed estimation procedures are applicable for various formulations on how the longitudinal data are collected. Here we consider the collected data as a random sample from the population process {y(t), x(t), z(t)}, t ∈ [0, T]. To facilitate the presentation, we assume that J_i, i = 1, ···, n are independent and identically distributed with 0 < E(J_i) < ∞, and for a given J_i, t_ij, j = 1, ···, J_i are independent and identically distributed according to a density f(t). Furthermore, suppose that the weight matrix W in (3.5) is block diagonal, i.e., W = diag{W₁, ···, W_n}, where W_i is a J_i × J_i matrix. Moreover, assume that the (u, v)-element of W_i is set to be w(t_iu, t_iv) for a bivariate positive function w(·, ·). When the weight function w(·, ·) is data-dependent, assume that it tends to a positive definite function in probability. Thus, for simplicity, assume that w(·, ·) is deterministic.

Let G(t) = Ex(t)x^T(t), Ψ(t) = Ex(t)z^T(t), and denote by

$${\stackrel{\sim }{\mathbf{X}}}_i={(\mathrm{\Psi }(t_{i1}){\mathbf{G}}^{-1}(t_{i1}){\mathbf{x}}_i(t_{i1}),\cdots ,\mathrm{\Psi }(t_{i{J}_i}){\mathbf{G}}^{-1}(t_{i{J}_i}){\mathbf{x}}_i(t_{i{J}_i}))}^T.$$

Set

$${\mathrm{\sum }}_n=\frac{1}{n}\sum _{i=1}^n{\{{\mathbf{Z}}_i-{\stackrel{\sim }{\mathbf{X}}}_i\}}^T{\mathbf{W}}_i\{{\mathbf{Z}}_i-{\stackrel{\sim }{\mathbf{X}}}_i\},\text{and}{\xi }_n=\frac{1}{n}\sum _{i=1}^n{\{{\mathbf{Z}}_i-{\stackrel{\sim }{\mathbf{X}}}_i\}}^T{\mathbf{W}}_i{\mathit{\epsilon }}_i,$$

where ε_i = (ε_i(t_i1), ···, ε_i(t_iJi))^T. Let

$$\mathbf{A}=E\{{({\mathbf{Z}}_1-{\stackrel{\sim }{\mathbf{X}}}_1)}^T{\mathbf{W}}_1({\mathbf{Z}}_1-{\stackrel{\sim }{\mathbf{X}}}_1)\},\text{and}\mathbf{B}=E\{{({\mathbf{Z}}_1-{\stackrel{\sim }{\mathbf{X}}}_1)}^T{\mathbf{W}}_1{\mathit{\epsilon }}_1{\mathit{\epsilon }}_1^T{\mathbf{W}}_1({\mathbf{Z}}_1-{\stackrel{\sim }{\mathbf{X}}}_1)\}.$$

Denote by α₀(t) and β₀ the true values of α(t) and β, respectively.

Theorem 4.1

Under the regularity conditions (1)–(5) in the Appendix, if the matrices A and B exist, and if A is positive definite, then as n → ∞,

$$\sqrt{n}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)=\sqrt{n}{\mathrm{\sum }}_n^{-1}{\xi }_n+o_P(1)\stackrel{\mathcal{L}}{\to }N(0,{\mathbf{A}}^{-1}\mathbf{B}{\mathbf{A}}^{-1}),$$

where n is the number of subjects.

When W_i is taken to be the inverse of the conditional variance-covariance matrix of ε_i given x_i(t_ij) and z_i(t_ij) for j = 1, ···, J_i, then A = B. In this case,

$$\sqrt{n}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)\stackrel{\mathcal{L}}{\to }N(0,{\mathbf{B}}_0^{-1}),$$

where B₀ = E{Z₁ − X̃₁)^T cov⁻¹ (ε₁|X₁, Z₁)(Z₁ − X̃₁)}. It will be shown in the Appendix that for any weight matrix W_i,

$${\mathbf{A}}^{-1}\mathbf{B}{\mathbf{A}}^{-1}-{\mathbf{B}}_0^{-1}\ge 0,$$ (4.1)

where the symbol D ≥ 0 means that the matrix D is nonnegative definite. Thus, the most efficient estimator for β among the profile weighted least-squares estimates given in (3.6) is the one that uses the inverse of the true variance-covariance matrix of ε_i as the weight matrix W_i.

One could also use a working independence correlation structure, i.e. let W be a diagonal matrix. Under conditions of Theorem 4.1, the resulting estimate of β is still root n consistent.

Let μ_i = ∫uⁱK(u)du and ν_i = ∫uⁱK²(u)du. For a vector of functions α(u) of u, denote α̇ (u) = dα(u)/du and α̈(u) = d²α(u)/du², which are the componentwise derivatives. The following theorem presents the asymptotic normality for α̂ (t) and σ̂²(t), and its proof was given the earlier version of this paper (Fan, Huang and Li, 2005).

Theorem 4.2

Suppose that conditions of Theorem 4.1 hold.

