Analysis of Informatively Interval-Censored Case–Cohort Studies with the Application to HIV Vaccine Trials

Abstract

Case–cohort studies are commonly used in various investigations, and many methods have been proposed for their analyses. However, most of the available methods are for right-censored data or assume that the censoring is independent of the underlying failure time of interest. In addition, they usually apply only to a specific model such as the Cox model that may often be restrictive or violated in practice. To relax these assumptions, we discuss regression analysis of interval-censored data, which arise more naturally in case–cohort studies than and include right-censored data as a special case, and propose a two-step inverse probability weighting estimation procedure under a general class of semiparametric transformation models. Among other features, the approach allows for informative censoring. In addition, an EM algorithm is developed for the determination of the proposed estimators and the asymptotic properties of the proposed estimators are established. Simulation results indicate that the approach works well for practical situations and it is applied to a HIV vaccine trial that motivated this investigation.

Appendix: Proofs of Theorems 4.1 and 4.2

In the following, we will sketch the proofs of Theorems 4.1 and 4.2. Let ${\mathbb {P}}_n$ denote the empirical measure for n independent observations, ${\mathbb {P}}$ the true probability measure, and ${\mathbb {G}}_n=n^{1/2}({\mathbb {P}}_n-{\mathbb {P}})$ the empirical process. Let $l(\beta ,\Lambda |u)$ be the log-likelihood for a single subject based on the complete data O, given by

$$\begin{aligned} l(\beta ,\Lambda |u)= & {} \sum _{k=1}^{K+1} \delta _k \log \left\{ \exp (-G[\Lambda (U_{k-1})\exp (\beta _1^T z + \beta _2 u)]) \right. \\{} & {} \left. - \exp (-G[\Lambda (U_k)\exp (\beta _1^T z + \beta _2 u)]) \right\} , \end{aligned}$$

and let $l^w(\beta ,\Lambda |u) = w \,l(\beta ,\Lambda |u)$ be the weighted log-likelihood for a single subject based on the observed data $O^\xi$ under the case–cohort design, where the weight w is given by $w=\xi /\pi _q(\delta _1,\ldots ,\delta _{K+1})$. Since $E(w|\delta _1,\ldots ,\delta _{K+1})=1$, we have ${\mathbb {P}}\{l^w(\beta ,\Lambda |u)\}={\mathbb {P}}\{l(\beta ,\Lambda |u)\}$.

Proof of Theorem 4.1

We first show that $\limsup _n{\hat{\Lambda }}(\tau _0-\epsilon )<\infty$ with probability 1 for any $\epsilon >0$. By the definition of $({{\hat{\beta }}},{{\hat{\Lambda }}})$, we have

$$\begin{aligned} {\mathbb {P}}_n l^w({{\hat{\beta }}},{{\hat{\Lambda }}}|{\hat{u}})\ge {\mathbb {P}}_n l^w(\beta _0,\Lambda _0|{\hat{u}}). \end{aligned}$$

From the consistency of $({{\hat{\alpha }}},{\hat{\Lambda }}_{h})$ established by Wang et al. [29], we can show that

$$\begin{aligned} \liminf _n {\mathbb {P}}_n l^w({{\hat{\beta }}},{{\hat{\Lambda }}}|{\hat{u}}) \ge \liminf _n {\mathbb {P}}_n l^w(\beta _0,\Lambda _0|{\hat{u}})={\mathbb {P}} l(\beta _0,\Lambda _0|u) = O(1) \end{aligned}$$

with probability 1. Define $u(\alpha ,\Lambda _h;\tau ,K,z)=\log \{K/[\Lambda _h(\tau )\exp (\alpha ^T z)]\}$. Let $\eta >0$ be such that $\exp \{\beta _1^T z + \beta _2 u(\alpha ,\Lambda _h;\tau ,K,z)\}\ge \eta$ for $\beta \in {\mathcal {B}}$, $\alpha \in {\mathcal {A}}$, $\tau \in [\zeta _0,\tau _0]$, $1\le K \le k_0$, and nondecreasing functions $\Lambda _h$ such that $\Lambda _h(\zeta _0)\ge \Lambda _{0h}(\zeta _0)-c_0>0$ and $\Lambda _h(\tau _0)\le 1$, where $k_0>1$ and $c_0$ are positive constants. Then, we have

