Semiparametric regression of panel count data with informative terminal event

Abstract

We study a semiparametric model for robust analysis of panel count data with an informative terminal event. To explore the explicit effect of the terminal event on recurrent events of interest, we propose a conditional mean model for a reversed counting process anchoring at the terminal event. Treating the distribution function of the terminal event as a nuisance functional parameter, we develop a predicted least squares-based two-stage estimation procedure with the spline-based sieve estimation technique, and derive the convergence rate of the proposed estimator. Furthermore, overcoming the difficulties caused by the convergence rate slower than $1∕\sqrt{n}$, we establish the asymptotic normality for the estimator of the finite-dimensional parameter and a functional of the estimator of the infinite-dimensional parameter. The proposed method is evaluated through extensive simulation studies and illustrated with an application to the Longitudinal Healthy Longevity Survey study on elder people in China.

1. Introduction

In many longitudinal follow-up studies, the observations of recurrent events usually occur at some random discrete time points, and only the event counts between the adjacent observation times are possibly recorded. Such data are referred to as panel count data (Kalbfleisch and Lawless, 1985). We take the number of serious diseases in a dataset of the Chinese Longitudinal Healthy Longevity Survey (CLHLS) (Zeng et al., 2017) as an example. In this study, the population-based survey on individuals who were at least 65 years old started in 1998 followed by six other waves in 2000, 2002, 2005, 2008, 2011, and 2014. During this longitudinal survey, the individuals were reached out for the information of severe illness since the last survey. For each individual, the survey dates were different, and the occurrences of severe illness between two adjacent survey dates were recorded resulting in panel count data on the counting process of occurrences of severe illness.

There were many studies for analysis of panel count data. Using the isotonic regression, Sun and Kalbfleisch (1995) first investigated the nonparametric estimation for the mean function of the counting process with panel count data. Wellner and Zhang (2000) and Lu, Zhang and Huang (2007) proposed the nonparametric maximum pseudo-likelihood and maximum likelihood estimation procedures for the mean function. Considering the covariate effect, Zhang (2002) and Wellner and Zhang (2007) studied the semiparametric maximum pseudo-likelihood and maximum likelihood estimations with the unknown baseline function estimated by the step function. Introducing the monotone spline approximation, Lu, Zhang and Huang (2009) improved the convergence rate of the estimators proposed by Zhang (2002) and Wellner and Zhang (2007). Zhang (2006), Balakrishnan and Zhao (2009), Zhao and Sun (2011), and Zhao and Zhang (2017) considered the two-sample or multi-sample hypothesis test for the mean function of the counting process with panel count data. For the problem of variable selection with panel count data, Tong et al. (2009) and Zhang, Sun and Wang (2013) studied the regularized estimation with the non-concave penalty and the seamless-$L_0$ (SELO) penalty, respectively. Recently, the statistical estimation and analysis approaches with panel count data were summarized in Sun and Zhao (2013) and Chiou et al. (2019).

In the above studies of panel count data, researchers modeled the counting process in a forward manner with the event count being 0 at the starting time of the study. However, this approach is not practically convenient when one is particularly interested in exploring the recurrent events near an informative terminal event, such as the severe disease profiles near death as exemplified in CLHLS study. In the early studies, the frailty model (Sun, Tong and He, 2007; Sun et al., 2012; Zhao, Li and Sun, 2013a; 2013b; Zhou et al., 2017) was the most commonly used method to describe the effect of an informative terminal event. This model introduced a latent frailty variable to characterize the correlation between the counting process and the terminal event and supposed that they were conditionally independent given the latent variable. In the CLHLS study, elder people tended to encounter more serious diseases before death, and a high incidence of severe illness can, in turn, lead them to death. However, in the frailty model, the frailty, the latent random variable while useful to acknowledge the association between the recurrent event process and the terminal event, lacks an explicit interpretation of the relationship between them. Recently, to evaluate the rate of recurrent events prior to the terminal events, Chan and Wang (2010) considered the time-backward processes, which started at the terminal event time and counted in a time-going-backward way. Then their model can be used to directly illustrate the effect of the terminal event through an unspecified nonparametric function. Recognizing the special data structure, the backward model for longitudinal data with the starting time anchoring at the informative terminal event seems more relevant in such applications. Treating the terminal event time as a fixed effect covariate and following the backward model in Chan and Wang (2010), Li et al. (2017) studied the semiparametric model for longitudinal data with an informative terminal event, and Shen et al. (2021) further investigated the model with an additional effect of the number of observations. Kong et al. (2018) established the asymptotic properties of the backward semiparametric model for longitudinal data with an informative terminal event. Li et al. (2018) also treated the terminal event time in the longitudinal data as a covariate, and they proposed a more general two-dimensional nonparametric function to represent the effect of the terminal event. To the best of our knowledge, there was no study for the backward semiparametric model with panel count data. For the backward nonparametric model, Liu et al. (2022) extended the method of Kong et al. (2018) to model panel count data with an informative terminal event backwards. They proposed a two-stage spline-based sieve nonparametric maximum likelihood estimation procedure for the inference of underlying recurrent events with panel count data associated with an informative right-censored terminal event.

This paper studies a semiparametric regression model for panel count data with an informative terminal event, where the stochastic mechanism for the underlying counting process arising from recurrent events is completely unspecified. We use the monotone I-spline function to approximate the unknown mean function of the process, and propose a least squares-based two-stage estimation by treating the distribution of the terminal event as a nuisance functional parameter. In stage 1, we estimate the conditional distribution of the terminal event given covariates under a conventional semiparametric model such as the Cox model (Cox, 1972; Breslow, 1972). In stage 2, we construct a loss function based on predicted least squares for estimation of the reversed counting process model. Furthermore, we establish the asymptotic properties of the proposed two-stage estimator.

The rest of this paper is organized as follows. In Section 2, we introduce the semiparametric model and the loss function for model estimation. We establish the consistency, the convergence rate, and the asymptotic normality of the proposed estimator in Section 3. Simulation and application results are reported in Sections 4 and 5, respectively. Some concluding remarks are made in Section 6. All technical proofs are given in the Appendix.

2. Model setting and estimation procedure

Denote the number of recurrent events of interest occurred up to time $t$ by the counting process $\{N(t):0\le t\le \tau \}$, where $\tau$ is a fixed time point. Set $K$ to be the total number of observations and $\mathbf{T}=(T_1,\dots ,T_K)$ to be the observation times of $N(t)$. Then the observed panel counts on the counting process are represented by $\mathbf{N}=(N(T_1),\dots ,N(T_K))$. Let $U$ and $C$ denote the terminal event time and the censoring time, respectively. We can only observe $Y=U\wedge C$ and $\Delta =1_{\{U\subset C\}}$. Let $\mathbf{Z}$ be a covariate vector associated with recurrent events and the terminal event. Let $X=(Y,\Delta ,K,\mathbf{T},\mathbf{N},\mathbf{Z})$. Then for a panel count data study with $n$ subjects, the observed data consist of $\mathbf{X}=\{X_i,i=1,\dots ,n\}$, where $X_i=(Y_i,{\Delta }_i,K_i,{\mathbf{T}}_i,{\mathbf{N}}_i,{\mathbf{Z}}_i)$ with ${\mathbf{T}}_i=(T_{i,1},\dots ,T_{i,K_i})$ and ${\mathbf{N}}_i=\{N_i(T_{i,1}),\dots ,N_i(T_{i,K_i})\}$ for $i=1,\dots ,n$.

Setting the number of recurrent events from time $t$ to the terminal event $U$ to be $\tilde{N}(t;U)$, we suppose that given the covariate and the terminal event time, the conditional expectation of $\tilde{N}(t;U)$ is

$$E(\tilde{N}(t;U)\mid U=u,\mathbf{Z}=\mathbf{z})=e^{{\beta }^T\mathbf{z}}\Lambda (u-t),0\le t\le u\le \tau ,$$ (1)

where $\Lambda$ is a non-negative and non-decreasing function with $\Lambda (0)=0$. Noting that $N(t_2)-N(t_1)=\tilde{N}(t_1;U)-\tilde{N}(t_2;U)$ for all $0\le t_1\le t_2\le u\le \tau$, then

$$E((N(t_2)-N(t_1))\mid U=u,\mathbf{Z}=\mathbf{z})=e^{{\beta }^T\mathbf{z}}(\Lambda (u-t_1)-\Lambda (u-t_2)).$$ (2)

Suppose that the conditional distribution function of $U$ given $\mathbf{Z}$ satisfies the Cox model

$$F(u\mid \mathbf{Z}=\mathbf{z})=P(U\le u\mid \mathbf{Z}=\mathbf{z})=1-e^{-H(u)e^{{\gamma }^T\mathbf{z}}},$$ (3)

where $H(u)$ is the baseline cumulative hazard function of $U$. Then the unknown parameters and functions to be estimated under models (1) and (3) are ($\beta$, $\Lambda$; $H$, $\gamma$). Although the covariates associated with the terminal event may not be the same as the covariates associated with the counting process, we would like to include all the potential covariates $\mathbf{Z}$ for exploring their effects on both the counting process and terminal events and for the sake of easy notation. That is, we denote the union of covariates associated with the counting process and the terminal event by $\mathbf{Z}$. In models (1) and (3), the coefficients for unrelated covariates are 0. We need the following basic assumptions before the analysis: (i) given $\mathbf{Z}$, $C$ and $U$ are independent; (ii) given $\mathbf{Z}$, $C$ is noninformative to $\Lambda$; and (iii) given ($Y$, $\Delta$, $\mathbf{Z}$), ($K$, $\mathbf{T}$) is noninformative to $\Lambda$. In the following, we let $𝒫$ and ${\mathbb{P}}_n$ denote the probability measure and the empirical measure, respectively.

Set $\Delta N_j=N(T_j)-N(T_{j-1})$ and $\Delta {\Lambda }_j(u)=\Lambda (u-T_{j-1})-\Lambda (u-T_j)$ for $j=1,\dots ,K$ with $T_0=0$. To use the least squares approach, we consider

$${\mathbb{P}}_n\left[\sum _{j=1}^K{\left\{\Delta N_j-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)\right\}}^2\right]=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{K_i}{\left\{\Delta N_{i,j}-e^{{\beta }^T{\mathbf{Z}}_i}\Delta {\Lambda }_{i,j}(U_i)\right\}}^2.$$

However, some $U_i$ are unknown due to censoring. We turn to consider the predicted least squares as a loss function. Define

$$\begin{gathered} m(\beta ,\Lambda ,F;X)≔E\left[\sum _{j=1}^K{\left\{\Delta N_j-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)\right\}}^2\mid Y,\Delta ,K,\mathbf{T},\mathbf{N},\mathbf{Z}\right] \\ =\sum _{j=1}^K\left[\Delta {\left\{\Delta N_j-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(Y)\right\}}^2+\frac{1-\Delta }{1-F(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{\left\{\Delta N_j-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)\right\}}^{2}dF(u\mid \mathbf{Z})\right]. \end{gathered}$$ (4)

By (4), we propose the empirical predicted least squares-based loss function as

$$\begin{gathered} {\ell }_n(\beta ,\Lambda ,F;\mathbf{X})={\mathbb{P}}_{n}m(\beta ,\Lambda ,F;X)=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{K_i}{\Delta }_i{\left\{\Delta N_{i,j}-e^{{\beta }^T{\mathbf{Z}}_i}\Delta {\Lambda }_{i,j}(Y_i)\right\}}^2 \\ +\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{K_i}\frac{1-{\Delta }_i}{1-F(Y_i\mid {\mathbf{Z}}_i)}{\int }_{Y_i}^{\infty }{\left\{\Delta N_{i,j}-e^{{\beta }^T{\mathbf{Z}}_i}\Delta {\Lambda }_{i,j}(u)\right\}}^{2}dF(u\mid {\mathbf{Z}}_i), \end{gathered}$$ (5)

where $\Delta N_{i,j}=N_i(T_{i,j})-N_i(T_{i,j-1})$ and $\Delta {\Lambda }_{i,j}(Y_i)=\Lambda (Y_i-T_{i,j-1})-\Lambda (Y_i-T_{i,j})$.

Replacing $F(u\mid {\mathbf{Z}}_i)$ by $1-\exp \{-H(u)\exp ({\gamma }^T{\mathbf{Z}}_i)\}$, a reasonable estimator is the minimizer of the loss function (5). Nevertheless, since the loss function consists of the unknown distribution function $F$, it is difficult to obtain the minimizer directly. Then we consider the two-stage estimation procedure by treating $F$ as the nuisance functional parameter. In stage 1, we estimate $\gamma$ and $H$ by the partial likelihood estimator ${\widehat{\gamma }}_n$ and the Breslow estimator ${\widehat{H}}_n$ (Breslow, 1972), respectively. Then we obtain the estimator of the conditional distribution function of the terminal event $\widehat{F}(u\mid \mathbf{z})=1-\exp \{-{\widehat{H}}_n(u)\exp ({\widehat{\gamma }}_n^T\mathbf{z})\}$. In stage 2, replacing $F$ by ${\widehat{F}}_n$ in (5), (${\widehat{\beta }}_n$, ${\widehat{\Lambda }}_n$) is obtained by minimizing the loss function ${\ell }_n(\beta ,\Lambda ,{\widehat{F}}_n;\mathbf{X})$.

To estimate $\Lambda$, we use the monotone I-spline function approximation. To this end, we divide [0, $\tau$] into $m_n+1$ subintervals by $0=t_1=\cdots =t_d<t_{d+1}<\cdots <t_{m_n+d}<t_{m_n+d+1}=\cdots =<t_{m_n+2d}=\tau$ with knots $\{t_i:i=1,\cdots ,m_n+2d\}$, where $d$ represents the order of I-spline functions. Let the I-spline basis functions be $\{I_l(s),l=1,\cdots ,q_n\}$, where $q_n=m_n+d$. Then we define the functional space for $\Lambda$:

$${\Phi }_n=\left\{\sum _{l=1}^{q_n}{\alpha }_l{I}_l(s):{\alpha }_l\ge 0,l=1,\cdots ,q_n\right\}.$$

Define $\mathbf{I}(s)={(I_1(s),\dots ,I_{q_n}(s))}^T$ and $\alpha ={({\alpha }_1,\cdots ,{\alpha }_{q_n})}^T$, and replace $\Lambda (s)$ by $\mathbf{I}{(s)}^T\alpha$. Then we can minimize the loss function by the constrained BFGS algorithm (Lange, 2001). Setting the minimizer of the loss function to be $({\widehat{\beta }}_n,{\widehat{\alpha }}_n)$, the spline estimator of $\Lambda (s)$ is ${\widehat{\Lambda }}_n(s)=\mathbf{I}{(s)}^T{\widehat{\alpha }}_n$.