1. If nh⁵ = O(1) as n → ∞, then

   $$\sqrt{nh}(\widehat{\mathit{\alpha }}(t)-\mathit{\alpha }(t)-\frac{1}{2}{\mu }_2{h}^2\ddot{\mathit{\alpha }}(t))\stackrel{\mathcal{L}}{\to }N\left(0,\frac{{\nu }_0}{f(t)E(J_1)}{\sigma }^2(t){\mathrm{\Gamma }}^{-1}(t)\right).$$

2. Under conditions (5) and (6) in the Appendix, if $c<n{h}_1^5<C$, and c < h/h₁ < C for some positive constants c and C, then, as n → ∞,

   $$\sqrt{n{h}_1}({\widehat{\sigma }}^2(t)-{\sigma }^2(t)-b(t))\stackrel{\mathcal{L}}{\to }N(0,v(t)),$$

   where the bias

   $$b(t)=\frac{h_1^2}{2}\left\{{\ddot{\sigma }}^2(t)+\frac{2{\stackrel{.}{\sigma }}^2(t)f^{\prime }(t)}{f(t)}\right\}{\mu }_2,$$

   and the variance

   $$v(t)=\frac{\mathit{var}\{{\epsilon }^2(t)\}{\nu }_0}{f(t)E(J_1)}.$$

Since the parametric convergence rate of β̂ is faster than the nonparametric convergence rate of α̂ (t), the asymptotic bias and variance have similar forms to those of varying coefficient model (Cai, Fan and Li, 2000). The choice of the weight matrix W determines the efficiency of β̂, but it does not affect the asymptotic bias and variance of α(t).

From Theorem 4.2 (B), the asymptotic bias and variance do not depend on the choice of the weight matrix W. Therefore, one may use the residuals obtained by using the working independence correlation matrix to estimate σ²(t). This is consistent with our empirical findings from the simulation studies. Therefore, in next section σ²(t) will be estimated using residuals obtained under working independence. Theorem 4.2 (B) implies that we may choose a bandwidth by modifying one of existing bandwidth selectors used for independent data.

5 Numerical comparison and application

In this section, we investigate finite sample properties of the proposed estimators in Sections 2 and 3 via Monte Carlo simulation. All simulation studies are conducted using Matlab code. We have examined the finite sample performance and numerical comparisons for the proposed estimate σ̂²(t), β̂ and α̂ (t) in the earlier version of this paper. See technical report (Fan, Li and Huang, 2005) for details. To save space, we focus on the inference on β in this section.

5.1 Simulation study

We generate 1000 data sets, each consisting of n = 50 subjects, from the following model:

$$y(t)=\mathbf{x}{(t)}^T\mathit{\alpha }(t)+\mathbf{z}{(t)}^T\mathit{\beta }+\epsilon (t).$$ (5.1)

In practice, observation times are usually scheduled but may be randomly missed. Thus, we generate the observation times in the following way. Each individual has a set of ‘scheduled’ time points, {0,1,2,…,12}, and each scheduled time, except time 0, has a 20% probability of being skipped. The actual observation time is a random perturbation of a scheduled time: a uniform [0, 1] random variable is added to a non-skipped scheduled time. This results in different observed time points t_ij per subject.

In our simulation, the random error process ε(t) in (5.1) is taken to be a Gaussian process with zero mean, variance function

$${\sigma }^2(t)=0.5exp(t/12),$$

and ARMA(1,1) correlation structure

$$\text{corr}(\epsilon (s),\epsilon (t))=\gamma {\rho }^{\mid t-s\mid }$$

for s ≠ t. We consider three pairs of (γ, ρ), namely, (0.85, 0.9), (0.85, 0.6) and (0.85, 0.3), which correspond to strongly, moderately and weakly correlated errors, respectively.

We let the coefficients of α(t) and β both be two-dimensional in our simulation, and further set x₁(t) ≡ 1 to include an intercept term. We generate the covariates in the following way: for a given t, (x₂(t), z₁(t))^T follows a bivariate normal distribution with means zero, variances one and correlation 0.5, and z₂(t) is a Bernoulli-distributed random variable with success probability 0.5 and independent of x₂(t) and z₁(t). In this simulation, we set β = (1, 2)^T,

$${\alpha }_1(t)=\sqrt{t/12},\text{and}{\alpha }_2(t)=sin(2\pi t/12).$$

Presumably we can gain some efficiency by incorporating the correlation structure, and it is of interest to study the size of gain. We consider the case in which the working correlation structure is taken to be the true one, which is ARMA(1,1) correlation structure. For comparison, we also estimate β using working independence correlation structure and using the true correlation structure in which the parameter (γ, ρ)^T is set to be the true value. Profile weighted least squares estimate using the true correlation is shown to be the most efficient estimate among the profile weighted least squares estimates and serves as a benchmark, while the working independence correlation structure is supposed to be a commonly used one in practice.

Table 1 presents a summary of the results over 1000 simulations. In Table 1, “bias” stands for the sample average over 1000 estimates subtracting the true value of β, “SD” stands for the sample standard deviation over 1000 estimates. “Median” represents the median of the 1000 estimates subtracting the true value and “MAD” represents the median absolute deviation of the 1000 estimates divided by a factor of 0.6745. From Table 1, both QL and MGV approaches yield estimates for β as good as the estimate using the true correlation function, and is much better than the estimate using working independence correlation structure. The relative efficiency (MAD(Indep.)/MAD(QL)) is about 3 for high correlation random error, 2 for moderately correlated error and 1.3 for weakly correlated error.

Table 1.