$$\begin{aligned} \begin{aligned}&\liminf _n {\mathbb {P}}_n l^w({{\hat{\beta }}},{{\hat{\Lambda }}}|{\hat{u}}) \\&\le -\limsup _n {\mathbb {P}}_n \Big \{w \delta _{K+1} G[{{\hat{\Lambda }}}(U_K)\exp ({\hat{\beta }}_1^T z + {\hat{\beta }}_2 {\hat{u}})]\Big \}\\&\le -\limsup _n {\mathbb {P}}_n \Big \{w \delta _{K+1} I(1\le K\le k_0) G[{{\hat{\Lambda }}}(U_K)\eta ]\Big \}\\&\le -\limsup _n {\mathbb {P}}_n \Big \{w \delta _{K+1} I(1\le K\le k_0, U_K\ge \tau _0-\epsilon )G[{{\hat{\Lambda }}}(\tau _0-\epsilon )\eta ]\Big \}. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \limsup _n {\mathbb {P}}_n \Big \{w \delta _{K+1} I(1\le K\le k_0, U_K\ge \tau _0-\epsilon )G[{{\hat{\Lambda }}}(\tau _0-\epsilon )\eta ]\Big \}=O(1). \end{aligned}$$

Note that as $n\rightarrow \infty$, ${\mathbb {P}}_n \{w \delta _{K+1} I(1\le K\le k_0, U_K\ge \tau _0-\epsilon )\}\rightarrow {\mathbb {P}} \{w \delta _{K+1} I(1\le K\le k_0, U_K\ge \tau _0-\epsilon )\}$, which is positive under Condition (C3). Thus, by Condition (C4), $\limsup _n{\hat{\Lambda }}(\tau _0-\epsilon )<\infty$ with probability 1 for any $\epsilon >0$. By Helly’s selection theorem and arguing as in the proof of Theorem 4.1 of Zeng et al. [32], for any subsequence of $({{\hat{\beta }}},{{\hat{\Lambda }}})$, we can choose a further subsequence such that ${{\hat{\Lambda }}}$ converges weakly to some function $\Lambda ^*$ on $[0,\tau _0]$ almost everywhere and ${{\hat{\beta }}}$ converges to some constant $\beta ^*$. The remaining is to show $(\beta ^*,\Lambda ^*)=(\beta _0,\Lambda _0)$.

Define

$$\begin{aligned} m(\beta ,\Lambda |u) = w\,\log \left\{ \frac{p(\beta ,\Lambda |u)+p(\beta _0,\Lambda _0|u)}{2}\right\} , \end{aligned}$$

where $p(\beta ,\Lambda |u)=\exp (l(\beta ,\Lambda |u))$. Since ${\mathbb {P}}_n l^w({{\hat{\beta }}},{{\hat{\Lambda }}}|{\hat{u}})\ge {\mathbb {P}}_n l^w(\beta _0,\Lambda _0|{\hat{u}})$, we have

$$\begin{aligned} {\mathbb {P}}_n m({{\hat{\beta }}},{{\hat{\Lambda }}}|{\hat{u}}) \ge {\mathbb {P}}_n l^w(\beta _0,\Lambda _0|{\hat{u}}) = {\mathbb {P}}_n m(\beta _0,\Lambda _0|{\hat{u}}) \end{aligned}$$

and thereby

$$\begin{aligned} \begin{aligned}&[{\mathbb {P}}_n m({{\hat{\beta }}},{{\hat{\Lambda }}}|{\hat{u}}) - {\mathbb {P}} m(\beta ^*,\Lambda ^*|u)] + {\mathbb {P}} m(\beta ^*,\Lambda ^*|u)\\&\quad \ge \, [{\mathbb {P}}_n m(\beta _0,\Lambda _0|{\hat{u}}) -{\mathbb {P}} m(\beta _0,\Lambda _0|u)] + {\mathbb {P}} m(\beta _0,\Lambda _0|u). \end{aligned} \end{aligned}$$