3. Asymptotic properties

In this section, we establish the asymptotic properties of $({\widehat{\beta }}_n,{\widehat{\Lambda }}_n)$. First, we define the following function classes

$$\begin{gathered} {\mathscr{H}}_r=\left\{g:\mid g^{(r-1)}(s)-g^{(r-1)}(t)\mid \le c_0\mid s-t\mid \text{for all}0\le s,t\le \tau \right\}, \\ \Phi =\{\Lambda \in {\mathscr{H}}_r:\Lambda \text{is a nondecreasing continuous function on}[0,\tau ]\text{with}\Lambda (0)=0\}, \\ ℱ=\{F:F(\cdot \mid \mathbf{z})\text{is a distribution function on}[0,\infty )\text{for}\mathbf{z}\in 𝒵\}, \end{gathered}$$

where $g^{(r)}$ is the $r$ derivative of $g$ for $r\ge 1$ and $𝒵\subset {\mathbb{R}}^p$. For a bounded and convex set $ℛ\subset {\mathbb{R}}^p$, denote the interior of $ℛ$ by ${ℛ}^{\circ }$. Set $F_{\mathbf{Z}}$ to be the distribution function of $\mathbf{Z}$ with a bounded support $𝒵$, and $({\beta }_0,{\Lambda }_0,F_0)\in {ℛ}^{\circ }\times \Phi \times ℱ$ to be the true value of ($\beta$, $\Lambda$, $F$). Let $ℬ$ and ${ℬ}^p$ be the collection of Borel sets in $\mathbb{R}$ and ${\mathbb{R}}^p$, respectively. Then for $B_1$, $B_2\in {ℬ}_{[0,\tau ]}≔\{B\cap [0,\tau ]:B\in ℬ\}$ and $C\in {ℬ}^p$, we define

$$\begin{gathered} {\mu }_1(B_1\times B_2\times C)={\int }_C\int \sum _{k=1}^{\infty }P(K=k\mid U=u,\mathbf{Z}=\mathbf{z}) \\ \times \sum _{j=1}^{k}P\left((u-T_j)\in B_1,(u-T_{j-1})\in B_2\mid K=k,U=u,\mathbf{Z}=\mathbf{z}\right)d{F}_0(u\mid \mathbf{z})d{F}_{\mathbf{Z}}(\mathbf{z}), \\ {\mu }_2(B_1\times B_2)={\mu }_1(B_1\times B_2\times {\mathbb{R}}^p). \end{gathered}$$

Setting $\Delta \Lambda (s_1,s_2)=\Lambda (s_2)-\Lambda (s_1)$, for any functions ${\Lambda }_1$, ${\Lambda }_2\in \Phi$, we define the metric

$$\begin{gathered} d_1^2({\Lambda }_1,{\Lambda }_2)={\mid \mid \Delta {\Lambda }_1(s_1,s_2)-\Delta {\Lambda }_2(s_1,s_2)\mid \mid }_{L_2({\mu }_2)}^2=E\left[\sum _{j=1}^K{\left(\Delta {\Lambda }_{1,j}(U)-\Delta {\Lambda }_{2,j}(U)\right)}^2\right] \\ =E\left[\sum _{j=1}^K\left\{\Delta {(\Delta {\Lambda }_{1,j}(Y)-\Delta {\Lambda }_{2,j}(Y))}^2+\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{(\Delta {\Lambda }_{1,j}(u)-\Delta {\Lambda }_{2,j}(u))}^{2}d{F}_0(u\mid \mathbf{Z})\right\}\right]. \end{gathered}$$

For any functions $F_1$, $F_2\in ℱ$, we define the metric

$$d_2(F_1,F_2)=\underset{u,\mathbf{z}}{\text{sup}}\mid F_1(u\mid \mathbf{z})-F_2(u\mid \mathbf{z})\mid .$$

For any (${\beta }_1$, ${\Lambda }_1$) and (${\beta }_2$, ${\Lambda }_2$) in the space $ℛ\times \Phi$, we define the metric

$$d_3(({\beta }_1,{\Lambda }_1),({\beta }_2,{\Lambda }_2))={\{{\Vert {\beta }_1-{\beta }_2\Vert }_2^2+d_1^2({\Lambda }_1,{\Lambda }_2)\}}^{1∕2}.$$

To establish the asymptotic properties of the proposed estimator, we need the following regularity conditions.

(C1) $0<{\Lambda }_0(\tau )<\infty$.

(C2) The true values of $\gamma$ and $H$ satisfy ${\gamma }_0\in {ℛ}^{\circ }$ and $H_0(\tau )<\infty$, respectively. Furthermore, the derivative of $H_0(u)$ has a uniform positive lower bound for all $u\in [M_1,\tau ]$, where $M_1<\tau$ is a constant representing the minimum value of the support of $U$.

(C3) $E[{\{N(T_K)\}}^2]<\infty$.

(C4) The probability of censoring $\varrho =P(Y<U)$ satisfies that $0<\varrho <1$.

(C5) The measure ${\mu }_2\times F_{\mathbf{Z}}$ is absolutely continuous with respect to ${\mu }_1$.

(C6) $P({\mathbf{a}}^T\mathbf{Z}\ne c)>0$ for all $\mathbf{a}\ne 0\in {\mathbb{R}}^p$ and for all $c\in \mathbb{R}$.

(C7) There is a constant $M_2>0$ such that $P(K\le M_2)=1$.

(C8) The number of subinterval in [0, $\tau$] satisfies $m_n=O(n^{\nu })$ for $0<\nu <1∕2$. Furthermore,

$$\underset{d+1\le i\le m_n+d+1}{\max }\mid t_i-t_{i-1}\mid =O(n^{-\nu })\text{and}\frac{{\max }_{d+1\le i\le m_n+d+1}\mid t_i-t_{i-1}\mid }{{\text{min}}_{d+1\le i\le m_n+d+1}\mid t_i-t_{i-1}\mid }\le M_3,$$

uniformly for $n$ with a constant $M_3>0$.

(C9) $P(T_j-T_{j-1}\ge M_4)$ for all $j=1,\dots ,K)=1$ and $P(Y\ge M_4)=1$ with a constant $M_4>0$.

Remark 1. Conditions (C1) and (C3) are mild for the statistical analysis of panel count data. Conditions (C2) and (C4) are common in survival data analysis. According to Wellner and Zhang (2007), Conditions (C5) and (C6) are necessary for the identifiability of the semiparametric model. Condition (C7) indicates that the number of observations is bounded, which is standard in many applications for panel count data. Condition (C8) is a regularity condition for the spline approximation by Lu, Zhang and Huang (2007, 2009). By Wellner and Zhang (2007), Condition (C9) is regular in applications of panel count data, meaning that the adjacent observation times are separable.

Theorem 3.1 (Consistency). Suppose Conditions (C1)–(C9) hold. Then we have the following:

1. (${\beta }_0$, ${\Lambda }_0$) is the unique minimum of $𝒫m(\beta ,\Lambda ,F_0;X)$.

2. $d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))\to 0$ almost surely.

We need the following additional conditions to derive the convergence rate and establish asymptotic normality.

(C10) ${\inf }_{\mathbf{z}\in 𝒵}P(U\ge \tau \mid \mathbf{Z}=\mathbf{z})>0$ and $P(C\ge \tau )>0$.

(C11) ${\mu }_2$ is absolutely continuous with respect to Lebesgue measure with a derivative ${\stackrel{.}{\mu }}_2$, and ${\stackrel{.}{\mu }}_2$ has a uniform positive lower bound.

(C12) There is a constant $0<M_5<\infty$ such that $1∕M_5<{\Lambda }_0^{\prime }(s)<M_5$ for all $s\in [{\tau }^{\prime },\tau ]$, where $0<{\tau }^{\prime }\le \tau$ such that ${\Lambda }_0({\tau }^{\prime })>0$.

(C13) There is a sufficiently large constant $c$ such that $E[\exp (cN(\tau ))\mid \mathbf{Z}]$ is uniformly bounded for $\mathbf{Z}\in 𝒞$.

(C14) There is a constant $\eta \in (0,1)$ such that for all $\mathbf{a}\in {\mathbb{R}}^p$, we have ${\mathbf{a}}^T\mathrm{Var}(\mathbf{Z}\mid S_1,S_2)\mathbf{a}\ge \eta {\mathbf{a}}^{T}E(\mathbf{Z}{\mathbf{Z}}^T\mid S_1,S_2)\mathbf{a}$ a.e. for ($S_1$, $S_2$, $\mathbf{Z}$) having the distribution ${\mu }_1$.

Remark 2. By Kong et al. (2018), Condition (C10) is necessary for the uniform weak convergence rate of ${\widehat{F}}_n$ on a finite interval. According to Wellner and Zhang (2007) and Lu, Zhang and Huang (2009), Conditions (C11)–(C14) are common in analysis of panel count data.

Theorem 3.2 (Convergence Rate). Suppose that Conditions (C1)–(C14) hold. Then, taking $\nu =1∕(1+2r)$, we have $d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))=O_p(n^{-r∕(1+2r)})$.

Remark 3. Although the overall convergence rate of (${\widehat{\beta }}_n$, ${\widehat{\Lambda }}_n$) is slower than $1∕\sqrt{n}$, the convergence rate of ${\widehat{\beta }}_n$ is still $1∕\sqrt{n}$, and we can also find a functional of ${\widehat{\Lambda }}_n$ having the convergence rate $1∕\sqrt{n}$.

Theorem 3.3 (Asymptotic Normality). Suppose that Conditions (C1)–(C14) hold, and ${\Lambda }_0\in {\mathscr{H}}_r$ with $r\ge 2$. Then we have the following:

• (i)For all ${\mathbf{h}}_1\in ℛ$ and ${\mathbf{h}}_2\in {\mathscr{H}}_r$, we have

$$\sqrt{n}{R}_1({\mathbf{h}}_1,h_2)({\widehat{\beta }}_n-{\beta }_0)+\sqrt{n}{R}_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0)⇝N(0,{\sigma }_0{[{\mathbf{h}}_1,h_2]}^2),$$

where $R_1({\mathbf{h}}_1,h_2)({\widehat{\beta }}_n-{\beta }_0)$, $R_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0)$, and ${\sigma }_0{[{\mathbf{h}}_1,h_2]}^2$ are defined in the Appendix.

• (ii)Furthermore, we have

$$\sqrt{n}({\widehat{\beta }}_n-{\beta }_0)⇝N(0,{(A^{\ast })}^{-1}{B}^{\ast }{({(A^{\ast })}^{-1})}^T),$$

where $A^{\ast }$ and $B^{\ast }$ are defined in the Appendix.

The result of the theorem can be used for making statistical inference of covariate effects on the recurrent event process.

4. Simulation studies

In this section, we conducted some simulation studies to evaluate the finite-sample performance of the proposed method. We generated the covariate vector ${\mathbf{Z}}_i={(Z_{i1},Z_{i2})}^T$ by $Z_{i1}\sim \mathrm{Unif}(0,1)$ and $Z_{i2}\sim \text{Bernoulli}(0.5)$. Given the covariate vector ${\mathbf{Z}}_i$, the terminal event $U_i$, satisfied model (3) with ${\gamma }_0={({\gamma }_1,{\gamma }_2)}^T={(-1,-2)}^T$ and $H_0(u)=u^2-6$ for $u\in [6,\infty )$. The censoring time $C_i$ was generated from the Cox model with covariates ${\mathbf{Z}}_i$ along with the coefficients ${(\kappa ,2\kappa )}^T$ and the baseline cumulative hazard function $c-6$ for $c\in [6,\infty )$, where $\kappa$ was −1.611 and −0.594 to yield 20% and 40% censoring rate, respectively. Then we had $Y_i=U_i\wedge C_i$ and ${\Delta }_i=1_{\{U_i\le C_i\}}$ with $\tau =10$. For the observation time process, we generated a sequence of independent times $\Delta T_{ij}\sim \mathrm{Unif}(0.1,3)$ for $j=1,2,\cdots$, and $K_i$ was the maximum number of $k$ such that $T_{ik}={\sum }_{j=1}^k\Delta T_{ij}\le Y_i$. Then we obtained the observation time points $\{T_{i1},\cdots ,T_{i{K}_i}\}$. Under model (1), we considered the following two different cases of ${\Lambda }_0$

$$\text{Case}1:{\Lambda }_0(s)=s\text{and Case}2:{\Lambda }_0(s)=\frac{10s}{s+1}$$

to generate the counting process ${\mathbf{N}}_i=\{N_i(T_{i1}),\cdots ,N_i(T_{i{K}_i})\}$ from the Poisson process with ${\beta }_0={({\beta }_1,{\beta }_2)}^T={(1,1.5)}^T$. That is $N_i(T_{i1})$ was generated from the Poisson distribution with mean $\{{\Lambda }_0(U_i)-{\Lambda }_0(U_i-T_{i1})\}\exp ({\beta }_0{\mathbf{Z}}_i^T)$ was generated from the Poisson distribution with mean $\{{\Lambda }_0(U_i-T_{i(j-1)})-{\Lambda }_0(U_i-T_{ij})\}\exp ({\beta }_0{\mathbf{Z}}_i^T)$ for $j=2,\cdots ,K_i$. For the knots of the I-spline basis functions, we set $t_{d+1},\cdots ,t_{d+m_n}$ to be the $1∕(m_n+1),\cdots ,m_n(m_n+1)$ percentiles of $\{Y_i-T_{ij}\}:j=1,\cdots ,K_i;i=1,\cdots ,n$ with $d=4$ and $m_n=[n^{1∕3}]$. Since it was difficult to empirically estimate the asymptotic variance given in Theorem 3.3, the standard error of the proposed estimate of regression parameter ${\beta }_0$ was estimated based on 100 bootstrap samples. The initial value of the BFGS iteration was taken as $\alpha =1_{q_n}$ and $\beta =0$, where $1_{q_n}$ was the $q_n$-dimensional vector with elements 1. The simulation results were summarized based on 500 replications with sample size $n=100$ and 200.