Performance of β̂^*

	β̂1				β̂2
Method	SD	Bias	MAD	Median	SD	Bias	MAD	Median
(γ, ρ) = (0.85, 0.9)
Indep.	47.780	−1.9730	44.575	−1.2802	82.488	−1.7276	79.580	−2.7890
True	25.061	−1.2565	25.905	−0.7676	45.003	0.1211	45.543	−0.1568
QL	25.156	−1.2545	25.536	−0.7709	44.932	0.1749	44.654	−0.6489
MGV	25.205	−1.2040	25.575	−0.9126	45.585	0.2663	45.033	−0.5308
(γ, ρ) = (0.85, 0.6)
Indep.	47.499	−2.6415	49.465	−0.8980	82.094	−1.1161	82.553	−3.0444
True	34.308	−1.6807	34.569	−1.5081	62.596	−0.2047	61.871	−0.3016
QL	46.365	−0.2651	34.807	−1.2672	62.650	−0.0023	62.485	−0.3322
MGV	34.634	−1.3411	35.450	−0.5676	64.393	−0.2691	61.090	−1.8051
(γ, ρ) = (0.85, 0.3)
Indep.	46.991	−2.8990	47.457	−1.6817	81.798	−1.0896	83.991	−1.2721
True	40.123	−1.9687	40.184	−2.1143	73.031	−0.5122	73.278	0.1861
QL	95.506	−6.7632	41.841	−1.9187	288.389	−5.7357	77.514	0.1459
MGV	40.389	−1.6740	40.685	−1.4153	74.798	−0.5055	73.465	0.1435

^*Values in the columns of SD, bias, MAD and median are multiplied by a factor of 1000

The simulation results also indicate that the MGV method is more stable and robust than the QL method. This is evidenced in the case of low correlated random error, in which for a few realizations, the estimates were apparently quite bad (the SD is much higher than the MAD). Note the object function to optimize in (2.2) may not be a concave function of θ. Thus, the numerical algorithm may not converge when it stops. This may yield a bad estimate for β and contributes to the issues of the robustness of the algorithm. In addition, the QL criterion is similar to the least-squares criterion and hence is not very robust. On the other hand, the MGV method, aiming directly at minimizing the precision of estimated standard errors, does not allow estimates to have large SEs.

We next study the impact of misspecification of correlation structure, by comparing the performance of β̂ using independent and AR(1) working correlation structures, when the true correlation structure is ARMA(1,1). The top panel of Table 2 summarizes the simulation results. From Table 2, we can see that AR(1) working correlation structure produces much more efficient estimate than working independence correlation structure. For example, the relative efficiency for high correlated random error is about (30.066/19.975)² ≈ 2.3. Thus, even when the true correlation structure is unavailable, it is still quite desirable to choose a structure close to the truth.

Table 2.

Impacts of Misspecification of Correlation on β̂^*

	β̂1				β̂2
Method	SD	Bias	MAD	Median	SD	Bias	MAD	Median
Optimization Algorithm Search
(γ, ρ) = (0.85, 0.9)
Indep.	47.7800	− 1.9730	44.5759	− 1.2802	82.4880	− 1.7276	79.5815	− 2.7890
QL	31.8570	− 0.4859	29.6149	− 0.0837	60.8860	− 0.0684	54.6946	0.2275
MGV	33.1210	− 0.5275	31.8003	0.0535	63.4840	0.3800	58.0557	0.7224
(γ, ρ) = (0.85, 0.6)
Indep.	47.4990	− 2.6415	49.4655	− 0.8980	82.0940	− 1.1161	82.5541	− 3.0444
QL	37.0470	− 1.0667	36.2184	− 0.8100	68.9890	− 0.0925	61.8111	− 1.6883
MGV	37.9660	− 1.1648	36.9805	− 1.0777	71.3970	0.1137	65.4138	− 2.2338
(γ, ρ) = (0.85, 0.3)
Indep.	46.9910	− 2.8990	47.4580	− 1.6817	81.7980	− 1.0896	83.9923	− 1.2721
QL	41.0240	− 1.6139	40.7671	− 1.0700	74.8700	− 0.3320	73.7801	0.0931
MGV	42.4130	− 1.6264	42.2526	− 0.7556	79.1190	− 0.1797	73.2301	− 2.0012
Rough Grid Point Search
(γ, ρ) = (0.85, 0.9)
Indep.	47.7800	− 1.9730	44.5759	− 1.2802	82.4880	− 1.7276	79.5815	− 2.7890
QL	31.9390	− 0.4489	29.3436	− 0.0714	60.7410	0.0272	54.9896	0.7295
MGV	33.2930	− 0.5232	31.4578	− 0.2297	63.8040	0.4463	58.2365	0.6303
(γ, ρ) = (0.85, 0.6)
Indep.	47.4990	− 2.6415	49.4655	− 0.8980	82.0940	− 1.1161	82.5541	− 3.0444
QL	37.2570	− 1.1533	36.4912	− 1.1077	6.9254	0.2263	63.1202	− 1.2543
MGV	40.6740	− 1.1390	39.7678	− 1.5885	77.0800	0.5339	71.0284	0.7412
(γ, ρ) = (0.85, 0.3)
Indep.	46.9910	− 2.8990	47.4580	− 1.6817	81.7980	− 1.0896	83.9923	− 1.2721
QL	41.3200	− 1.6483	40.9850	− 1.8832	75.1910	0.0079	73.4095	0.3885
MGV	48.4380	− 1.5369	47.8895	− 1.9910	91.9430	0.2399	84.2413	1.8811

^*Values in the columns of SD, bias, MAD and median are multiplied by a factor of 1000.