Arguing as in Zeng et al. [32], we can show that ${\mathcal {M}} = \{m(\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z)):\, \beta \in {\mathcal {B}},\,\alpha \in {\mathcal {A}},\,\Lambda \in {\mathcal {L}},\,\Lambda _h\in {\mathcal {L}}_h\}$ is a Glivenko–Cantelli class, where ${\mathcal {L}}$ is the set of nondecreasing functions $\Lambda$ on $[0,\tau _0]$ satisfying $\Lambda (0)=0$ and ${\mathcal {L}}_h$ is the set of nondecreasing functions $\Lambda _h$ on $[0,\tau _0]$ satisfying $\Lambda _h(0)=0$, $\Lambda _h(\zeta _0)\ge \Lambda _{0h}(\zeta _0)-c_0>0$ for some positive constant $c_0$ and $\Lambda _h(\tau _0)\le 1$. Furthermore, based on the asymptotic properties of $({{\hat{\alpha }}},{\hat{\Lambda }}_h)$ established by Wang et al. [29], we can show that ${\mathbb {P}}_n m(\beta ,\Lambda |{\hat{u}})$ converges to ${\mathbb {P}} m(\beta ,\Lambda |u)$ almost surely for any fixed $(\beta ,\Lambda )$. Therefore, we have ${\mathbb {P}} m(\beta ^*,\Lambda ^*|u)\ge {\mathbb {P}} m(\beta _0,\Lambda _0|u)$ and further

$$\begin{aligned} {\mathbb {P}} \log \left\{ \frac{p(\beta ^*,\Lambda ^*|u)+p(\beta _0,\Lambda _0|u)}{2}\right\} \ge {\mathbb {P}} \log p(\beta _0,\Lambda _0|u). \end{aligned}$$

By the properties of the Kullback–Leibler information, $p(\beta ^*,\Lambda ^*|u)=p(\beta _0,\Lambda _0|u)$ with probability 1. Thus, for any $t\in [0,\tau _0]$, $\log \{\Lambda ^*(t)\}+\beta _1^{*T} z +\beta _2^* u=\log \{\Lambda _0(t)\}+\beta _{01}^T z +\beta _{02} u$. Under Condition (C2), we obtain $\beta ^*=\beta _0$ and $\Lambda ^*=\Lambda _0$. This completes the proof. $\square$

Proof of Theorem 4.2

Let $\beta =(\beta _1^T,\beta _2)^T$ and $x=(z^T,u)^T$. The score function for $\beta$ based on the log-likelihood $l(\beta ,\Lambda |u)$ is

$$\begin{aligned} \begin{aligned}&l_\beta (\beta ,\Lambda |u) \\&= \sum _{k=1}^{K+1} \delta _k \, \left\{ \frac{-\exp (-G[\Lambda (U_{k-1})\exp (\beta ^T x)]) G'[\Lambda (U_{k-1})\exp (\beta ^T x)] \Lambda (U_{k-1})}{M(U_{k-1},U_k;\beta ,\Lambda ,x)} \right. \\&\quad \left. + \frac{\exp (-G[\Lambda (U_k)\exp (\beta ^T x)]) G'[\Lambda (U_k)\exp (\beta ^T x)] \Lambda (U_k)}{M(U_{k-1},U_k;\beta ,\Lambda ,x)}\right\} \, \exp (\beta ^T x) x, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} M(u,v;\beta ,\Lambda ,x) = \exp (-G[\Lambda (u)\exp (\beta ^T x)]) - \exp (-G[\Lambda (v)\exp (\beta ^T x)]). \end{aligned}$$

The score function for $\beta$ based on the weighted log-likelihood $l^w(\beta ,\Lambda |u)$ is given by

$$\begin{aligned} l_\beta ^w(\beta ,\Lambda |u)=w\,l_\beta (\beta ,\Lambda |u). \end{aligned}$$