For comparison purposes, we also applied the forward proportional mean model with panel count data (Wellner and Zhang, 2007; Lu, Zhang and Huang, 2009). We implemented the maximum pseudolikelihood spline (MPLS) and the maximum likelihood spline (MLS) by “panelReg” with methods “MPLs” and “MLs” in the R package “spef”.

We show the estimation results of $\Lambda$ under our reversed mean model for Case 1 in Figure 1. The plots for Case 2 are similar, and we show them in Figure 2. In those figures, the dash lines display the averages of the estimated functions, the solid lines are the true value ${\Lambda }_0$ for comparison, and the dotted-dash lines are the 2.5% and 97.5% pointwise percentiles of the estimated functions, which reveal the uncertainty of the estimated functions. We can see that the averages of the estimated functions are close to ${\Lambda }_0$, meaning that ${\widehat{\Lambda }}_n$ is consistent. The simulation results for the regression parameter $\beta$ based on MPLS, MLS, and our proposed model are summarized in Table 1. Note that in Case 1, we set ${\Lambda }_0(s)=s$, for which the proposed model and the forward proportional mean model with panel count data (Wellner and Zhang, 2007; Lu, Zhang and Huang, 2009) are essentially the same. Hence, Table 1 shows that MPLS and MLS were valid for Case 1 as expected and had slightly better estimation efficiency than the proposed method due to the fact that no estimation for the conditional distribution function was needed. It is also interesting to note that for Case 1, the censoring rate does not seem to impact the estimation results of the counting process much. We believe it is due to the fact that our model is equivalent to the forward proportional mean model with no effect of the terminal event being considered, for which the censoring rate for the terminal event is not even relevant. However, for Case 2, the forward proportional mean model was misspecified, which resulted in biased inferences for MPLS and MLS. In addition, Table 1 also shows that the biases of our estimates are small and the sample standard deviations (SSD) are close to the estimated standard errors (ESE). Both of them decrease as the sample size increases, and they also decrease as the censoring rate decreases in Case 2. The empirical coverage probabilities (CP) of the 95% Wald confidence intervals are close to 0.95. The simulation studies provide the numerical evidence to support the asymptotic properties depicted in Section 3. It appears that the inference based on the asymptotic theory for our proposed method is valid in finite sample with moderate sample size, say $n>100$.

Figure 1.

Simulation results for $\Lambda$ in Case 1. The solid lines are the true functions, the dash lines are the mean of the estimates, and the dotted-dash lines are the 2.5% and 97.5% pointwise percentiles.

Figure 2.

Simulation results for $\Lambda$ in Case 2. The solid lines are the true functions, the dash lines are the mean of the estimates, and the dotted-dash lines are the 2.5% and 97.5% pointwise percentiles.

Table 1.

Simulation results for the estimation of parameter $\beta$.

	Censoring rate = 20%			Censoring rate = 40%
	Proposed	MPLS	MLS	Proposed	MPLS	MLS
	Case 1, $n=100$
Bias	(0.001,0.001)	(−0.001,−0.003)	(0.001,0.001)	(−0.001,0.001)	(−0.001,−0.001)	(−0.002,0.002)
SSD	(0.079,0.051)	(0.078,0.052)	(0.067,0.045)	(0.084,0.057)	(0.079,0.058)	(0.071,0.051)
ESE	(0.075,0.054)	(0.073,0.055)	(0.065,0.049)	(0.076,0.055)	(0.074,0.055)	(0.066,0.049)
CP	(0.934,0.946)	(0.916,0.940)	(0.938,0.956)	(0.910,0.932)	(0.922,0.914)	(0.922,0.906)
	Case 1, $n=200$
Bias	(0.006,0.001)	(0.006,0.001)	(0.005,−0.001)	(0.001,0.001)	(0.001,−0.001)	(0.001,−0.001)
SSD	(0.052,0.038)	(0.053,0.039)	(0.047,0.034)	(0.052,0.037)	(0.050,0.041)	(0.045,0.036)
ESE	(0.052,0.038)	(0.050,0.039)	(0.045,0.035)	(0.054,0.039)	(0.052,0.039)	(0.046,0.035)
CP	(0.954,0.952)	(0.928,0.948)	(0.938,0.942)	(0.966,0.932)	(0.948,0.912)	(0.962,0.932)
	Case 2, $n=100$
Bias	(0.003,0.015)	(−0.148,−0.256)	(−0.172,−0.305)	(−0.002,0.007)	(−0.155,−0.256)	(−0.176,−0.300)
SSD	(0.111,0.080)	(0.176,0.102)	(0.191,0.098)	(0.140,0.089)	(0.175,0.101)	(0.193,0.101)
ESE	(0.106,0.078)	(0.180,0.099)	(0.193,0.101)	(0.131,0.087)	(0.184,0.100)	(0.197,0.102)
CP	(0.922,0.922)	(0.870,0.272)	(0.860,0.156)	(0.936,0.940)	(0.852,0.286)	(0.842,0.184)
	Case 2, $n=200$
Bias	(0.001,0.012)	(−0.141,−0.260)	(−0.166,−0.308)	(0.010,0.014)	(−0.135,−0.248)	(−0.160,−0.292)
SSD	(0.079,0.055)	(0.127,0.073)	(0.135,0.073)	(0.092,0.066)	(0.123,0.073)	(0.133,0.072)
ESE	(0.073,0.054)	(0.124,0.069)	(0.136,0.070)	(0.089,0.061)	(0.129,0.071)	(0.138,0.070)
CP	(0.932,0.932)	(0.784,0.066)	(0.762,0.018)	(0.938,0.916)	(0.818,0.080)	(0.774,0.018)

5. Application

In this section, we used the proposed semiparametric approach to analyze the incidence of serious diseases for elder people in China based on the datasets of the Chinese Longitudinal Healthy Longevity Survey (CLHLS) in the period 1998 to 2014 (Zeng et al., 2017). The CLHLS was conducted by the Center for Healthy Aging and Development Studies (CHADS) of the National School of Development at Peking University and the Chinese Center for Disease Control and Prevention (CDC), starting in 1998 with 6 follow-up waves in 2000, 2002, 2005, 2008, 2011 and 2014. Aiming to provide a better understanding of the determinants of health and longevity, the CLHLS interviewed a large number of elder people in the 22 provinces of China, who were at least 65 years old at the interviews, and collected the information about their medical history, socioeconomic status, lifestyles, family and demographic profile.

In this study, for the $i\mathrm{th}$ elder person, we took the number of months from the date of the first survey to the date of the $j\mathrm{th}$ follow-up wave of survey to be $T_{ij}$ for $j=1,\cdots ,K_i$, where $K_i\le 6$ representing the number of follow-up surveys. $\tau =197$ is the longest follow-up time possibly occurred in this study. Denote the incidence of serious diseases for this subject before the $j\mathrm{th}$ follow-up survey to be $N(T_{ij})$, the incidence of serious diseases from the $j\mathrm{th}$ follow-up survey to death to be $\tilde{N}(T_{ij})$, the terminal event time due to death to be $U_i$, and the censoring time due to loss-of-connection to be $C_i$. Then $Y_i=U_i\wedge C_i$ is the follow-up time, and ${\Delta }_i=1_{\{U_i\le C_i\}}$ the indicator of the observation of death.

We focused on the difference of the incidence of serious diseases between elders living in urban and rural. For this analysis, we considered 5 covariates that include three demographic variables: residence status ($Z_1=1$ for urban and $Z_1=0$ for rural), age ($Z_2$), and gender ($Z_3=1$ for male and $Z_3=0$ for female); and two clinical variables: indicator of hypertension ($Z_4=1$ for systolic blood pressure ≥ 140 mmHg and $Z_4=0$ for others), and peak lung flow ($Z_5$) at the first interview. We chose the individuals who had at least one follow-up survey. Hence a total of 4831 individuals interviewed in both 1998 and 2000 were selected for analysis. After removing 1099 individuals with missing or erroneous records, and 1160 individuals who had lived in both areas during the study period, we finally included 2572 individuals in the analysis, among which 73.7% had the terminal event, death. Table 2 shows the number of elders with different categories of age, gender, blood pressure, and peak lung flow stratified by urban and rural, respectively. In this table, the p-values of ${𝒳}^2$ tests reveal that the age of elders living in urban is different from elders living in rural at significance level 0.01; and the gender, the hypertension, and the degree of peak lung flow are not different between elders living in urban and rural at significance level 0.01.

Table 2.

The number of participants with different types of covariates for different residence status.

	Total	Age				Gender
	–	≤79	80-89	90-99	≥100	Male	Female
Urban	1152	15	636	381	120	507	645
Rural	1420	13	689	406	312	590	830
Total	2572	28	1325	787	432	1097	1475
p-value	–	< 0.001***				0.224
	Systolic Pressure		Peak Lung Flow
	≥140	≤139	≤99	100-199	200-299	300-399	≥400
Urban	820	332	373	484	232	53	10
Rural	1005	415	423	467	267	71	12
Total	1825	747	796	1131	499	124	22
p-value	0.856		0.406

^***represents significance level of 0.01.

Although the unit of peak lung flow ($Z_5$) was not specified in the dataset, it was clinically important because people with larger peak lung flow value generally have higher functional cardiorespiratory system capacity. We standardized the covariates $Z_2$ and $Z_5$ to put them on the same scale before the analysis. We considered the I-spline sieve estimation for $\Lambda (\cdot )$ with order $d=4$ and seven internal knots located at $t_{d+1}=\tau ∕8,\cdots t_{d+7}=7\tau ∕8$ for the I-spline basis functions. We chose the initial value $\alpha =1$ and $\beta =0$ in the BFGS algorithm. Similar to Section 4, we used MLS for comparison and obtained the bootstrap standard error of the proposed estimates of the regression parameters based on 100 bootstrap samples.

In Figure 3, the solid line represents the estimate of $\Lambda$ under our backward model. Table 3 summarizes the inferences on the covariates effects on the incidence of serious diseases under our reversed mean model and the MLS. Compared to the MLS method, for which only $Z_1$ showed a significant positive effect at the 0.01 level, our proposed model seemed to have a better power to pick out more statistically significant covariates. $Z_1$ is significant at the 0.05 level; $Z_3$ and $Z_5$ are marginally significant at the 0.1 level. Specifically, $Z_1$ has positive effect on the incidence of serious diseases, which may reflect the fact that people living in urban have a better opportunity to access advanced healthcare services than people living in rural that allow then to have more serious diseases identified. That $Z_3$ has negative effect on the incidence of serious diseases may be due to the longer lifetime in females. The positive effect of $Z_5$ is reasonable because of the higher functional cardiorespiratory system capacity for people with larger peak lung flow.

Figure 3.

Estimate of $\Lambda$ for the CLHLS data.

Table 3.

Inference results for the CLHLS data.

	Reversed Mean Model					MLS
	$Z_1$	$Z_2$	$Z_3$	$Z_4$	$Z_5$	$Z_1$	$Z_2$	$Z_3$	$Z_4$	$Z_5$
Estimates	0.246	−0.061	−0.175	−0.065	0.106	0.284	0.068	−0.073	0.106	0.040
ESE	0.100	0.063	0.101	0.122	0.056	0.077	0.044	0.076	0.083	0.039
p-value	0.014**	0.333	0.084*	0.595	0.059*	0.001***	0.130	0.330	0.200	0.310

^*represents a significant inference at level of 0.1

^**represents a significant inference at level of 0.05

^***represents a significant inference at level of 0.01.

6. Concluding remarks

For analyzing complex panel count data with an informative terminal event, we proposed a reversed mean model to depict its explicit relationship with recurrent events. For estimating unknown parameters of the proposed model, we developed a two-stage spline-based sieve estimation procedure to reduce the computation burden. Overcoming the theoretical challenges from the estimator having the overall convergence rate slower than the standard rate, we established the joint asymptotic normality for a functional of the estimator, and further concluded that the finite-dimensional estimator still achieves the standard convergence rate and is asymptotically normal.

Note that the proposed estimation procedure is robust in the sense that the stochastic mechanism of the recurrent event process is completely unspecified. When the underlying counting process is a Poisson-type process, we can use the maximum likelihood approach to improve the estimation efficiency. Since the likelihood function in this situation is much more complicated, extra efforts are needed to study the theoretical properties, which are currently under investigation. Though Cox model (3) and Breslow-induced estimator ${\widehat{F}}_n(t\mid \mathbf{Z})$ were adopted in our implementation for stage 1 due to their popularity and good asymptotic properties, they are not the only choice. Indeed, our theories only require that the estimator of the conditional distribution function of the terminal event time can be asymptotically represented by the sum of a series of i.i.d. terms, such as the representation in Lemma 1. Then the asymptotic properties for ${\widehat{\Lambda }}_n$ and ${\widehat{\beta }}_n$ in Theorems 3.1-3.3 still hold.

The proposed method focuses on modeling the data with some conditions that the observation times are independent of the recurrent events and the covariates are time-independent. Though these conditions were commonly adopted in analysis for panel count data, the use of the proposed method is somehow restricted in view of real-world applications. A further direction is to consider the informative observation times and time-dependent covariates for analysis of panel count data with an informative terminal event.