In practice, one may try several values for ρ and choose the best one using the QL or MGV method rather than using an optimization algorithm. We refer to such search as the rough grid point search. We next examine how such search works in practice, using the points {0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95} for ρ. The bottom panel of Table 2 presents the simulation results. Comparing the bottom panel with the top panel of Table 2, the performance of the resulting estimates using the rough grid point search is very close to that using an optimization algorithm.

Now we test the accuracy of the proposed standard error formula (3.7). Table 3 depicts the simulation results for the case (γ, ρ) = (0.85, 0.9). Results for other cases are similar. In Table 3, “SD” stands for the sample standard deviation of 1000 estimates of β and can be viewed as the true standard deviation of the resulting estimate. “SE” stands for the sample average of 1000 estimated standard errors using formula (3.7), and “Std” presents the standard deviation of these 1000 standard errors. From Table 3, the standard error formula works very well for both correctly specified and misspecified correlation structures.

Table 3.

Standard Errors

	β̂1		β̂2
	SD	SE (Std)	SD	SE (Std)
ARMA(1,1) working correlation matrix
Independence	0.0478	0.0464(0.0065)	0.0825	0.0800 (0.0108)
QL	0.0252	0.0254 (0.0030)	0.0449	0.0440 (0.0047)
MGV	0.0252	0.0257 (0.0031)	0.0456	0.0446 (0.0049)
AR(1) working correlation matrix
QL	0.0319	0.0307 (0.0078)	0.0609	0.0541 (0.0131)
MGV	0.0331	0.0316 (0.0084)	0.0635	0.0557 (0.0141)

5.2 Some comparison with traditional approach

The purpose of this section is to demonstrate the flexibility and efficiency of model (1.1) by comparing its performance with linear models for longitudinal data:

$$y(t)=\mathbf{x}{(t)}^T\mathit{\alpha }+\mathbf{z}{(t)}^T\mathit{\beta }+\epsilon (t),$$ (5.2)

which can be viewed as a special case of model (1.1) with constant function α(·). We employed the weighted least squares method to estimate α and β in model (5.2). To make a fair comparison, we generated 1000 data sets, each consisting of n = 50 samples, from model (5.1) with

Case I: ${\alpha }_1(t)=\sqrt{t/12}$ and α₂(t) = sin(2πt/12). This is exactly the same as those in Section 5.1.

Case II: α₁(t) = 2 and α₂(t) = 1. That is, both α₁(t) and α₂(t) are constant functions.

and all others parameters and generation scheme of observation times are the same as those in Section 5.1.

To illustrate the flexibility of model (1.1), we fit data generated under the setting of Case I using the linear model (5.2). The error correlation structure is no longer to be ARMA if model (5.2) is fitted under the setting of Case I. Thus, we did not include “True” correlation structure in our simulation. Simulation results are summarized in the top panel of Table 4, in which the caption is the same as that in Tables 1 and 2. To save space, we present only the simulation results with (γ, ρ) = (0.85, 0.6). Results for other (γ, ρ) pairs are similar. Compared with results in Tables 1 and 2, it can be found that misspecification α(t) may yield an estimate with larger bias and less efficient.

Table 4.

Comparison to linear model^*

			β̂1		β̂2
Model	Correlation	Method	MAD	Median(bias)	MAD	Median(bias)
Case I: ${\alpha }_1(t)=\sqrt{t/12}$, α2(t) = sin(2πt/12) and (γ, ρ) = (0.85, 0.6)
(5.2)	Indep.		62.5252	− 3.7274	102.7234	2.3116
	AMRA(1,1)	QL	43.3809	− 5.2141	75.1942	1.0293
	ARMA(1,1)	MGV	60.7346	− 4.1006	98.3291	1.8330
	AR(1)	QL	52.8866	− 2.2510	93.2711	− 3.9622
	AR(1)	MGV	59.9004	− 3.2324	96.4753	1.1836
Case II: α1(t) = 2, α2(t) = 1 and (γ, ρ) = (0.85, 0.6)
(5.2)	Indep.		47.7871	− 3.1878	82.1597	− 2.2803
	AMRA(1,1)	True	32.9404	− 1.9984	61.6268	0.5491
	ARMA(1,1)	QL	33.1803	− 2.8015	61.8600	0.1782
	ARMA(1,1)	MGV	47.0792	− 1.2353	76.6334	− 0.6911
	AR(1)	QL	35.1901	− 0.8354	64.2013	− 0.3883
	AR(1)	MGV	47.0576	− 1.4820	76.8226	− 0.8559
(1.1)	Indep.		49.4474	− 1.0333	82.7413	− 3.0255
	AMRA(1,1)	True	34.3453	− 1.6239	63.0509	0.2820
	ARMA(1,1)	QL	35.3995	− 1.7503	62.9548	− 0.5040
	ARMA(1,1)	MGV	35.6286	− 0.3130	62.1033	− 2.5856
	AR(1)	QL	36.2746	− 0.8732	63.2304	− 1.2967
	AR(1)	MGV	39.8883	− 0.8650	72.1075	1.2003

^*Values in the columns of MAD and median are multiplied by a factor of 1000

Simulation results of models (5.2) and (1.1) for Case II are summarized in the middle panel and the bottom panel of Table 4, respectively. The bias of the resulting estimates for all estimation procedures are in the same magnitude. Comparing the simulation of models (5.2) and (1.1) with independent working correlation matrix and with the true/QL ARMA(1,1) correlation matrix, the proposed models do not lose much efficiency. In summary, the proposed estimation procedure with the model (1.1) offers us a good balance between model flexibility and estimation efficiency.