To obtain the score operator for $\Lambda$, we consider a parametric submodel of $\Lambda$ defined by $d\Lambda _{\epsilon ,h}=(1+\epsilon h)d\Lambda$ for $h\in L_2([0,\tau _0])$. The score function along this submodel based on the log-likelihood $l(\beta ,\Lambda |u)$ is

$$\begin{aligned} \begin{aligned}&l_\Lambda (\beta ,\Lambda |u)(h) \\&= \frac{\partial }{\partial \epsilon } \, l(\beta ,\Lambda _{\epsilon ,h}|u) \Big |_{\epsilon =0}\\&\quad = \sum _{k=1}^{K+1} \delta _k \, \left\{ \frac{-\exp (-G[\Lambda (U_{k-1})\exp (\beta ^T x)]) G'[\Lambda (U_{k-1})\exp (\beta ^T x)]}{M(U_{k-1},U_k;\beta ,\Lambda ,x)} \int _0^{U_{k-1}} h(t) \textrm{d}\Lambda (t) \right. \\&\qquad \left. + \frac{\exp (-G[\Lambda (U_k)\exp (\beta ^T x)]) G'[\Lambda (U_k)\exp (\beta ^T x)]}{M(U_{k-1},U_k;\beta ,\Lambda ,x)} \int _0^{U_k} h(t) \textrm{d}\Lambda (t)\right\} \, \exp (\beta ^T x). \end{aligned} \end{aligned}$$

The score function along this submodel based on the weighted log-likelihood $l^w(\beta ,\Lambda |u)$ is

$$\begin{aligned} l_\Lambda ^w(\beta ,\Lambda |u)(h) = w\,l_\Lambda (\beta ,\Lambda |u)(h). \end{aligned}$$

By the definition of $({\hat{\beta }},{\hat{\Lambda }})$, we have ${\mathbb {P}}_n\{l_\beta ^w({\hat{\beta }},{\hat{\Lambda }}|{\hat{u}})\}=0$ and ${\mathbb {P}}_n\{l_\Lambda ^w({\hat{\beta }},{\hat{\Lambda }}|{\hat{u}})(h)\}=0$. Also, ${\mathbb {P}}\{l_\beta ^w(\beta _0,\Lambda _0|u)\}={\mathbb {P}}\{l_\beta (\beta _0,\Lambda _0|u)\}=0$ and ${\mathbb {P}}\{l_\Lambda ^w(\beta _0,\Lambda _0|u)(h)\}={\mathbb {P}}\{l_\Lambda (\beta _0,\Lambda _0|u)(h)\}=0$. Therefore,

$$\begin{aligned}{} & {} n^{1/2}[{\mathbb {P}}_n\{{l}_\beta ^w({\hat{\beta }},{\hat{\Lambda }}|{\hat{u}})\}-{\mathbb {P}}\{l_\beta ({\hat{\beta }},{\hat{\Lambda }}|u)\}]\nonumber \\{} & {} \quad = - n^{1/2}[{\mathbb {P}}\{l_\beta ({\hat{\beta }},{\hat{\Lambda }}|u)\}-{\mathbb {P}}\{l_\beta (\beta _0,\Lambda _0|u)\}] \end{aligned}$$

(8.1)

and

$$\begin{aligned}{} & {} n^{1/2}[{\mathbb {P}}_n\{{l}_\Lambda ^w({\hat{\beta }},{\hat{\Lambda }}|{\hat{u}})(h)\}-{\mathbb {P}}\{l_\Lambda ({\hat{\beta }},{\hat{\Lambda }}|u)(h)\}]\nonumber \\{} & {} \quad = - n^{1/2}[{\mathbb {P}}\{l_\Lambda ({\hat{\beta }},{\hat{\Lambda }}|u)(h)\}-{\mathbb {P}}\{l_\Lambda (\beta _0,\Lambda _0|u)(h)\}]. \end{aligned}$$

(8.2)