Appendix: Proofs of main results

A.1. Proof of Theorem 3.1

Proof. (i) We show that (${\beta }_0$, ${\Lambda }_0$) is the unique minimum of ${𝒫}_m(\beta ,\Lambda ,F_0;X)$. After some algebraic calculations, we have

$$𝒫m(\beta ,\Lambda ,F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X)=𝒫\left[\sum _{j=1}^K{\left\{e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)\right\}}^2\right]\ge 0.$$

It follows that ${𝒫}_m(\beta ,\Lambda ,F_0;X)\ge {𝒫}_m({\beta }_0,{\Lambda }_0,F_0;X)$, and ${𝒫}_m(\beta ,\Lambda ,F_0;X)={𝒫}_m({\beta }_0,{\Lambda }_0,F_0;X)$ if and only if $\Delta \Lambda (s_1,s_2)\exp ({\beta }^T\mathbf{z})=\Delta {\Lambda }_0(s_1,s_2)\exp ({\beta }_0^T\mathbf{z})$ a.e. with respect to ${\mu }_1$. This implies that $(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))\exp ({\beta }^T\mathbf{z})=\Delta {\Lambda }_0(s_1,s_2)(\exp ({\beta }_0^T\mathbf{z}-\exp ({\beta }^T\mathbf{z}))$ a.e. with respect to ${\mu }_1$. By Condition (C5), ${\mu }_2\times F_{\mathbf{Z}}$ is absolutely continuous with respect to ${\mu }_1$. Using Fubini’s theorem, we obtain

$$\begin{gathered} \int \{a(s_1,s_2)(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))\}d{\mu }_2\int \left\{b(\mathbf{z})e^{{\beta }^T\mathbf{z}}\right\}d{F}_{\mathbf{Z}} \\ =\int \{a(s_1,s_2)\Delta {\Lambda }_0(s_1,s_2)\}d{\mu }_2\int \left\{b(\mathbf{z})\left(e^{{\beta }_0^T\mathbf{z}}-e^{{\beta }^T\mathbf{z}}\right)\right\}d{F}_{\mathbf{Z}}, \end{gathered}$$

for all ${\mu }_2$ measurable function $a(s_1,s_2)$ and $F_{\mathbf{Z}}$ measurable function $b(\mathbf{z})$. Taking $a(s_1,s_2)=(\Delta \Lambda (s_1,s_2)-{\Delta }_0\Lambda (s_1,s_2))1_A$ and $b(\mathbf{z})=\exp ({\beta }^T\mathbf{z})1_B$ for $A\in {ℬ}_{[0,\tau ]}^2$ and $B\in {ℬ}^p$, we have

$$\begin{gathered} {\int }_A{(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))}^{2}d{\mu }_2{\int }_B{e}^{2{\beta }^T\mathbf{z}}d{F}_{\mathbf{Z}} \\ ={\int }_A\{(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))\Delta {\Lambda }_0(s_1,s_2)\}d{\mu }_2{\int }_B\left\{\left(e^{{\beta }_0^T\mathbf{z}}-e^{{\beta }^T\mathbf{z}}\right)e^{{\beta }^T\mathbf{z}}\right\}d{F}_{\mathbf{Z}}. \end{gathered}$$

Similarly, $a(s_1,s_2)=(\Delta {\Lambda }_0(s_1,s_2)1_A$ and $b(\mathbf{z})=(\exp ({\beta }_0^T\mathbf{z}))1_B$ yield that

$$\begin{gathered} {\int }_A\{\Delta {\Lambda }_0(s_1,s_2)(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))\}d{\mu }_2{\int }_B\left\{\left(e^{{\beta }_0^T\mathbf{z}}-e^{{\beta }^T\mathbf{z}}\right)e^{{\beta }^T\mathbf{z}}\right\}d{F}_{\mathbf{Z}} \\ ={\int }_A{(\Delta {\Lambda }_0(s_1,s_2))}^{2}d{\mu }_2{\int }_B{\left(e^{{\beta }_0^T\mathbf{z}}-e^{{\beta }^T\mathbf{z}}\right)}^{2}d{F}_{\mathbf{Z}}. \end{gathered}$$

Then for all the product sets $A\times B$, we obtain

$${\int }_{A\times B}{(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))}^2{e}^{2{\beta }^T\mathbf{z}}d{\mu }_2\times F_{\mathbf{Z}}={\int }_{A\times B}{(\Delta {\Lambda }_0(s_1,s_2))}^2{(e^{{\beta }_0^T\mathbf{z}}-e^{{\beta }^T\mathbf{z}})}^{2}d{\mu }_2\times F_{\mathbf{Z}}.$$

That is ${(\Delta \Lambda (s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2))}^2\exp (2{\beta }^T\mathbf{z})={(\Delta {\Lambda }_0(s_1,s_2))}^2{(\exp ({\beta }_0^T\mathbf{z})-\exp ({\beta }^T\mathbf{z}))}^2$ a.e. with respect to ${\mu }_2\times F_{\mathbf{Z}}$, which is equivalent to ${(\Delta \Lambda (s_1,s_2)∕\Delta {\Lambda }_0(s_1,s_2)-1)}^2={(\exp ({({\beta }_0-\beta )}^T\mathbf{z}-1))}^2$ a.e. with respect to ${\mu }_2\times F_{\mathbf{Z}}$. Intergrading the above equality with respect to ${\mu }_2$, we obtain that the right hand side is a constant a.e. with respect to $F_{\mathbf{Z}}$. Then Condition (C6) implies that $\beta ={\beta }_0$ and $\Delta \Lambda (s_1,s_2)=\Delta {\Lambda }_0(s_1,s_2)$ a.e. with respect to ${\mu }_2$.

(ii) To prove the consistency, we first show that ${\widehat{\lambda }}_n$ is uniformly bounded. By Lemma A1 of Lu, Zhang and Huang (2007), under Condition (C8), there is a ${\Lambda }_n^{\ast }\in {\Phi }_n$ such that ${\Vert {\Lambda }_n^{\ast }-{\Lambda }_0\Vert }_{\infty }=O(n^{-\nu r})$. Consider a direction vector ${\mathbf{h}}_{1,n}\in ℛ$ with ${\Vert {\mathbf{h}}_{1,n}\Vert }_2^2=O(n^{-a})$ for a constant $0<a<1∕2$ and a bounded positive monotone nondecreasing direction function $h_{2,n}\in {\Phi }_n$ with ${\Vert \Delta h_{2,n}(s_1,s_2)\Vert }_{L_2({\mu }_2)}^2=O(n^{-\nu r}+n^{-(1-\nu ∕2)})$, where $\Delta h_{2,n}(s_1,s_2)=h_{2,n}(s_2)-h_{2,n}(s_1)$. Then for any constant $\alpha >0$, we have ${\Vert \Delta {\Lambda }_n^{\ast }(s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2)+\alpha \Delta h_{2,n}(s_1,s_2)\Vert }_{L_2({\mu }_2)}^2=O(n^{-\nu r}+n^{-(1-\nu )∕2})$ and ${\inf }_{s_2-s_1\ge M_4}(\Delta {\Lambda }_n^{\ast }(s_1,s_2)-\Delta {\Lambda }_0(s_1,s_2)+\alpha \Delta h_{2,n}(s_1,s_2))>0$ for sufficiently large $n$ with $M_4$ defined in Condition (C9). By some direct calculations,

$$\stackrel{.}{m}(\alpha )=\partial m({\beta }_0+\alpha {\mathbf{h}}_{1,n},{\Lambda }_n^{\ast }+\alpha h_{2,n},{\widehat{F}}_n;X)∕\partial \alpha =-2\psi ({\beta }_0+\alpha {\mathbf{h}}_{1,n},{\Lambda }_n^{\ast }+\alpha h_{2,n},{\widehat{F}}_n;X)[{\mathbf{h}}_{1,n},h_{2,n}],$$

where

$$\begin{gathered} \psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]=\sum _{j=1}^K[\Delta \{\Delta N_j-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(Y)\}{e}^{{\beta }^T\mathbf{Z}}\{\Delta {\Lambda }_j(Y){\mathbf{h}}_1^T\mathbf{Z}+\Delta h_{2,j}(Y)\}\phantom{]} \\ +\phantom{[}\frac{1-\Delta }{1-F(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\{\Delta N_j-e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)\}{e}^{{\beta }^T\mathbf{Z}}\{\Delta {\Lambda }_j(u){\mathbf{h}}_1^T\mathbf{Z}+\Delta h_{2,j}(u)\}dF(u\mid \mathbf{Z})]. \end{gathered}$$ (6)

Note that we obtain (${\widehat{\beta }}_n$, ${\widehat{\Lambda }}_n$) by minimizing ${\mathbb{P}}_{n}m(\beta ,\Lambda ,{\widehat{F}}_n;X)$ under the constraint that $(\beta ,\Lambda )\in ℛ\times {\Phi }_n$. Then we can verify $d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))=o_p(1)$ by showing that ${\mathbb{P}}_n\stackrel{.}{m}(\alpha )>0$ and ${\mathbb{P}}_n\stackrel{.}{m}(-\alpha )<0$ for any constant $\alpha >0$. Similar to Lemma 3, we have

$$\{\psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]:(\beta ,\Lambda )\in ℛ\times \Phi ,F\in ℱ,({\mathbf{h}}_1,h_2)\in ℛ\times \Phi ,\Lambda \text{and}{h}_2\text{are uniformly bounded}\}$$

is Donsker. Therefore, we obtain $({\mathbb{P}}_n-𝒫)\stackrel{.}{m}(\alpha )=O_p(1∕\sqrt{n})$. Furthermore, Lemma 2 implies that

$$\begin{gathered} 𝒫\stackrel{.}{m}(\alpha )\gtrsim -2𝒫\psi ({\beta }_0+\alpha {\mathbf{h}}_{1,n},{\Lambda }_n^{\ast }+\alpha h_{2,n},F_0;X)[{\mathbf{h}}_{1,n},h_{2,n}]-d_2({\widehat{F}}_n,F_0) \\ =-2𝒫[\sum _{j=1}^K\{\left(e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)-e^{{({\beta }_0+\alpha {\mathbf{h}}_{1,n})}^T\mathbf{Z}}\left(\Delta {\Lambda }_{n,j}^{\ast }(U)+\alpha \Delta h_{2,n,j}(U)\right)\right)\phantom{\}}\phantom{]} \\ \times e^{{({\beta }_0+\alpha {\mathbf{h}}_{1,n})}^T\mathbf{Z}}\phantom{[}\phantom{\{}\left(\left(\Delta {\Lambda }_{n,j}^{\ast }(U)+\alpha \Delta h_{2,n,j}(U)\right){\mathbf{h}}_{1,n}^T\mathbf{Z}+\Delta h_{2,n,j}(U)\right)\}]-d_2({\widehat{F}}_n,F_0) \\ =2𝒫\left[\sum _{j=1}^K\left\{(b(1)-b(0))e^{{({\beta }_0+\alpha {\mathbf{h}}_{1,n})}^T\mathbf{Z}}\left(\left(\Delta {\Lambda }_{n,j}^{\ast }(U)+\alpha \Delta h_{2,n,j}(U)\right){\mathbf{h}}_{1,n}^T\mathbf{Z}+\Delta h_{2,n,j}(U)\right)\right\}\right] \\ -d_2({\widehat{F}}_n,F_0), \end{gathered}$$

where $b(\xi )=\exp ({({\beta }_0+\xi \alpha {\mathbf{h}}_{1,n})}^T\mathbf{Z})(\Delta {\Lambda }_{0,j}(U)+\xi (\Delta {\Lambda }_{n,j}^{\ast }(U)-\Delta {\Lambda }_{0,j}(U)+\alpha \Delta h_{2,n,j}(U)))$, and the notation $c_1\gtrsim c_2$ means that $c_1\ge c{c}_2$ for a constant $c$. By the mean value theorem, there exists a $0<{\xi }^{\ast }<1$ such that $b(1)-b(0)=b^{\prime }({\xi }^{\ast })$, where

$$\begin{gathered} b^{\prime }(\xi )=e^{{({\beta }_0+\xi \alpha {\mathbf{h}}_{1,n})}^T\mathbf{Z}}\left(\Delta {\Lambda }_{n,j}^{\ast }(U)-\Delta {\Lambda }_{0,j}(U)+\alpha \Delta h_{2,n,j}(U)\right) \\ +e^{{({\beta }_0+\xi \alpha {\mathbf{h}}_{1,n})}^T\mathbf{Z}}\alpha {\mathbf{h}}_{1,n}^T\mathbf{Z}\left(\Delta {\Lambda }_{0,j}(U)+\xi \left(\Delta {\Lambda }_{n,j}^{\ast }(U)-\Delta {\Lambda }_{0,j}(U)+\alpha \Delta h_{2,n,j}(U)\right)\right). \end{gathered}$$

Since ${\Lambda }_0$, ${\Lambda }_n^{\ast }$, and $h_{2,n}$ are bound on $[0,\tau ]$, and ${\beta }_0$ and ${\mathbf{h}}_{1,n}$ are bounded vectors, it follows that $b^{\prime }({\xi }^{\ast })\gtrsim (\Delta {\Lambda }_{n,j}^{\ast }(U)-\Delta {\Lambda }_{0,j}(U)+\alpha \Delta h_{2,n,j}(U))+{\mathbf{h}}_{1,n}^T\mathbf{Z}$. Thus,

$$\begin{gathered} 𝒫\stackrel{.}{m}(\alpha )\ge c𝒫\left[\sum _{j=1}^K\left\{\left((\Delta {\Lambda }_{n,j}^{\ast }(U)-\Delta {\Lambda }_{0,j}(U)+\alpha \Delta h_{2,n,j}(U))+{\mathbf{h}}_{1,n}^T\mathbf{Z}\right)\left({\mathbf{h}}_1^T\mathbf{Z}+\Delta h_{2,j}(U)\right)\right\}\right] \\ -d_2({\widehat{F}}_n,F_0)\ge cO(n^{-\nu r}+n^{-(1-\nu )∕2}+n^{-a})-d_2({\widehat{F}}_n,F_0), \end{gathered}$$

for a constant $c$. Note that $n^{-\nu r}+n^{-(1-\nu ∕2)}\ge n^{-r∕(1+2r)}>(1∕\sqrt{n}$, $0<a<1∕2$, and $d_2({\widehat{F}}_n,{\widehat{F}}_0)=O_p(1∕\sqrt{n})$. This yields that ${\mathbb{P}}_n\stackrel{.}{m}(\alpha )\ge ({\mathbb{P}}_n-𝒫)\stackrel{.}{m}(\alpha )+cO(n^{-\nu r}+n^{-(1-\nu )∕2}+n^{-a})-d_2({\widehat{F}}_n,F_0)>0$ with probability converging to one. We can similarly show that ${\mathbb{P}}_n\stackrel{.}{m}(-\alpha )<0$ except on an event with probability converging to zero. Therefore, for all $\epsilon >0$, we have $P(d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))>\epsilon )\to 0$ as $n\to \infty$. It follows that for any $∊>0$, there exists a measurable set $\Xi$ with $P(\Xi )\ge 1-∊$ such that ${\widehat{\Lambda }}_n(s)$ is uniformly bounded for $s\in [0,\tau ]$ on $\Xi$.