5.3 An application

We next demonstrate the newly proposed procedures by an analysis of a subset of data from the Multi-Center AIDS Cohort study. The data set contains the human immunodeficiency virus (HIV) status of 283 homosexual men who were infected with HIV during the following-up period between 1984 and 1991. This data set has been analyzed by Fan and Zhang (2000) and Huang, Wu and Zhou (2002) using functional linear models. Details of the study design, methods, and medical implications are given by Kaslow et al. (1987).

All participants were scheduled to have their measurements taken during semiannual visits, but, because many participants missed some of their scheduled visits and the HIV infections occurred randomly during the study, there are unequal numbers of repeated measurements and different measurement times per individual. Our interest is to describe the trend of the mean CD4 percentage depletion over time and to evaluate the effects of cigarette smoking, pre-HIV infection CD4 percentage, and age at infection on the mean CD4 percentage after the infection. Huang, Wu and Zhou (2002) took the response y(t) to be CD4 cell percentage and considered the functional linear model,

$$y(t)={\beta }_0(t)+{\beta }_1(t)\text{Smoking}+{\beta }_2(t)\text{Age}+{\beta }_3(t)\text{PreCD}4+\epsilon (t).$$ (5.3)

The results of the hypothesis testing in Huang, Wu and Zhou (2002) indicate that the baseline function varies over time; neither Smoking nor Age has a significant impact on the mean CD4 percentage; and it is unclear whether PreCD4 has a constant effect over time or not. The P-value for testing whether β₃(t) varies over time or not is 0.059. Thus, we fit the data using a simpler semiparametric varying coefficient partially linear model

$$y(t)={\alpha }_1(t)+{\alpha }_2(t)X_1+{\beta }_1{Z}_1+{\beta }_2{Z}_2+\epsilon (t),$$

where, for numerical stability, X₁ is the standardized variable for PreCD4, Z₁ is the smoking status (1 for a smoker and 0 for a nonsmoker), Z₂ is the standardized variable for age, and the unit for observation time t is one month.

Bandwidth selection

We employ a multifold cross-validation method to select a bandwidth for α̂ (t). We partition the data into Q groups, each of which has approximately the same number of subjects. For each k, k = 1, ···, Q, model (5.3) is fitted for the data excluding the k-group of data. Cross-validation score is defined as the sum of residual squares:

$$CV(h)=\sum _{k=1}^Q\sum _{i\in d_k}\sum _{j=1}^{J_i}{\{y_i(t_{ij})-{\widehat{y}}_{-d_k}(t_{ij})\}}^2,$$

where ŷ_−dk (t_ij) is the fitted value for the i-th subject at observed time t_ij with the data in d_k being deleted, using a working independence correlation matrix. In the implementation, we choose Q = 15. Figure 1(a) depicts the cross-validation score function CV (h) which gives the optimal bandwidth h = 21.8052. Note that σ̂²(t) is a one-dimensional kernel regression of the squared residuals over time. Thus, various bandwidth selectors for one-dimensional smoothing can be used to choose a bandwidth for σ̂²(t). In this application, we directly use the plug-in bandwidth selector (Ruppert, Sheather and Wand, 1995), and the bandwidth h₁ = 12.7700 is chosen.

Figure 1.

(a) Plot of the cross-validation score against the bandwidth. (b) and (c) are plots of estimate of α₁(t) and α₂(t) with bandwidth 21.8052, chosen by the cross-validation method. (d) Plot of estimated σ (t) with bandwidth 12.7700, chosen by the plug-in method.

Estimation

The resulting estimate of α(t) is depicted in Figures 1(b) and (c). The intercept function decreases as time increases. This implies that the overall trend of CD4 cell percentage decreases over time. The trend of α₂(t) implies that the impact of PreCD4 on CD4 cell percentage decreases gradually as time evolves. The results are consistent with our expectation. They quantify the extent to which the mean CD4 percentage depletes over time and how the association between CD4 percentage and PreCD4 varies as time evolves. The resulting estimate of σ̂ (t) is depicted in Figure 1(d), from which we can see that σ (t) seems to be constant during the first and half year, and then increases as time increases. This shows that the CD4 percentage gets harder to predict as time evolves.

We next estimate β. Here we consider ARMA(1,1) correlation structure. The proposed estimation procedures in Section 2 were applied for estimating (γ, ρ). The resulting estimates are displayed in the top panel of Table 5, and the corresponding estimates for β are depicted in the bottom panel of Table 5. The quasi-likelihood approach yields a correlation structure with moderate correlation, and the standard error for the resulting estimate of β is smaller than that using independence correlation structure. The minimum generalized variance method results in a correlation structure with low correlation, but the corresponding standard error is still smaller than that of independence correlation structure. From Table 5, the effects of smoking status and age are not significant under the three estimation schemes.