We first consider ${\mathbb {P}}_n\{{l}_\beta ^w(\beta ,\Lambda |{\hat{u}})\}-{\mathbb {P}}\{l_\beta (\beta ,\Lambda |u)\}$ and ${\mathbb {P}}_n\{{l}_\Lambda ^w(\beta ,\Lambda |{\hat{u}})(h)\}-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}$ for fixed $(\beta ,\Lambda )$. Define the functions $H(t)=E[\exp (u) I(\tau \ge t)]$, $R(t)=H(t)\Lambda _{0h}(t)$, $Q(t)=\int _0^t H(s)\textrm{d}\Lambda _{0h}(s)$, and for $i=1,\ldots ,n$,

$$\begin{aligned} b_i(t)=\sum _{k=1}^{K_i} \left\{ \int _t^{\tau _0} \frac{I(U_{ik}\le s\le \tau _i)}{R^2(s)} \textrm{d}Q(s) -\frac{I(t\le U_{ik}\le \tau _0)}{R(U_{ik})}\right\} . \end{aligned}$$

In addition, for $i=1,\ldots ,n$, define

$$\begin{aligned} e_i=-\int \frac{{\tilde{w}}{\tilde{z}}kb_i(\tau )}{\Lambda _{0h}(\tau )}\textrm{d}{\mathcal {P}}({\tilde{w}},{\tilde{z}},k,\tau )+{\tilde{w}}_i{\tilde{z}}_i\{K_i\Lambda _{0h}^{-1}(\tau _i)-\exp (\gamma ^T {\tilde{z}}_i)\}, \end{aligned}$$

where ${\tilde{z}}_i=(1,z_i^T)^T$, $\gamma =(\log \{E[\exp (u)]\},\alpha ^T)^T$, ${\tilde{w}}_i$ is the weight given in the estimating equations for $\alpha$, and ${\mathcal {P}}(\cdot )$ denotes the joint probability measure of $({\tilde{w}},{\tilde{z}},K,\tau )$. From Wang et al. [29], we have ${\hat{\Lambda }}_h(t)-\Lambda _{0h}(t)=n^{-1}\sum _{i=1}^n \Lambda _{0h}(t)b_i(t)+o_p(n^{-1/2})$ for $\inf \{s:\Lambda _{0h}(s)>0\}<t<\tau _0$ and ${\hat{\alpha }}-\alpha _0=n^{-1}\sum _{i=1}^n f_i(\alpha _0)+o_p(n^{-1/2})$, where $f_i(\alpha )=E[-\partial e_1/\partial \gamma ]^{-1}e_i$ without the first entry. Define the function $u(\alpha ,\Lambda _h;\tau ,K,z)=\log \{K/[\Lambda _h(\tau )\exp (\alpha ^T z)]\}$. Then ${\hat{u}}=u({\hat{\alpha }},{\hat{\Lambda }}_h;\tau ,K,z)$. Furthermore, define

$$\begin{aligned} l_{\beta \alpha }(\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z)) = \frac{\partial }{\partial \alpha } l_\beta (\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z)) \end{aligned}$$

and

$$\begin{aligned} l_{\beta \Lambda _h}(\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z)) = \frac{\partial }{\partial s} l_\beta (\beta ,\Lambda |u(\alpha ,s;\tau ,K,z))\Big |_{s=\Lambda _h(\tau )}. \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {P}}_n\{{l}_\beta ^w(\beta ,\Lambda |{\hat{u}})\}-{\mathbb {P}}\{l_\beta (\beta ,\Lambda |u)\}\\&\quad =\, {\mathbb {P}}_n\{l_\beta ^w(\beta ,\Lambda |u({\hat{\alpha }},{\hat{\Lambda }}_h;\tau ,K,z))\}-{\mathbb {P}}_n\{l_\beta ^w(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau ,K,z))\}\\&\qquad +{\mathbb {P}}_n\{l_\beta ^w(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau ,K,z))\}-{\mathbb {P}}\{l_\beta (\beta ,\Lambda |u)\}\\&\quad =\frac{1}{n}\sum _{i=1}^n \Big \{{\mathbb {P}}\{l_{\beta \alpha }(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau _i,K_i,z_i)) f_i(\alpha _0)\}\\&\qquad +{\mathbb {P}}\{l_{\beta \Lambda _h}(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau _i,K_i,z_i))\Lambda _{0h}(\tau _i)b_i(\tau _i)\}\\&\qquad +w_i \, l_\beta (\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau _i,K_i,z_i))-{\mathbb {P}}\{l_\beta (\beta ,\Lambda |u)\}\Big \} + o_p(n^{-1/2})\\&\quad =\, \frac{1}{n}\sum _{i=1}^n c_{\beta i}(\beta ,\Lambda )+ o_p(n^{-1/2}). \end{aligned} \end{aligned}$$