We restrict us on the measurable set $\Xi$ at the moment. By the Cauchy–Schwarz inequality, under Conditions (C1) and (C7), we have

$$\begin{gathered} 𝒫m(\beta ,\Lambda ,F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X) \\ =E\left[\sum _{j=1}^K{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)\right)}^2\right]\lesssim E\left[\sum _{j=1}^K{(\Delta {\Lambda }_j(U)-\Delta {\Lambda }_{0,j}(U))}^2\right] \\ +E\left[\sum _{j=1}^K{\left(e^{{\beta }^T\mathbf{Z}}-e^{{\beta }_0^T\mathbf{Z}}\right)}^2\right]+2{\left\{E\left[\sum _{j=1}^K{(\Delta {\Lambda }_j(U)-\Delta {\Lambda }_{0,j}(U))}^2\right]\right\}}^{\frac{1}{2}}{\left\{E\left[\sum _{j=1}^K{\left(e^{{\beta }^T\mathbf{Z}}-e^{{\beta }_0^T\mathbf{Z}}\right)}^2\right]\right\}}^{\frac{1}{2}} \\ \lesssim E\left[\sum _{j=1}^K{(\Delta {\Lambda }_j(U)-\Delta {\Lambda }_{0,j}(U))}^2\right]+E\left[{\left(e^{{\beta }^T\mathbf{Z}}-e^{{\beta }_0^T\mathbf{Z}}\right)}^2\right]. \end{gathered}$$

By the mean value theorem, there exists a ${\beta }_{\zeta }\in ℛ$ such that $E[{(\exp ({\beta }^T\mathbf{Z})-\exp ({\beta }_0^T\mathbf{Z}))}^2]=E[\exp (2{\beta }_{\zeta }^T\mathbf{Z}){\{{\mathbf{Z}}^T(\beta -{\beta }_0)\}}^2]\lesssim {\left|\right|\beta -{\beta }_0\left|\right|}_2^2$. It follows that

$$𝒫m(\beta ,\Lambda ,F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X)\lesssim {\Vert \beta -{\beta }_0\Vert }_2^2+d_1^2(\Lambda ,{\Lambda }_0)=d_3^2((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0)).$$

Furthermore, Note that ${𝒫}_m(\beta ,\Lambda ,F_0;X)-{𝒫}_m({\beta }_0,{\Lambda }_0,F_0;X)\ge 0$ with equality if and only if $\beta ={\beta }_0$ and $\Delta \Lambda (s_1,s_2)=\Delta {\Lambda }_0(s_1,s_2)$ a.e. with respect to ${\mu }_2$. Hence, for every $\delta >0$, there exists an $\epsilon >0$ such that $\{d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))\ge \delta \}\subset \{{𝒫}_m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0;X)-{𝒫}_m({\beta }_0,{\Lambda }_0,F_0;X)>\epsilon \}$. Note that

$$\begin{gathered} 0\le 𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X) \\ =𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0;X)-𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)+𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-{\mathbb{P}}_{n}m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X) \\ +{\mathbb{P}}_{n}m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-{\mathbb{P}}_{n}m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)+{\mathbb{P}}_{n}m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-𝒫m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X) \\ +𝒫m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-𝒫m({\beta }_0,{\Lambda }_n^{\ast },F_0;X)+𝒫m({\beta }_0,{\Lambda }_n^{\ast },F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X). \end{gathered}$$ (7)

According to Conditions (C2) and (C7), $0\le 𝒫m({\beta }_0,{\Lambda }_n^{\ast },F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X)\lesssim {\Vert {\Lambda }_n^{\ast }-{\Lambda }_0\Vert }_{\infty }^2=o(1)$. The definition of $({\widehat{\beta }}_n,{\widehat{\Lambda }}_n)$ yields that ${\mathbb{P}}_{n}m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)\le {\mathbb{P}}_{n}m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)$. By Lemma 2 of the online Supplementary Material (Hu et al., 2023), we have $𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)=o_p(1)$ and $𝒫m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-𝒫m({\beta }_0,{\Lambda }_n^{\ast },F_0;X)=o_p(1)$. By Lemma 3 of the online Supplementary Material (Hu et al., 2023), {$m(\beta ,\Lambda ,F;X):\beta \in ℛ$, $\Lambda \in \Phi$, $\Lambda$ is uniformly bounded, $F\in ℱ$, $d_2(F,F_0)\le \delta$} is Donsker, meaning that it is Glivenko-Cantelli. Noting that $d_2({\widehat{F}}_n,F_0)=O_p(n^{-1∕2})$, we have $({\mathbb{P}}_n-𝒫)m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)=o_p(1)$ and $({\mathbb{P}}_n-𝒫)m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)=o_p(1)$. Combining them with (7), we have $0\le 𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0;X)-𝒫m({\beta }_0,{\Lambda }_0,F_0;X)\le o_p(1)$. Therefore, $\{𝒫m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0;X)>𝒫m({\beta }_n,{\Lambda }_n,F_0;X)+\epsilon \}$ goes into a null set as $n\to \infty$. Then $({\widehat{\beta }}_n,{\widehat{\Lambda }}_n)\to ({\beta }_0,{\Lambda }_0)$ almost uniformly, recalling that the relation holds on the measurable set $\Xi$ with $P(\Xi )\ge 1-∊$. Thus, the almost sure convergence of (${\widehat{\beta }}_n$, ${\widehat{\Lambda }}_n$) follows by Lemma 1.9.2 of van der Vaart and Wellner (1996). □

A.2. Proof of Theorem 3.2

Proof. We use Lemma 5 of the online Supplementary Material (Hu et al., 2023) to prove the rate of convergence.

First, by some direct calculations, we have

$$\begin{gathered} 𝒫(m({\beta }_0,{\Lambda }_0,F;X)-m(\beta ,\Lambda ,F;X))=-E\left[\sum _{j=1}^K{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)\right)}^2\right] \\ +𝒫[\sum _{j=1}^K\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right)}^2\phantom{]}d{F}_0(u\mid \mathbf{Z}) \\ -\phantom{[}\sum _{j=1}^K\frac{1-\Delta }{1-F(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right)}^{2}dF(u\mid \mathbf{Z})]. \end{gathered}$$

The bound of $E[{\sum }_{j=1}^K{\{\exp ({\beta }^T\mathbf{Z})\Delta {\Lambda }_j(U)-\exp ({\beta }_0^T\mathbf{Z})\Delta {\Lambda }_{0,j}(u)\}}^2]$ can be assessed by the arguments similar to those for the proof of Theorem 3.2 in Wellner and Zhang (2007). For $\Lambda \in \Phi$, $\beta \in {\mathbb{R}}^p$ and $(S_1,S_2,\mathbf{Z})\sim {\mu }_1$, let $g(\xi )=\exp ({\beta }_0^T\mathbf{Z})\Delta {\Lambda }_{\xi }(S_1,S_2)$, where $\Delta {\Lambda }_{\xi }(S_1,S_2)=\xi \Delta \Lambda (S_1,S_2)+(1-\xi )\Delta {\Lambda }_0(S_1,S_2)$ and ${\beta }_{\xi }=\xi \beta +(1-\xi ){\beta }_0$ with $\xi \in (0,1)$. Then we have $\exp ({\beta }^T\mathbf{Z})\Delta \Lambda (S_1,S_2)-\exp ({\beta }_0^T\mathbf{Z})\Delta {\Lambda }_0(S_1,S_2)=g(1)-g(0)$. By the mean value theorem, there is a $\xi \in (0,1)$ such that

$$\begin{gathered} g(1)-g(0)=g^{\prime }(\xi )=e^{{\beta }_{\xi }^T\mathbf{Z}}[(\Delta \Lambda (S_1,S_2)-\Delta {\Lambda }_0(S_1,S_2))+\Delta {\Lambda }_{\xi }(S_1,S_2){(\beta -{\beta }_0)}^T\mathbf{Z}] \\ =e^{{\beta }_{\xi }^T\mathbf{Z}}\left[\left\{1+\frac{(\Delta \Lambda (S_1,S_2)-\Delta {\Lambda }_0(S_1,S_2))}{\Delta {\Lambda }_0(S_1,S_2)∕\xi }\right\}{(\beta -{\beta }_0)}^T\mathbf{Z}\Delta {\Lambda }_0(S_1,S_2)+(\Delta \Lambda (S_1,S_2)-\Delta {\Lambda }_0(S_1,S_2))\right], \end{gathered}$$

where $g^{\prime }$ is the derivative of $g$. Setting $g_1={(\beta -{\beta }_0)}^T\mathbf{Z}\Delta {\Lambda }_0(S_1,S_2)$, $g_2=(\Delta \Lambda (S_1,S_2)-\Delta {\Lambda }_0(S_1,S_2))$ and $g_3=1+\xi \left(\Delta \Lambda (S_1,S_2)-\Delta {\Lambda }_0(S_1,S_2)\right)∕\Delta {\Lambda }_0(S_1,S_2)$, we have $g(1)-g(0)=\exp ({\beta }_{\xi }^{T}Z)(g_1{g}_3+g_2)$ This yields that

$$E\left[\sum _{j=1}^K{\left\{e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)\right\}}^2\right]=E_{{\mu }_1}\left[{\{g(1)-g(0)\}}^2\right]\gtrsim E_{{\mu }_1}\left[{(g_1{g}_3+g_2)}^2\right].$$

Similar to the proof of Theorem 3.2 in Wellner and Zhang (2007), Condition (C14) implies that $E_{{\mu }_1}^2[g_1{g}_2]\le (1-\eta )E_{{\mu }_1}[{(g_1)}^2]E_{{\mu }_1}[{(g_2)}^2]$. According to Lemma 8.8 of van der Vaart (2002), we have $E_{{\mu }_1}[{(g_1{g}_3+g_2)}^2]\gtrsim E_{{\mu }_1}[{(g_1)}^2]+E_{{\mu }_1}[{(g_2)}^2]\gtrsim d_3^2((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))$. Therefore,

$$\begin{gathered} 𝒫(m({\beta }_0,{\Lambda }_0,F;X)-m(\beta ,\Lambda ,F;X))\lesssim -d_3^2((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0)) \\ +𝒫[\sum _{j=1}^K\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{\left\{e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right\}}^{2}d{F}_0(u\mid \mathbf{Z})\phantom{]} \\ -\phantom{[}\sum _{j=1}^K\frac{1-\Delta }{1-F(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{\left\{e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right\}}^{2}dF(u\mid \mathbf{Z})]. \end{gathered}$$

By Conditions (C1), (C2) and (C7), Cauchy–Schwarz inequality and Lemma 2 of the online Supplementary Material (Hu et al., 2023),

$$\begin{gathered} \mid 𝒫[\sum _{j=1}^K(1-\Delta )\{\frac{{\int }_Y^{\infty }{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right)}^{2}d{F}_0(u\mid \mathbf{Z})}{1-F_0(Y\mid \mathbf{Z})}\phantom{\}}\phantom{]}\phantom{\mid } \\ -\phantom{\mid }\phantom{[}\phantom{\{}\frac{{\int }_Y^{\infty }{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right)}^{2}dF(u\mid \mathbf{Z})}{1-F(Y\mid \mathbf{Z})}\}]\mid \lesssim \{𝒫\left[\sum _{j=1}^K{\left(e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)\right)}^2\right]\phantom{\}} \\ +𝒫\phantom{\{}\left[\sum _{j=1}^{K}2\mid e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(U)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(U)\mid \mid e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j^{\prime }(U)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}^{\prime }(U)\mid \right]\}{d}_2(F,F_0) \\ \lesssim d_3((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))d_2(F,F_0)+d_3^2((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))d_2(F,F_0). \end{gathered}$$

This yields that

$$\begin{gathered} 𝒫(m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)-m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X))\lesssim -d_3^2(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0)) \\ +d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))d_2({\widehat{F}}_n,F_0)+d_3^2(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))d_2({\widehat{F}}_n,F_0). \end{gathered}$$

Second, we need to find a ${\varphi }_n(\eta )$ such that

$$E\underset{\{(\beta ,\Lambda )\in ℛ\times {\Phi }_n:d_3((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))<\eta \}}{\text{sup}}\mid ({\mathbb{P}}_n-𝒫)(m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X))\mid \lesssim \frac{{\varphi }_n(\eta )}{\sqrt{n}}.$$

By Lemma 4 of the online Supplementary Material (Hu et al., 2023), for sufficiently large $n$, we have

$$\log N_{[]}(\epsilon ,{ℳ}_{\eta }({\widehat{F}}_n),{\Vert \cdot \Vert }_{P,B})\lesssim q_n\log (\eta ∕\epsilon ),$$

where $ℳ({\widehat{F}}_n)=\{m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X):\beta \in ℛ,\Lambda \in {\Phi }_n,d_3^2((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))\le {\eta }^2\}$. For $(\beta ,\Lambda )\in ℛ\times {\Phi }_n$ satisfying $d_3((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))<\eta$, similar to the proof of Lemma 4 of the online Supplementary Material (Hu et al., 2023), we have

$$\begin{gathered} \mid m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)\mid \lesssim \sum _{j=1}^K[(\Delta N_j+1)\{\Delta \mid e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(Y)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\mid \phantom{\}}\phantom{]} \\ +\phantom{[}\phantom{\{}\frac{1-\Delta }{1-{\widehat{F}}_n(Y\mid \mathbf{Z})}\sum _{j=1}^K{\int }_Y^{\infty }\mid e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\mid d{\widehat{F}}_n(u\mid \mathbf{Z})\}]. \end{gathered}$$