Table 5.

Estimates of (γ, ρ) and β

	Independence	QL	MGV
γ̂		0.8575	0.5334
ρ̂		0.9852	0.0804
β̂1	0.8726(1.1545)	0.6848(0.9972)	0.6328(1.0864)
β̂2	− 0.5143(0.6110)	0.0556(0.4718)	− 0.3658(0.5488)

Prediction of individual trajectory

We now illustrate how to incorporate correlation information into prediction. Let us assume that given the covariates x(t) and z(t), the error process ε (t) is a Gaussian process with zero mean and covariance function c(t, s). Denote by μ (t) = x(t)^Tα(t)+z(t)^Tβ. Suppose that data for an individual are collected at t = t_1, ···, t_J and we want to predict his/her y(t) at t = t* with covariates x(t*) and z(t*). Let y_o = (y(t₁), ···, y(t_J ))^T be the observed response and μ = (μ(t₁), ···, μ(t_J ))^T be its associated mean. Let Σ be the covariance matrix of (ε (t₁), ···, ε (t_J ))^T, and c* = (c(t₁, t*), ···, c(t_J, t*))^T. Then, by the properties of the multivariate normal distribution, we have

$$E\{y(t^{\ast })\mid {\mathbf{y}}_o\}=\mu (t^{\ast })+{\mathbf{c}}^{\ast T}{\mathrm{\sum }}^{-1}({\mathbf{y}}_o-\mathit{\mu }),$$

and

$$var\{y(t^{\ast })\mid {\mathbf{y}}_o\}={\sigma }^2(t^{\ast })-{\mathbf{c}}^{\ast T}{\mathrm{\sum }}^{-1}{\mathbf{c}}^{\ast }.$$

Thus, the prediction of y(t*) is

$$\widehat{y}(t^{\ast })=\widehat{\mu }(t^{\ast })+{\widehat{\mathbf{c}}}^{\ast T}{\widehat{\mathrm{\sum }}}^{-1}({\mathbf{y}}_o-\widehat{\mathit{\mu }}).$$

Since the errors in estimating the unknown regression coefficients and parameters of covariance matrix are negligible relative to random error, the (1 − α)100% predictive interval is

$$\widehat{y}(t^{\ast })\pm z_{1-\alpha /2}\sqrt{{\widehat{\sigma }}^2(t^{\ast })-{\widehat{\mathbf{c}}}^{\ast T}{\widehat{\mathrm{\sum }}}^{-1}{\widehat{\mathbf{c}}}^{\ast }},$$

where z_{1− α/2} is the 1 − α/2 quantile of the standard normal distribution. In particular, it is easy to verify that when t* is one of the observed time points, the prediction error is zero, a desired property.

We now apply the prediction procedure for this application. Assume that ε (t) has AMRA(1,1) correlation structure. As an illustration, here we only consider the prediction with (γ, ρ) estimated by quasi-likelihood approach. That is, (γ̂, ρ̂) = (0.8575, 0.9852). Predictions and their 95% predictive intervals for 4 typical subjects are displayed in Figure 2.

Figure 2.

Plot of pointwise predictions and their 95% predictive intervals for 4 typical subjects. Solid line is the prediction, dashdot lines stand for the limits for the 95% pointwise predictive confidence interval, and “o” is the observed value of y(t).

6 Discussions

In this paper, we proposed a class of semiparametric models for the covariance function of longitudinal data. We further developed an estimation procedure for σ²(t) using kernel regression, estimation procedures for θ in correlation matrix using quasi-likelihood and minimum generalized variance approaches, and estimation procedure for regression coefficients α(t) and β using profile weighted least squares. Robust method estimation procedures have been proposed for semiparametric regression modeling with longitudinal data (He, Zhu and Fung, 2002; He, Fung and Zhu, 2005). In the presence of outliers, one should consider robust method to estimate α(t) and β.

Although misspecification of the correlation structure ρ (s, t, θ) does not affect the consistency of the resulting estimate of α(t) and β, it may lead to nonexistence or inconsistency of the estimates of θ. Thus, it is of interest to check whether the imposed correlation structure is approximately correct. To address this issue, we may consider a full nonparametric estimate for the correlation function ρ (s, t)

$$\rho (s,t)=\frac{{\sum }_{i=1}^n{\sum }_{j\ne j^{\prime }}^{J_i}{\widehat{e}}_i(t_{ij}){\widehat{e}}_i(t_{i{j}^{\prime }})K_{h_2}(s-t_{ij})K_{h_2}(t-t_{i{j}^{\prime }})}{{\sum }_{i=1}^n{\sum }_{j\ne j^{\prime }}^{J_i}{K}_{h_2}(s-t_{ij})K_{h_2}(t-t_{i{j}^{\prime }})}$$ (6.1)

for s ≠ t, where ê (t_ij) = r̂_ij/σ̂ (t_ij), the standardized residual.

The nonparametric covariance estimator cannot be guaranteed to be positive definite, but it may be useful in specifying an approximate correlation structure, or checking whether the imposed correlation structure ρ (s, t, θ) its approximately correct. This is a two-dimensional smoothing problem, but the effective data points in (6.1) can be small unless the time points for each subject are nearly balanced.