(8.3)

The $c_{\beta i}(\beta ,\Lambda )$’s are independent random variables because $c_{\beta i}(\beta ,\Lambda )$ depends only on the observed data from the ith subject. It follows from the law of large numbers that for fixed $(\beta ,\Lambda )$, ${\mathbb {P}}_n\{{l}_\beta ^w(\beta ,\Lambda |{\hat{u}})\}-{\mathbb {P}}\{l_\beta (\beta ,\Lambda |u)\}\rightarrow 0$ almost surely as $n\rightarrow \infty$. Furthermore, by the central limit theorem, $n^{1/2}[{\mathbb {P}}_n\{{l}_\beta ^w(\beta ,\Lambda |{\hat{u}})\}-{\mathbb {P}}\{l_\beta (\beta ,\Lambda |u)\}]$ converges in distribution to a zero-mean normal random vector. Similarly, we can derive the asymptotic properties of ${\mathbb {P}}_n\{{l}_\Lambda ^w(\beta ,\Lambda |{\hat{u}})(h)\}-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}$. In particular, define

$$\begin{aligned} l_{\Lambda \alpha }(\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z))(h) = \frac{\partial }{\partial \alpha } l_\Lambda (\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z))(h) \end{aligned}$$

and

$$\begin{aligned} l_{\Lambda \Lambda _h}(\beta ,\Lambda |u(\alpha ,\Lambda _h;\tau ,K,z))(h) = \frac{\partial }{\partial s} l_\Lambda (\beta ,\Lambda |u(\alpha ,s;\tau ,K,z))(h)\Big |_{s=\Lambda _h(\tau )}. \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {P}}_n\{{l}_\Lambda ^w(\beta ,\Lambda |{\hat{u}})(h)\}-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}\\&\quad = {\mathbb {P}}_n\{l_\Lambda ^w(\beta ,\Lambda |u({\hat{\alpha }},{\hat{\Lambda }}_h;\tau ,K,z))(h)\}-{\mathbb {P}}_n\{l_\Lambda ^w(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau ,K,z))(h)\}\\&\qquad +{\mathbb {P}}_n\{l_\Lambda ^w(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau ,K,z))(h)\}-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}\\&\quad =\frac{1}{n}\sum _{i=1}^n \left\{ {\mathbb {P}}\{[l_{\Lambda \alpha }(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau _i,K_i,z_i))(h)] f_i(\alpha _0)\}\right. \\&\qquad +{\mathbb {P}}\{[l_{\Lambda \Lambda _h}(\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau _i,K_i,z_i))(h)]\Lambda _{0h}(\tau _i)b_i(\tau _i)\}\\&\left. \qquad +w_i \, [l_\Lambda (\beta ,\Lambda |u(\alpha _0,\Lambda _{0h};\tau _i,K_i,z_i))(h)]-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}\right\} + o_p(n^{-1/2})\\&\quad = \frac{1}{n}\sum _{i=1}^n c_{\Lambda i}(\beta ,\Lambda )(h)+ o_p(n^{-1/2}). \end{aligned} \end{aligned}$$

(8.4)

The $c_{\Lambda i}(\beta ,\Lambda )(h)$’s are independent random variables because $c_{\Lambda i}(\beta ,\Lambda )(h)$ depends only on the observed data from the ith subject. By the law of large numbers, ${\mathbb {P}}_n\{{l}_\Lambda ^w(\beta ,\Lambda |{\hat{u}})(h)\}-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}\rightarrow 0$ almost surely as $n\rightarrow \infty$, for fixed $(\beta ,\Lambda )$. By the central limit theorem, $n^{1/2}[{\mathbb {P}}_n\{{l}_\Lambda ^w(\beta ,\Lambda |{\hat{u}})(h)\}-{\mathbb {P}}\{l_\Lambda (\beta ,\Lambda |u)(h)\}]$ converges in distribution to a zero-mean normal random vector.