Furthermore, since $\exp ({\beta }^T\mathbf{Z})$, $\Delta {\Lambda }_j$, $\exp ({\beta }_0^T\mathbf{Z})$ and $\Delta {\Lambda }_{0,j}$ are bounded and $d_2({\widehat{F}}_n,F_0)=o_p(1)$, we have $\exp (\mid m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0{\widehat{F}}_n;X)\mid )\lesssim \exp (CN(T_K))$. The above two inequalities yield that

$$\begin{gathered} 𝒫\left[e^{\mid m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)\mid }{\mid m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)\mid }^2\right] \\ \lesssim 𝒫[\sum _{j=1}^K\Delta {\left\{e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(Y)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\right\}}^2\phantom{]} \\ +\sum _{j=1}^K\frac{1-\Delta }{1-{\widehat{F}}_n(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\phantom{[}{\left\{e^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_j(u)-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right\}}^{2}d{\widehat{F}}_n(u\mid \mathbf{Z})] \\ \lesssim d_3^2((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))+d_3((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))d_2({\widehat{F}}_n,F_0). \end{gathered}$$

That means for sufficiently large $n$, ${\Vert m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0{\widehat{F}}_n;X)\Vert }_{P,B}^2\lesssim {\eta }^2$. By Lemma 3.4.3 of van der Vaart and Wellner (1996),

$$E{\Vert n^{1∕2}({\mathbb{P}}_n-𝒫)\Vert }_{{ℳ}_n({\widehat{F}}_n)}\lesssim J_{[]}(\eta ,{ℳ}_{\eta }({\widehat{F}}_n),{\Vert \cdot \Vert }_{P,B})\left\{1+J_{[]}(\eta ,{ℳ}_n({\widehat{F}}_n),{\Vert \cdot \Vert }_{P,B})∕({\eta }^2{n}^{1∕2})\right\},$$

where $J_{[]}(\eta ,{ℳ}_{\eta }({\widehat{F}}_n),{\Vert \cdot \Vert }_{P,B})≔{\int }_0^{\eta }{\{1+\log N_{[]}(\epsilon ,{ℳ}_n({\widehat{F}}_n),{\Vert \cdot \Vert }_{P,B})\}}^{1∕2}d\epsilon \lesssim q_n^{1∕2}\eta$. It follows that

$$E\underset{\{(\beta ,\Lambda )\in ℛ\times {\Phi }_n:d_3((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))<\eta \}}{\text{sup}}\mid ({\mathbb{P}}_n-𝒫)(m(\beta ,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X))\mid \lesssim \frac{\sqrt{n{q}_n}\eta +q_n}{n}.$$

Setting ${\varphi }_n(\eta )=\sqrt{q_n}\eta +q_n∕\sqrt{n}$ such that ${\varphi }_n(\eta )∕\eta$ decreases about $\eta$, for a sequence $r_n=O(n^a)$, we have $r_n^2\varphi (1∕r_n)=\sqrt{q_n}{r}_n+q_n{r}_n^2∕\sqrt{n}$. Note that $q_n=O(n^{\nu })$, $0<\nu <1∕2$. This yields that $r_n^2\varphi (1∕r_n)=O(n^{a+\nu ∕2}+n^{2a+\nu -1∕2})$. Since $a\le (1-\nu )∕2$ ensures $r_n^2\varphi (1∕r_n)\lesssim \sqrt{n}$, we choose $r_n=O(n^{(1-\nu )∕2})$.

Finally, we determine $\nu$ satisfying ${\mathbb{P}}_n(m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X))\le O_p(1∕r_n^2)$. Note that for ${\Lambda }_0\in {\mathscr{H}}_r$, there is a ${\Lambda }_n^{\ast }\in {\Phi }_n$ such that ${\Vert {\Lambda }_n^{\ast }-{\Lambda }_0\Vert }_{\infty }=O(n^{-\nu r})$. By the definition of $({\widehat{\beta }}_n,{\widehat{\Lambda }}_n)$ and $0\le {𝒫}_m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-{𝒫}_m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)\lesssim {\Vert {\Lambda }_n^{\ast }-{\Lambda }_0\Vert }_{\infty }^2$, we have

$$\begin{gathered} {\mathbb{P}}_n(m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)) \\ ={\mathbb{P}}_{n}m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-{\mathbb{P}}_n({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)+{\mathbb{P}}_{n}m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-𝒫m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X) \\ +𝒫m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-𝒫m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)+𝒫m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)-{\mathbb{P}}_{n}m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X) \\ \le n^{-\nu r+\epsilon }({\mathbb{P}}_n-𝒫)[\{m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X)\}∕n^{-\nu r+\epsilon }]+O_p(n^{-2\nu r}). \end{gathered}$$

Set $\widetilde{ℳ}({\widehat{F}}_n)=\{m({\beta }_0,\Lambda ,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X):\Lambda \in {\Phi }_n,{\Vert \Lambda -{\Lambda }_0\Vert }_{\infty }\le O(n^{-\nu r})\}$. According to Lemma 4 of the online Supplementary Material (Hu et al., 2023), $\widetilde{ℳ}({\widehat{F}}_n)$ is Donsker. After some algebraic calculations, for any $f\in \widetilde{ℳ}({\widehat{F}}_n)$, we have $P{(f∕n^{-\nu r+\epsilon })}^2\to 0$ as $n\to 0$ for any $\epsilon >0$. Using Corollary 2.3.12 of van der Vaart and Wellner (1996), we have $({\mathbb{P}}_n-𝒫)(m({\beta }_0,{\Lambda }_n^{\ast },{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X))=o_p(n^{-\nu r+\epsilon -1∕2})$. When $0<\epsilon \le 1∕2-r\nu$, we have ${\mathbb{P}}_n(m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X))\le O_p(n^{-2\nu r})$, meaning that $\nu \ge 1∕(1+2r)$ ensures ${\mathbb{P}}_n(m({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)-m({\beta }_0,{\Lambda }_0,{\widehat{F}}_n;X))\le O_p(r_n^{-2})$. Thus, taking $\nu =1∕(1+2r)$, we have $d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))=O_p(n^{-r∕(1+2r)})$. □

A.3. Proof of Theorem 3.3

Proof. (i) Define $\tilde{\mathscr{H}}=\{({\mathbf{h}}_1,h_2):{\mathbf{h}}_1\in ℛ,h_2\in {\mathscr{H}}_r\}$. For $({\mathbf{h}}_1,h_2)\in \tilde{\mathscr{H}}$, let $Q_n(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]={\mathbb{P}}_n\psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]$ and $Q(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]=𝒫\psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]$ with $\psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]$ given in (6). Following Theorem 1 of Zhao and Zhang (2017), it suffices to verify the following conditions (B1)-(B5) to prove this theorem.

(B1) $Q({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]=0$ and $Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]=o_p(n^{-1∕2})$.

(B2) $\sqrt{n}(Q_n-Q)({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,\widehat{F})[{\mathbf{h}}_1,h_2]-\sqrt{n}(Q_n-Q)({\beta }_0,{\Lambda }_0,F_0)=[{\mathbf{h}}_1,h_2]=o_p(1)$.

(B3) $Q(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]$ is Fréchet-differentiable with respect to ($\beta$, $\Lambda$) at (${\beta }_0$, ${\Lambda }_0$, $F_0$) with a continuous derivative ${\stackrel{.}{Q}}_{1,{\beta }_0,{\Lambda }_0,F_0}[{\mathbf{h}}_1,h_2]$; $Q(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]$ is Fréchet-differentiable with respect to $F$ at (${\beta }_0$, ${\Lambda }_0$, $F_0$) with a continuous derivative ${\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}[{\mathbf{h}}_1,h_2]$.

(B4) $Q({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]-Q({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]-{\stackrel{.}{Q}}_{1,{\beta }_0,{\Lambda }_0,F_0}({\widehat{\beta }}_n-{\beta }_0,{\widehat{\Lambda }}_n-{\Lambda }_0)[{\mathbf{h}}_1,h_2]-{\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}({\widehat{F}}_n-F_0)[{\mathbf{h}}_1,h_2]=o_p(n^{-1∕2})$.

(B5) $\sqrt{n}{Q}_n({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]+\sqrt{n}{\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}({\widehat{F}}_n-F_0)[{\mathbf{h}}_1,h_2]$ converges in distribution to a tight Gaussian progress.

For (B1), under model (1), we have $Q({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]=0$. By the definition of (${\widehat{\beta }}_n$, ${\widehat{\Lambda }}_n$), for all $({\mathbf{h}}_1,h_2)\in ℛ\times {\Phi }_n$, we obtain ${\lim }_{\eta \to 0}{\mathbb{P}}_{n}m({\widehat{\beta }}_n+\eta {\mathbf{h}}_1,{\widehat{\Lambda }}_n+\eta h_2.{\widehat{F}}_n;X)∕\eta =0$. This implies that $Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]=0$ for all $({\mathbf{h}}_1,h_2)\in ℛ\times {\Phi }_n$. By Lemma A1 of Lu, Zhang and Huang (2007) and the properties of spline functions, for any $h_2\in {\mathscr{H}}_r$, we can find an $h_{2,n}\in {\Phi }_n$ satisfying ${\Vert h_{2,n}-h_2\Vert }_{\infty }=O(n^{-r∕(1+2r)})$ and ${\Vert h_{2,n}^{\prime }-h_2^{\prime }\Vert }_{\infty }=O(1)$, where $h_2^{\prime }$ is the derivative of $h_2$. Thus, for each $h_2\in {\mathscr{H}}_r$, we need to prove $Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[0,h_2-h_{2,n}]={\mathbb{P}}_n\psi ({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[0,h_2-h_{2,n}]=o_p(n^{-1∕2})$ to verify $Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]=o_p(n^{-1∕2})$. Note that

$$\begin{gathered} Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[0,h_2-h_{2,n}]=\{Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[0,h_2-h_{2,n}]-Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0)[0,h_2-h_{2,n}]\} \\ +\{Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0)[0,h_2-h_{2,n}]-Q_n({\beta }_0,{\Lambda }_0,F_0)[0,h_2-h_{2,n}]\}+Q_n({\beta }_0,{\Lambda }_0,F_0)[0,h_2-h_{2,n}] \\ ≔I_{1n}+I_{2n}+I_{3n}. \end{gathered}$$

For the first term $I_{1n}$, Lemma 2 of the online Supplementary Material (Hu et al., 2023) yields that

$$\begin{gathered} 𝒫\mid I_{1n}\mid =𝒫\mid Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[0,h_2-h_{2,n}]-Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0)[0,h_2-h_{2,n}]\mid \\ \le 𝒫[\sum _{j=1}^K(1-\Delta )\mid \frac{{\int }_Y^{\infty }\{\Delta N_j-e^{{\widehat{\beta }}_n^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(u)\}(\Delta h_{2,j}(u)-\Delta h_{2,n,j}(u))d{\widehat{F}}_n(u\mid \mathbf{Z})}{1-{\widehat{F}}_n(Y\mid \mathbf{Z})}\phantom{\mid }\phantom{]} \\ -\phantom{[}\phantom{\mid }\frac{{\int }_Y^{\infty }\{\Delta N_j-e^{{\widehat{\beta }}_n^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(u)\}(\Delta h_{2,j}(u)-\Delta h_{2,n,j}(u))d{F}_0(u\mid \mathbf{Z})}{1-F_0(Y\mid \mathbf{Z})}\mid e^{{\widehat{\beta }}_n^T\mathbf{Z}}] \\ \lesssim d_2({\widehat{F}}_n,F_0)({\Vert h_2-h_{2,n}\Vert }_{\infty }+{\Vert h_2^{\prime }-h_{2,n}^{\prime }\Vert }_{\infty })=o_p(n^{-1∕2}). \end{gathered}$$

For the second term $I_{2n}$, after some algebraic calculations, we have

$$\begin{gathered} 𝒫\mid I_{2n}\mid =𝒫\mid Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,F_0)[0,h_2-h_{2,n}]-Q_n({\beta }_0,{\Lambda }_0,F_0)[0,h_2-h_{2,n}]\mid \\ \le {\Vert h_2-h_{2,n}\Vert }_{\infty }𝒫[\sum _{k=1}^K\Delta \mid \{\Delta N_j-e^{{\widehat{\beta }}_n^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n_j}(Y)\}{e}^{{\widehat{\beta }}_n^T\mathbf{Z}}-\{\Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\}{e}^{{\beta }_0^T\mathbf{Z}}\mid \phantom{]} \\ +\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\phantom{[}\mid \{\Delta N_j-e^{{\widehat{\beta }}_n^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(u)\}{e}^{{\widehat{\beta }}_n^T\mathbf{Z}}-\{\Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\}{e}^{{\beta }_0^T\mathbf{Z}}\mid d{F}_0(u\mid \mathbf{Z})] \\ \lesssim \{{\Vert {\widehat{\beta }}_n-{\beta }_0\Vert }_2+d_3((2{\widehat{\beta }}_n,{\widehat{\Lambda }}_n),(2{\beta }_0,{\Lambda }_0))\}{\Vert h_2-h_{2,n}\Vert }_{\infty }=o_p(n^{-1∕2}). \end{gathered}$$

For the third term $I_{3n}$, note that $Q({\beta }_0,{\Lambda }_0,F_0;X)[0,h_2-h_{2,n}]=0$. By the independence of $X_i$ and $X_j$, it follows that

$$\begin{gathered} 𝒫I_{3n}^2=n^{-1}𝒫\left(\frac{1}{n}\sum _{i=1}^n{\psi }^2({\beta }_0,{\Lambda }_0,F_0;X_i)[0,h_2-h_{2,n}]\right)\lesssim n^{-1}𝒫[\sum _{j=1}^K\{\Delta \mid \Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\mid e^{{\beta }_0^T\mathbf{Z}}\phantom{\}}\phantom{]} \\ +\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }{\phantom{[}\phantom{\{}\mid \Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\mid e^{{\beta }_0^T\mathbf{Z}}d{F}_0(u\mid \mathbf{Z})\}]}^2{\Vert h_2-h_{2,n}\Vert }_{\infty }^2\lesssim n^{-1}{\Vert h_2-h_{2,n}\Vert }_{\infty }^2. \end{gathered}$$

Then we have $Q_n({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[0,h_2-h_{2,n}]=o_p(n^{-1∕2})$, and (B1) holds.