Some alternative estimation procedures for α(t) and β may also be considered. For example, an alternative strategy to estimate β is to first decorrelate data within subjects, and then apply the profile least squares techniques to the decorrelated data. Further research and comparison may be of interest.

In this paper, we have not discussed the sampling property of θ̂ derived by QL and MGV approaches. If the correlation function is correctly specified, the asymptotic property of θ̂ may be derived by following conventional techniques related to linear mixed effects models. It is an interesting topic to investigate the asymptotic behaviors of θ̂ when the correlation function is misspecified. Some new formulation may be needed to establish the asymptotic property of θ̂. This research topic is out of scope of this paper. Further research is needed.

Appendix

The following technical conditions are imposed. They are not the weakest possible conditions, but they are imposed to facilitate the proofs.

1. The density function f(·) is Lipschitz continuous and bounded away from 0. The function K(·) is a symmetric density function with a compact support.

2. nh⁸ → 0 and nh²/(log n)³ → ∞.

3. Ex(t)x(t)^T and Ex(t)z(t)^T are Lipschitz continuous.

4. J_i has a finite moment generating function. In addition, E||x(t)||⁴ +E||z(t)||² < ∞

5. α(t) has a continuous second derivative.

6. σ²(·) has a continuous second derivative.

Proof of Theorem 4.1

First, by condition (4), we can easily show almost surely that max_1≤i≤n J_i = O(log n). For each given β, the estimator α̂ (t; β) is a local linear estimator by minimizing (3.2) based on data

$$\{t_{ij},{\mathbf{x}}_i(t_{ij}),y_i^{\ast }(t_{ij})\},j=1,\cdots ,J_i,i=1,\cdots ,n.$$

Observe that { $y_i^{\ast }(t_{ij})$, j = 1, ···, J_i} is a realization from the process

$$y^{\ast }(t)=\mathbf{x}{(t)}^T{\mathit{\alpha }}_0(t)+\mathbf{z}{(t)}^T({\mathit{\beta }}_0-\mathit{\beta })+\epsilon (t).$$

Note that the consistency of α̂ (t; β) is not affected by ignoring the correlation within subjects. Following the proof of Fan and Huang (2005), α̂ (t; β) is a consistent estimator of the function

$$\mathit{\alpha }(t;\mathit{\beta })={\mathit{\alpha }}_0(t)-{\mathbf{G}}^{-1}(t)\mathrm{\Psi }(t)(\mathit{\beta }-{\mathit{\beta }}_0).$$ (A.1)

Indeed, uniformly in t,

$$\widehat{\mathit{\alpha }}(t;\mathit{\beta })-\mathit{\alpha }(t;\mathit{\beta })=O_P(c_n),$$ (A.2)

where c_n = h² + {− log h/(nh)}^1/2. Let m̂_ij(β) = x_i(t_ij)^Tα̂ (t_ij; β) and m̂_i(β) = (m̂_i1, ···, m̂_iJi)^T. Note that the profile weighted least squares estimate β̂ is the minimizer of the following weighted quadratic function:

$${\ell }_n(\mathit{\beta })=\frac{1}{n}\sum _{i=1}^n{({\mathbf{y}}_i-{\widehat{\mathbf{m}}}_i(\mathit{\beta })-{\mathbf{Z}}_i\mathit{\beta })}^T{\mathbf{W}}_i({\mathbf{y}}_i-{\widehat{\mathbf{m}}}_i(\mathit{\beta })-{\mathbf{Z}}_i\mathit{\beta }),$$ (A.3)

which is a convex and quadratic function of β. This allows us to apply the convexity lemma and the quadratic approximation lemma (see, for example, Fan and Gibjels, 1996, pp.209–210) to establish the asymptotic normality of β̂.

We next decompose ℓ_n(β). Denote

$$\begin{array}{l}{\mathbf{m}}_i(\mathit{\beta })={({\mathbf{x}}_i{(t_{i1})}^T\mathit{\alpha }(t_{i1};\mathit{\beta }),\cdots ,{\mathbf{x}}_i{(t_{i{J}_1})}^T\mathit{\alpha }(t_{i{J}_1};\mathit{\beta }))}^T,\\ I_{n,1}(\mathit{\beta })=\frac{1}{n}\sum _{i=1}^n{\{{\mathbf{y}}_i-{\mathbf{m}}_i(\mathit{\beta })-{\mathbf{Z}}_i\mathit{\beta }\}}^T{\mathbf{W}}_i\{{\mathbf{y}}_i-{\mathbf{m}}_i(\mathit{\beta })-{\mathbf{Z}}_i\mathit{\beta }\},\\ I_{n,2}(\mathit{\beta })=\frac{2}{n}\sum _{i=1}^n{\{{\mathbf{y}}_i-{\mathbf{m}}_i(\mathit{\beta })-{\mathbf{Z}}_i\mathit{\beta }\}}^T{\mathbf{W}}_i\{{\mathbf{m}}_i(\mathit{\beta })-{\widehat{\mathbf{m}}}_i(\mathit{\beta })\},\text{and}\\ I_{n,3}(\mathit{\beta })=\frac{1}{n}\sum _{i=1}^n{\{{\mathbf{m}}_i(\mathit{\beta })-{\widehat{\mathbf{m}}}_i(\mathit{\beta })\}}^T{\mathbf{W}}_i\{{\mathbf{m}}_i(\mathit{\beta })-{\widehat{\mathbf{m}}}_i(\mathit{\beta })\},\end{array}$$

where m_i(β) = (m_i1, ···, m_iji)^Tα(t_ij, β). Then

$${\ell }_n(\mathit{\beta })=I_{n,1}(\mathit{\beta })+I_{n,2}(\mathit{\beta })+I_{n,3}(\mathit{\beta }),$$ (A.4)