On the other hand, arguing as in the proof of Theorem 2 of Zeng et al. [33], we can show that

$$\begin{aligned}{} & {} -[{\mathbb {P}}\{l_\beta ({\hat{\beta }},{\hat{\Lambda }}|u)\}-{\mathbb {P}}\{l_\beta (\beta _0,\Lambda _0|u)\}] + [{\mathbb {P}}\{l_\Lambda ({\hat{\beta }},{\hat{\Lambda }}|u)(h^*)\}-{\mathbb {P}}\{l_\Lambda (\beta _0,\Lambda _0|u)(h^*)\}]\nonumber \\{} & {} \quad = E[\{l_\beta -l_\Lambda (h^*)\}\{l_\beta -l_\Lambda (h^*)\}^T]({{\hat{\beta }}}-\beta _0)+O_p(\Vert {{\hat{\beta }}}-\beta _0\Vert ^2+n^{-2/3}),\nonumber \\ \end{aligned}$$

(8.5)

where $l_\beta =l_\beta (\beta _0,\Lambda _0|u)$, $l_\Lambda (h^*)=l_\beta (\beta _0,\Lambda _0|u)(h^*)$, and $h^*$ is the least favourable direction, a $(p+1)$-vector with components in $L_2([0,\tau _0])$, that solves the normal equation $l_\Lambda ^*l_\Lambda (h^*)=l_\Lambda ^*l_\beta$ with $l_\Lambda ^*$ being the adjoint operator of $l_\Lambda$. The existence of $h^*$ can be established as in Zeng et al. [33]. From Eqs. (8.3)–(8.5), the difference between (8.1) and (8.2) yields

$$\begin{aligned}{} & {} n^{-1/2}\sum _{i=1}^n \{c_{\beta i}({\hat{\beta }},{\hat{\Lambda }})-c_{\Lambda i}({\hat{\beta }},{\hat{\Lambda }})(h^*)\}+o_p(1)\nonumber \\{} & {} \quad =n^{1/2}E[\{l_\beta -l_\Lambda (h^*)\}\{l_\beta -l_\Lambda (h^*)\}^T]({{\hat{\beta }}}-\beta _0)+O_p(n^{1/2}\Vert {{\hat{\beta }}}-\beta _0\Vert ^2+n^{-1/6}).\nonumber \\ \end{aligned}$$

(8.6)

The left-hand side of (8.6) can be written as ${\mathbb {G}}_n\{c_{\beta }({\hat{\beta }},{\hat{\Lambda }})-c_{\Lambda }({\hat{\beta }},{\hat{\Lambda }})(h^*)\}+o_p(1)$. As argued in Zeng et al. [26], we can show that $h^*(t)$ is continuously differentiable on $[0,\tau _0]$, and further we are able to prove that $c_{\beta }({\hat{\beta }},{\hat{\Lambda }})-c_{\Lambda }({\hat{\beta }},{\hat{\Lambda }})(h^*)$ belongs to a Donsker class and converges in the $L_2({\mathbb {P}})$-norm to $c_{\beta }-c_{\Lambda }(h^*)$, where $c_{\beta }$ and $c_{\Lambda }(h^*)$ are evaluated at $(\beta _0,\Lambda _0)$. In addition, it is easy to show via proof by contradiction that the matrix $E[\{l_\beta -l_\Lambda (h^*)\}\{l_\beta -l_\Lambda (h^*)\}^T]$ is invertible. Therefore, (8.6) entails $n^{1/2}({{\hat{\beta }}}-\beta _0)=O_p(1)$ and yields

$$\begin{aligned} n^{1/2}({{\hat{\beta }}}-\beta _0) = \left( E[\{l_\beta -l_\Lambda (h^*)\}\{l_\beta -l_\Lambda (h^*)\}^T]\right) ^{-1} {\mathbb {G}}_n\{c_{\beta }-c_{\Lambda }(h^*)\}+o_p(1). \end{aligned}$$

This implies that $n^{1/2}({{\hat{\beta }}}-\beta _0)$ converges to a zero-mean normal random vector. $\square$