For (B2), after some algebraic calculations, we have

$$\begin{gathered} \sqrt{n}(Q_n-Q)({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]-\sqrt{n}(Q_n-Q)({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2] \\ =\sqrt{n}({\mathbb{P}}_n-𝒫)(\psi ({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)[{\mathbf{h}}_1,h_2]-\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1,h_2]). \end{gathered}$$

For each fixed bounded $({\mathbf{h}}_1,h_2)\in \tilde{\mathscr{H}}$, set

$$\begin{gathered} {\stackrel{\text{-}}{\Psi }}_{\eta }({\mathbf{h}}_1,h_2)=\{\psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]-\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1,h_2]:\beta \in ℛ,\Lambda \in {\Phi }_n,F\in ℱ \\ {d}_3((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))<\eta ,d_2(F,F_0)<\eta ,\Lambda \text{is uniformly bounded}\}. \end{gathered}$$

Similar to Lemma 3 of the online Supplementary Material (Hu et al., 2023), it follows that ${\stackrel{\text{-}}{\Psi }}_{\eta }({\mathbf{h}}_1,h_2)$ is Donsker. By Condition (C6) and Lemma 2 of the online Supplementary Material (Hu et al., 2023), after some algebraic calculations, we obtain $𝒫{(\psi (\beta ,\Lambda ,F;X)[{\mathbf{h}}_1,h_2]-\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1,h_2])}^2\lesssim d_3{((\beta ,\Lambda ),({\beta }_0,{\Lambda }_0))}^2+d_2{(F,F_0)}^2$. Then Corollary 2.3.12 of van der Vaart and Wellner (1996) implies that

$$\sqrt{n}({\mathbb{P}}_n-𝒫)(\psi ({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n;X)[{\mathbf{h}}_1,h_2]-\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1,h_2])=o_p(1),$$

and (B2) holds.

For (B3), $Q(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]$ is Fréchet-differentiable with respect to ($\beta$, $\Lambda$) at (${\beta }_0$, ${\Lambda }_0$, $F_0$) because $Q(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]$ is a smooth functional with respect to ($\beta$, $\Lambda$, $F$). Similarly, $Q(\beta ,\Lambda ,F)[{\mathbf{h}}_1,h_2]$ is Fréchet-differentiable with respect to $F$ at (${\beta }_0$, ${\Lambda }_0$, $F_0$). By some direct calculations, we obtain

$$\begin{gathered} {\stackrel{.}{Q}}_{1,{\beta }_0,{\Lambda }_0,F_0}({\widehat{\beta }}_n-{\beta }_0,{\widehat{\Lambda }}_n-{\Lambda }_0)[{\mathbf{h}}_1,h_2] \\ =\frac{d}{d\epsilon }\{𝒫[\sum _{j=1}^K\{\Delta \left(\Delta N_j-(\Delta {\Lambda }_{0,j}(Y)+\epsilon (\Delta {\widehat{\Lambda }}_{n,j}(Y)-\Delta {\Lambda }_{0,j}(Y))\right)e^{{({\beta }_0+\epsilon ({\widehat{\beta }}_n-{\beta }_0))}^T\mathbf{Z}}\phantom{\}}\phantom{]}\phantom{\}} \\ \times \left(\Delta h_{2,j}(Y)+(\Delta {\Lambda }_{0,j}(Y)+\epsilon (\Delta {\widehat{\Lambda }}_{n,j}(Y)-\Delta {\Lambda }_{0,j}(Y)))h_1^T\mathbf{Z}\right)e^{{({\beta }_0+\epsilon ({\widehat{\beta }}_n-{\beta }_0))}^T\mathbf{Z}} \\ +\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\left(\Delta N_j-(\Delta {\Lambda }_{0,j}(u)+\epsilon (\Delta {\widehat{\Lambda }}_{n,j}(u)-\Delta {\Lambda }_{0,j}(u)))e^{{({\beta }_0+\epsilon ({\widehat{\beta }}_n-{\beta }_0))}^T\mathbf{Z}}\right) \\ {\phantom{\mid }\phantom{\{}\phantom{[}\phantom{\{}\times \left(\Delta h_{2,j}(u)+(\Delta {\Lambda }_{0,j}(u)+\epsilon (\Delta {\widehat{\Lambda }}_{n,j}(u)-\Delta {\Lambda }_{0,j}(u)))h_1^T\mathbf{Z}\right)e^{{({\beta }_0+\epsilon ({\widehat{\beta }}_n-{\beta }_0))}^T\mathbf{Z}}d{F}_0(u\mid \mathbf{Z})\}]\}\mid }_{\epsilon =0} \\ ≔-R_1({\mathbf{h}}_1,h_2)({\widehat{\beta }}_n-{\beta }_0)-R_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0), \end{gathered}$$

where

$$\begin{gathered} R_1({\mathbf{h}}_1,h_2)({\widehat{\beta }}_n-{\beta }_0)=-𝒫[e^{{\beta }_0^T\mathbf{Z}}\sum _{j=1}^K\{\Delta \left(\Delta N_j-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\right)\left(\Delta h_{2,j}(Y)+\Delta {\Lambda }_{0,j}(Y){\mathbf{h}}_1^T\mathbf{Z}\right)\phantom{\}}\phantom{]} \\ +\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\phantom{[}\phantom{\{}\left(\Delta N_j-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right)\left(\Delta h_{2,j}(u)+\Delta {\Lambda }_{0,j}(u){\mathbf{h}}_1^T\mathbf{Z}\right)d{F}_0(u\mid \mathbf{Z})\}{\mathbf{Z}}^T]({\widehat{\beta }}_n-{\beta }_0) \end{gathered}$$ (8)

and

$$\begin{gathered} R_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0)=-𝒫[e^{{\beta }_0^T\mathbf{Z}}\sum _{j=1}^K\{\Delta \left(\Delta N_j{\mathbf{h}}_1^T\mathbf{Z}-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y){\mathbf{h}}_1^T\mathbf{Z}-e^{{\beta }_0^T\mathbf{Z}}\Delta h_{2,j}(Y)\right)\phantom{\}}\phantom{]} \\ \times \left(\Delta {\widehat{\Lambda }}_{n,j}(Y)-\Delta {\Lambda }_{0,j}(Y)\right)+\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\left(\Delta {\widehat{\Lambda }}_{n,j}(u)-{\Lambda }_{0,j}(u)\right) \\ \times \phantom{[}\phantom{\{}\left(\Delta N_j{\mathbf{h}}_1^T\mathbf{Z}-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u){\mathbf{h}}_1^T\mathbf{Z}-e^{{\beta }_0^T\mathbf{Z}}\Delta h_{2,j}(u)\right)d{F}_0(u\mid \mathbf{Z})\}]. \end{gathered}$$ (9)

Since the equation

$$\begin{gathered} {\phantom{\mid }d\frac{{\int }_Y^{\infty }g(u-T_j)d(F_0+\epsilon ({\widehat{F}}_n-F_0))(u\mid \mathbf{Z})}{1-F_0(Y\mid \mathbf{Z})-\epsilon ({\widehat{F}}_n-F_0)(Y\mid \mathbf{Z})}∕d\epsilon \mid }_{\epsilon =0} \\ =\frac{(1-F_0(Y\mid \mathbf{Z})){\int }_Y^{\infty }g(u-T_j)d({\widehat{F}}_n-F_0)(u\mid \mathbf{Z})+({\widehat{F}}_n-F_0)(Y\mid \mathbf{Z}){\int }_Y^{\infty }g(u-T_j)d{F}_0(u\mid \mathbf{Z})}{{(1-F_0(Y\mid \mathbf{Z}))}^2} \\ =\frac{1}{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\left\{g(u-T_j)-{\int }_Y^{\infty }\frac{g(s-T_j)}{1-F_0(Y\mid \mathbf{Z})}d{F}_0(s\mid \mathbf{Z})\right\}d({\widehat{F}}_n-F_0)(u\mid \mathbf{Z}) \end{gathered}$$

holds for any differentiable function $g$, we obtain

$$\begin{gathered} {\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}({\widehat{F}}_n-F_0)[{\mathbf{h}}_1,h_2]={\phantom{\mid }\frac{d}{d\epsilon }\{Q({\beta }_0,{\Lambda }_0,F_0+\epsilon ({\widehat{F}}_n-F_0))[{\mathbf{h}}_1,h_2]\}\mid }_{\epsilon =0} \\ =\frac{d}{d\epsilon }\{𝒫[\sum _{j=1}^K\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})-\epsilon ({\widehat{F}}_n-F_0)(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\{\Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\}\phantom{]}\phantom{\}} \\ \times {\phantom{\mid }\phantom{\{}\phantom{[}{e}^{{\beta }_0^T\mathbf{Z}}\{\Delta h_{2,j}(u)+\Delta {\Lambda }_{0,j}(u){\mathbf{h}}_1^T\mathbf{Z}\}d(F_0+\epsilon ({\widehat{F}}_n-F_0))(u\mid \mathbf{Z})]\}\mid }_{\epsilon =0} \\ =𝒫\left[{\int }_Y^{\infty }{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d({\widehat{F}}_n-F_0)(u\mid \mathbf{Z})\right], \end{gathered}$$

where

$$\begin{gathered} {\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2] \\ =\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}\sum _{j=1}^K\left\{{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]-{\int }_Y^{\infty }\frac{{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(s;X)[{\mathbf{h}}_1,h_2]}{1-F_0(Y\mid \mathbf{Z})}d{F}_0(s\mid Z)\right\}, \end{gathered}$$

and ${\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]=\{\Delta N_j-\exp ({\beta }_0^T\mathbf{Z})\Delta {\Lambda }_{0,j}(u)\}\exp ({\beta }_0^T\mathbf{Z})\{\Delta h_{2,j}(u)+\Delta {\Lambda }_{0,j}(u){\mathbf{h}}_1^T\mathbf{Z}\}$. Then (B3) is verified.

For (B4), since $d_3(({\widehat{\beta }}_n,{\widehat{\Lambda }}_n),({\beta }_0,{\Lambda }_0))=O_p(n^{-r∕(1+2r)})$ and by the Taylor expansion, we have $\exp ({\widehat{\beta }}_n^T\mathbf{Z})=\exp ({\beta }_0^T\mathbf{Z})+\exp ({\beta }_0^T\mathbf{Z})Z^T({\widehat{\beta }}_n-{\beta }_0)+o_p(1∕\sqrt{n})$. By the above equation and Lemma 2 of the online Supplementary Material (Hu et al., 2023), we obtain

$$\begin{gathered} Q({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]-Q({\beta }_0,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2] \\ =𝒫[e^{{\beta }_0^T\mathbf{Z}}\sum _{j=1}^K\{\Delta \left(\Delta N_j-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(Y)\right)\left(\Delta h_{2,j}(Y)+\Delta {\widehat{\Lambda }}_{n,j}(Y){\mathbf{h}}_1^T\mathbf{Z}\right)\phantom{\}}+\frac{1-\Delta }{1-{\widehat{F}}_n(Y\mid \mathbf{Z})}\phantom{]} \\ \times {\int }_Y^{\infty }\phantom{[}\phantom{\{}\left(\Delta N_j-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(u)\right)\left(\Delta h_{2,j}(u)+\Delta {\widehat{\Lambda }}_{n,j}(u){\mathbf{h}}_1^T\mathbf{Z}\right)d{\widehat{F}}_n(u\mid \mathbf{Z})\}{\mathbf{Z}}^T]({\widehat{\beta }}_n-{\beta }_0) \\ =𝒫[e^{{\beta }_0^T\mathbf{Z}}\sum _{j=1}^K\{\Delta \left(\Delta N_j-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(Y)\right)\left(\Delta h_{2,j}(Y)+\Delta {\widehat{\Lambda }}_{n,j}(Y){\mathbf{h}}_1^T\mathbf{Z}\right)+\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}\phantom{\}}\phantom{]} \\ \times {\int }_Y^{\infty }\phantom{[}\phantom{\{}\left(\Delta N_j-2e^{{\beta }_0^T\mathbf{Z}}\Delta {\widehat{\Lambda }}_{n,j}(u)\right)\left(\Delta h_{2,j}(u)+\Delta {\widehat{\Lambda }}_{n,j}(u){\mathbf{h}}_1^T\mathbf{Z}\right)d{F}_0(u\mid \mathbf{Z})\}{\mathbf{Z}}^T]({\widehat{\beta }}_n-{\beta }_0) \\ +o_p(n^{-1∕2})=-R_1({\mathbf{h}}_1,h_2)({\widehat{\beta }}_n-{\beta }_0)+o_p(n^{-1∕2}). \end{gathered}$$ (10)

Similarly, since $d_1({\widehat{\Lambda }}_n,{\Lambda }_0)=O_p(n^{-r∕(1+2r)})$ and $d_1({\widehat{\Lambda }}_n^{\prime },{\Lambda }_0^{\prime })=o_p(1)$, using Lemma 2 of the online Supplementary Material (Hu et al., 2023), we have

$$\begin{gathered} Q({\beta }_0,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]-Q({\beta }_0,{\Lambda }_0,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2] \\ =𝒫[\sum _{j=1}^K\{\Delta (e^{2{\beta }_0^T\mathbf{Z}}\left(\Delta {\Lambda }_{0,j}{(Y)}^2-\Delta {\widehat{\Lambda }}_{n,j}{(Y)}^2\right)h_1^T\mathbf{Z}+\left(\Delta {\widehat{\Lambda }}_{n,j}(Y)-\Delta {\Lambda }_{0,j}(Y)\right)\phantom{)}\phantom{\}}\phantom{]} \\ \times \phantom{(}\left(\Delta N_j{e}^{{\beta }_0^T\mathbf{Z}}{h}_1^T\mathbf{Z}-e^{2{\beta }_0^T\mathbf{Z}}\Delta h_{2,j}(Y)\right))+\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }(e^{2{\beta }_0^T\mathbf{Z}}\left(\Delta {\widehat{\Lambda }}_{n,j}{(u)}^2-\Delta {\Lambda }_{0,j}{(u)}^2\right)\phantom{)}{h}_1^T\mathbf{Z} \\ +\phantom{[}\phantom{\{}\left(\Delta {\widehat{\Lambda }}_{n,j}(u)-\Delta {\Lambda }_{0,j}(u)\right)\phantom{(}\left(\Delta N_j{e}^{{\beta }_0^T\mathbf{Z}}{h}_1^T\mathbf{Z}-e^{2{\beta }_0^T\mathbf{Z}}\Delta h_{2,j}(u)\right))d{F}_0(u\mid \mathbf{Z})\}]+o_p(n^{-1∕2}) \\ =-R_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0)-𝒫[\sum _{j=1}^K\{\Delta e^{2{\beta }_0^T\mathbf{Z}}{(\Delta {\widehat{\Lambda }}_{n,j}(Y)-{\Delta \Lambda }_{0,j}(Y))}^2{h}_1^T\mathbf{Z}\phantom{\}}\phantom{]} \\ +\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}{\int }_Y^{\infty }\phantom{[}\phantom{\{}\left(e^{2{\beta }_0^T\mathbf{Z}}{(\Delta {\widehat{\Lambda }}_{n,j}(u)-\Delta {\Lambda }_{0,j}(u))}^2{h}_1^T\mathbf{Z}\right)d{F}_0(u\mid \mathbf{Z})\}]+o_p(n^{-1∕2}) \\ =-R_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0)+o_p(n^{-1∕2}). \end{gathered}$$ (11)