Note that I_n,2(β) and I_n,3(β) are quadratic in β. Using techniques related to Müller and Stadtmüller (1993) and Fan and Huang (2005), following some tedious calculations, it follows that for each given β

$$I_{n,2}(\mathit{\beta })=I_{n,3}(\mathit{\beta })=O(c_n^2)=o_P(n^{-1/2}).$$ (A.5)

We now deal with the main term I_n,1(β). Using the model

$$y(t)=\mathbf{x}{(t)}^T{\mathit{\alpha }}_0(t)+\mathbf{z}{(t)}^T{\mathit{\beta }}_0+\epsilon (t)$$

and (A.1), we have

$$I_{n,1}(\mathit{\beta })=\frac{1}{n}\sum _{i=1}^n{\epsilon }_i^T{W}_i{\epsilon }_i-2{(\mathit{\beta }-{\mathit{\beta }}_0)}^T{\xi }_n+{(\mathit{\beta }-{\mathit{\beta }}_0)}^T{\mathrm{\sum }}_n(\mathit{\beta }-{\mathit{\beta }}_0).$$ (A.6)

The minimization of I_n,1 is given by

$${\widehat{\mathit{\beta }}}_0={\mathit{\beta }}_0+{\mathrm{\sum }}_n{\xi }_n,$$

where Σ_n and ξ_n are defined before Theorem 4.1. By the WLLN and CLT,

$$\sqrt{n}({\widehat{\mathit{\beta }}}_0-{\mathit{\beta }}_0)\stackrel{\mathcal{L}}{\to }N(0,{\mathbf{A}}^{-1}\mathbf{B}{\mathbf{A}}^{-1}),$$ (A.7)

where A and B are defined in Section 3.2. Finally, we apply the convexity lemma to show that

$$\sqrt{n}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)=\sqrt{n}{\mathrm{\sum }}_n^{-1}{\xi }_n+o_P(1).$$ (A.8)

This together with (A.7) proves the results. To show (A.8), first of all, by the convexity lemma, β̂ is a consistent estimator of β₀. From (A.4), we have

$$0={\stackrel{.}{I}}_{n,1}(\widehat{\mathit{\beta }})+{\stackrel{.}{I}}_{n,2}(\widehat{\mathit{\beta }})+{\stackrel{.}{I}}_{n,3}(\widehat{\mathit{\beta }})=2{\mathrm{\sum }}_n(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)-2{\xi }_n+{\stackrel{.}{I}}_{n,2}(\widehat{\mathit{\beta }})+{\stackrel{.}{I}}_{n,3}(\widehat{\mathit{\beta }}).$$

Since I₂(β) and I₃(β) are quadratic in β, it follows from (A.5) that

$${\stackrel{.}{I}}_{n,2}(\widehat{\mathit{\beta }})=o_P(n^{-1/2})\text{and}{\stackrel{.}{I}}_{n,3}(\widehat{\mathit{\beta }})=o_P(n^{-1/2}).$$

This completes the proof of Theorem 4.1.

Proof of (4.1)

Denote U = (Z₁ − X̃₁), and W₀ = cov(ε|X₁, Z₁). Define

$$\mathbf{D}={\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}{\mathbf{U}}^T{\mathbf{W}}_1{\mathbf{W}}_0^{1/2}-{\{E({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U})\}}^{-1}{\mathbf{U}}^T{\mathbf{W}}_0^{-1/2}$$

Then

$$\begin{array}{l}\mathbf{D}{\mathbf{D}}^T={\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}({\mathbf{U}}^T{\mathbf{W}}_1{\mathbf{W}}_0{\mathbf{W}}_1\mathbf{U}){\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}\\ -{\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U}){\{E({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U})\}}^{-1}\\ -{\{E({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U})\}}^{-1}({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U}){\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}\\ +{\{E({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U})\}}^{-1}({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U}){\{E({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U})\}}^{-1}.\end{array}$$

Since DD^T is nonnegative definite, we have

$$E(\mathbf{D}{\mathbf{D}}^T)={\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}E({\mathbf{U}}^T{\mathbf{W}}_1{\mathbf{W}}_0{\mathbf{W}}_1\mathbf{U}){\{E({\mathbf{U}}^T{\mathbf{W}}_1\mathbf{U})\}}^{-1}-{\{E({\mathbf{U}}^T{\mathbf{W}}_0^{-1}\mathbf{U})\}}^{-1}$$

is nonnegative definite. Hence,

$${\mathbf{A}}^{-1}\mathbf{B}{\mathbf{A}}^{-1}-{\mathbf{B}}_0^{-1}\ge 0$$

The equality holds if and only if D = 0, which occurs when $\mathbf{W}={\mathbf{W}}_0^{-1}$