By (10) and (11), it follows that $Q({\widehat{\beta }}_n,{\widehat{\Lambda }}_n,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]-Q({\beta }_0,{\Lambda }_0,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]={\stackrel{.}{Q}}_{1,{\beta }_0,{\Lambda }_0,F_0}({\widehat{\beta }}_n-{\beta }_0,{\widehat{\Lambda }}_n-{\Lambda }_0)[{\mathbf{h}}_1,h_2]+o_p(1∕\sqrt{n})$. By Lemma 2 of the online Supplementary Material (Hu et al., 2023), we can obtain that

$$\begin{gathered} \mid Q({\beta }_0,{\Lambda }_0,{\widehat{F}}_n)[{\mathbf{h}}_1,h_2]-Q({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]-{\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}({\widehat{F}}_n-F_0)[{\mathbf{h}}_1,h_2]\mid \\ =\mid 𝒫[(1-\Delta )\frac{{\widehat{F}}_n(Y\mid \mathbf{Z})-F_0(Y\mid \mathbf{Z})}{1-F_0(Y\mid \mathbf{Z})}\sum _{j=1}^K\{\frac{{\int }_Y^{\infty }{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d{\widehat{F}}_n(u\mid \mathbf{Z})}{1-{\widehat{F}}_n(Y\mid \mathbf{Z})}\phantom{\}}\phantom{]}\phantom{\mid } \\ -\phantom{\mid }\phantom{[}\phantom{\{}\frac{{\int }_Y^{\infty }{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d{F}_0(u\mid \mathbf{Z})}{1-F_0(Y\mid \mathbf{Z})}\}]\mid \\ \lesssim {\Vert {\widehat{F}}_n-F_0\Vert }_{\infty }𝒫[(1-\Delta )\sum _{j=1}^K\mid \frac{{\int }_Y^{\infty }{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d{\widehat{F}}_n(u\mid \mathbf{Z})}{1-{\widehat{F}}_n(Y\mid \mathbf{Z})}\phantom{\mid }\phantom{]} \\ -\phantom{[}\phantom{\mid }\frac{{\int }_Y^{\infty }{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d{F}_0(u\mid \mathbf{Z})}{1-F_0(Y\mid \mathbf{Z})}\mid ]\lesssim {\Vert {\widehat{F}}_n-F_0\Vert }_{\infty }^2=o_p(n^{-1∕2}). \end{gathered}$$

Thus, (B4) holds.

Finally, we consider (B5). Note that

$$\begin{gathered} \sqrt{n}{Q}_n({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]+\sqrt{n}{\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}({\widehat{F}}_n-F_0)[{\mathbf{h}}_1,h_2] \\ =\sqrt{n}{\mathbb{P}}_n\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1,h_2]\mid +\sqrt{n}𝒫\left[{\int }_Y^{\infty }{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d({\widehat{F}}_n-F_0)(u\mid \mathbf{Z})\right]. \end{gathered}$$

According to Lemma 1 of the online Supplementary Material (Hu et al., 2023), we have

$$\begin{gathered} 𝒫\left[{\int }_Y^{\infty }{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]d({\widehat{F}}_n(u\mid \mathbf{Z})-F_0(u\mid \mathbf{Z}))\right] \\ =𝒫[{\int }_Y^{\infty }\frac{\partial {\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]}{\partial u}({\widehat{F}}_n(u\mid \mathbf{Z})-F_0(u\mid \mathbf{Z}))du\phantom{]} \\ -\phantom{[}{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(Y;X)[{\mathbf{h}}_1,h_2]({\widehat{F}}_n(Y\mid \mathbf{Z})-F_0(Y\mid \mathbf{Z}))] \\ =𝒫[\frac{1}{n}\sum _{i=1}^n\{{\int }_Y^{\infty }\frac{\partial {\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]}{\partial u}\Omega (u,\mathbf{Z};{\tilde{Y}}_i,{\tilde{\Delta }}_i,{\tilde{\mathbf{Z}}}_i)du\phantom{\}}\phantom{]} \\ -\phantom{[}\phantom{\{}{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(Y;X)[{\mathbf{h}}_1,h_2]\Omega (Y,\mathbf{Z};{\tilde{Y}}_i,{\tilde{\Delta }}_i,{\tilde{\mathbf{Z}}}_i)\}]≔{\mathbb{P}}_n\phi ({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})[{\mathbf{h}}_1,h_2], \end{gathered}$$

where

$$\begin{gathered} \phi ({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})[{\mathbf{h}}_1,h_2]={𝒫}_X[\{{\int }_Y^{\infty }\frac{\partial {\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(u;X)[{\mathbf{h}}_1,h_2]}{\partial u}\Omega (u,\mathbf{Z};\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})du\phantom{\}}\phantom{]} \\ -\phantom{[}\phantom{\{}{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}(Y;X)[{\mathbf{h}}_1,h_2]\Omega (Y,\mathbf{Z};\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})\}]. \end{gathered}$$

By the central limit theorem, setting

$${\sigma }_0{[{\mathbf{h}}_1,h_2]}^2=E\left[{\left\{\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1,h_2]+\phi ({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})[{\mathbf{h}}_1,h_2]\right\}}^2\right],$$ (12)

we have $\sqrt{n}{Q}_n({\beta }_0,{\Lambda }_0,F_0)[{\mathbf{h}}_1,h_2]+\sqrt{n}{\stackrel{.}{Q}}_{2,{\beta }_0,{\Lambda }_0,F_0}({\widehat{F}}_n-F_0)[{\mathbf{h}}_1,h_2]⇝N(0,{\sigma }_0{[{\mathbf{h}}_1,h_2]}^2)$, and (B5) holds.

By Theorem 1 of Zhao and Zhang (2017), (B1)-(B5) yields that

$$\sqrt{n}{R}_1({\mathbf{h}}_1,h_2)({\widehat{\beta }}_n-{\beta }_0)+\sqrt{n}{R}_2({\mathbf{h}}_1,h_2)({\widehat{\Lambda }}_n-{\Lambda }_0)⇝N(0,{\sigma }_0{[{\mathbf{h}}_1,h_2]}^2).$$

(ii) To prove the asymptotic normality of ${\widehat{\beta }}_n$, we need to find an (${\mathbf{h}}_1^{\ast }$, $h_2^{\ast }$) such that $R_2({\mathbf{h}}_1^{\ast },h_2^{\ast })({\widehat{\Lambda }}_n-{\Lambda }_0)=0$. After some algebraic calculations, we obtain

$$\begin{gathered} R_2({\mathbf{h}}_1^{\ast },h_2^{\ast })({\widehat{\Lambda }}_n-{\Lambda }_0) \\ =𝒫\left[\sum _{j=1}^K\left\{(\Delta {\widehat{\Lambda }}_{n,j}(U)-\Delta {\Lambda }_{0,j}(U))E\left[(\Delta h_{2,j}^{\ast }(U)+\Delta {\Lambda }_{0,j}(U){\mathbf{h}}_1^{\ast T}\mathbf{Z})e^{2{\beta }_0^T\mathbf{Z}}\mid U,K,\mathbf{T}\right]\right\}\right]. \end{gathered}$$

Setting ${\mathbf{R}}^{\ast }(U,K,\mathbf{T})=E[\exp (2{\beta }_0^T\mathbf{Z})\mathbf{Z}\mid U,K,\mathbf{T}]∕E[\exp (2{\beta }_0^T\mathbf{Z})\mid U,K,\mathbf{T}]$, the above equality implies that $\Delta h_{2,j}^{\ast }(U)=-{\mathbf{h}}_1^{\ast T}(U,K,\mathbf{T})\Delta {\Lambda }_{0,j}(U)$. Then we have

$$\Delta {\Lambda }_j(U){\mathbf{h}}_1^{\ast T}\mathbf{Z}+\Delta h_{2,j}^{\ast }(U)=\Delta {\Lambda }_j(U){\mathbf{h}}_1^{\ast T}(\mathbf{Z}-{\mathbf{R}}^{\ast }(U,K,\mathbf{T})).$$ (13)

It follows that $R_1({\mathbf{h}}_1^{\ast },h_2^{\ast })(({\widehat{\beta }}_n-{\beta }_0)={\mathbf{h}}_1^{\ast T}{\mathbf{A}}^{\ast }({\widehat{\beta }}_n-{\beta }_0)$, where

$${\mathbf{A}}^{\ast }=𝒫\left[\sum _{j=1}^K\left\{e^{2{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}{(U)}^2{(\mathbf{Z}-{\mathbf{R}}^{\ast }(U,K,\mathbf{T}))}^{\otimes 2}\right\}\right].$$

Furthermore, by (13), we obtain

$$\begin{gathered} \psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1^{\ast },h_2^{\ast }]={\mathbf{h}}_1^{\ast T}\sum _{j=1}^K[\Delta \left\{\Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)\right\}{e}^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(Y)(\mathbf{Z}-{\mathbf{R}}^{\ast }(Y,K,\mathbf{T}))\phantom{]} \\ +\phantom{[}\frac{1-\Delta }{1-F_0(Y\mid Z)}{\int }_Y^{\infty }\left\{\Delta N_j-e^{{\beta }_0^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)\right\}{e}^{{\beta }^T\mathbf{Z}}\Delta {\Lambda }_{0,j}(u)(\mathbf{Z}-{\mathbf{R}}^{\ast }(u,K,\mathbf{T}))d{F}_0(u\mid Z)] \\ ≔{\mathbf{h}}_1^{\ast T}{\psi }^{\ast }({\beta }_0,{\Lambda }_0,F_0;X), \\ \phi ({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})[{\mathbf{h}}_1^{\ast },h_2^{\ast }]={\mathbf{h}}_1^{\ast T}{𝒫}_X=[\{{\int }_Y^{\infty }\frac{\partial {\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}^{\ast }(u;X)}{\partial u}\Omega (u,\mathbf{Z};\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})du\phantom{\}}\phantom{]} \\ -\phantom{[}\phantom{\{}{\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}^{\ast }(Y;X)\Omega (Y,\mathbf{Z};\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})\}]≔{\mathbf{h}}_1^{\ast T}{\phi }^{\ast }({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}}), \end{gathered}$$

where

$${\stackrel{\text{-}}{\phi }}_{{\beta }_0,{\Lambda }_0,F_0}^{\ast }(u;X)=\frac{1-\Delta }{1-F_0(Y\mid \mathbf{Z})}\sum _{j=1}^K\left\{{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}^{\ast }(u;X)-{\int }_Y^{\infty }\frac{{\tilde{\phi }}_{j,{\beta }_0,{\Lambda }_0,F_0}^{\ast }(s;X)}{1-F_0(Y\mid \mathbf{Z})}d{F}_0(s\mid Z)\right\}$$

and ${{\tilde{\phi }}^{\ast }}_{j,{\beta }_0,{\Lambda }_0,F_0}(u;X)=\{\Delta N_j-\exp ({\beta }_0^T\mathbf{Z})\Delta {\Lambda }_{0,j}(u)\}\exp ({\beta }_0^T\mathbf{Z})\{\Delta {\Lambda }_j(u)(\mathbf{Z}-{\mathbf{R}}^{\ast }(u,K,\mathbf{T}))$. After some algebraic calculations, we have

$$\begin{gathered} {\sigma }_0{[{\mathbf{h}}_1^{\ast },h_2^{\ast }]}^2=E\left[{\left\{\psi ({\beta }_0,{\Lambda }_0,F_0;X)[{\mathbf{h}}_1^{\ast },h_2^{\ast }]+\phi ({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})[{\mathbf{h}}_1^{\ast },h_2^{\ast }]\right\}}^2\right] \\ ={\mathbf{h}}_1^{\ast T}E\left[{\left\{{\psi }^{\ast }({\beta }_0,{\Lambda }_0,F_0;X)+{\phi }^{\ast }({\beta }_0,{\Lambda }_0,F_0;\tilde{Y},\tilde{\Delta },\tilde{\mathbf{Z}})\right\}}^2\right]{\mathbf{h}}_1^{\ast }≔{\mathbf{h}}_1^{\ast T}{\mathbf{B}}^{\ast }{\mathbf{h}}_1^{\ast }. \end{gathered}$$

It follows that $\sqrt{n}{\mathbf{h}}_1^{\ast T}{\mathbf{A}}^{\ast }({\widehat{\beta }}_n-{\beta }_0)⇝N(0,{\mathbf{h}}_1^{\ast T}{\mathbf{B}}^{\ast }{\mathbf{h}}_1^{\ast })$ for all ${\mathbf{h}}_1^{\ast }\in ℛ$. Then we obtain $\sqrt{n}({\widehat{\beta }}_n-{\beta }_0)⇝N(0,{({\mathbf{A}}^{\ast })}^{-1}{\mathbf{B}}^{\ast }{({({\mathbf{A}}^{\ast })}^{-1})}^T)$. □