Estimation of complier causal treatment effects with informatively interval-censored failure time data

Abstract

Estimation of compiler causal treatment effects has been discussed by many authors under different situations but only limited literature exists for interval-censored failure time data, which often occur in many areas such as longitudinal or periodical follow-up studies. Particularly it does not seem to exist a method that can deal with informative interval censoring, which can happen naturally and make the analysis much more challenging. Also, it has been shown that when the informative censoring exists, the analysis without taking it into account would yield biased or misleading results. To address this, we propose an estimated sieve maximum likelihood approach with the use of instrumental variables. The asymptotic properties of the resulting estimators of regression parameters are established, and a simulation study is performed and suggests that it works well. Finally, it is applied to a set of real data that motivated this study.

Appendix: Proofs of Theorems 1–3

In this Appendix, we will sketch the proofs for the asymptotic properties of ${\hat{\xi }}_n$ given in Theorems 1–3. To establish the asymptotic properties of ${\hat{\xi }}_n$, we need the following regularity conditions, which are commonly used in the studies of interval-censored data and usually satisfied in practice (Huang and Wolfe 2002; Ma et al. 2016; Zhang et al. 2010; Zhou et al. 2016).

(A.1):

The true value for $\eta$, denoted as $\eta _0$, is in the interior of a compact set $\mathcal {B}$ in $R^{p_{\eta }}$, $\Vert \eta _0\Vert \le B$ for a constant $B>0$, and $P(R-L>\varepsilon )=1$ for some $\varepsilon >0$.

(A.2):

The distribution of the covariate X has a bounded support in $R^p$ and is not concentrated on any proper subspace of $R^p$.

(A.3):

The first derivative of $\Lambda _{t0}(\cdot )$ and $\Lambda _{w0}(\cdot )$, denoted by $\Lambda _{t0}^{(1)}(\cdot )$ and $\Lambda _{w0}^{(1)}(\cdot )$, is Holder continuous with exponent $s\in (0, 1]$. That is, there exists a constant $K > 0$ such that $|\Lambda _{t0}^{(1)}(t_1)- \Lambda _{t0}^{(1)}(t_2)|\le K|t_1-t_2|^{s}$ for all $t_1$, $t_2\in [\sigma _1,\tau _1]$, where $0<\sigma _1<\tau _1<\infty$, and $\Lambda _{w0}(\cdot )$ has the similar properties. Let $v=1+s$.

(A.4):

There exists a constant $K >0$ such that for every $\xi$ in a neighborhood of $\xi _0$, $P\{l(\xi ,O)-l(\xi _0,O)\}\preceq -Kd^2(\xi ,\xi _0)$, where O is the observation data and $\preceq$ means ‘smaller than, up to a constant.’

(A.5):

The matrix $E({l^*(\eta _0,O)}^{\otimes 2})$ is finite and positive definite, where $a^{\otimes 2}=aa^{T}$ for a vector a, and $l^*(\eta ,O)$ is the efficient score for $\eta$ based on the observation O and given in the proof of Theorem 3.

For the proof, we will mainly employ the empirical process theory and some nonparametric techniques. Let $Pf=\int f(y)dP$ denote the expectation of f(Y) under the probability measure P, and $P_nf = n^{-1}\sum _{i=1}^nf(Y_i)$, the expectation of f(Y) under the empirical measure $P_n$. Define the covering number of the class $\mathcal {L}_n =\{l(\xi ,O ):\xi \in \Psi _n\}$, where $l(\xi ,O)$ is the log-likelihood function based on a single observation O. Also for any $\epsilon >0$, define the covering number $N(\epsilon ,\mathcal {L}_n, L_1(P_n))$ as the smallest positive integer $\kappa$ for which there exists $\{\xi ^{(1)},\ldots ,\xi ^{(\kappa )}$} such that

$$\begin{aligned} \min _{j\in \{1,\ldots ,\kappa \}}\frac{1}{n}\sum _{i=1}^n|l(\xi ,O_i)-l(\xi ^{(j)},O_i)|<\epsilon \end{aligned}$$

for all $\xi \in \Psi _n$, where $\{O_1,\ldots ,O_n\}$ represent the observed data and for $j =1,\ldots ,\kappa$, $\xi ^{(j)}=(\eta ^{(j)},\Lambda _t^{(j)},\Lambda _w^{(j)})\in \Psi _n$. If no such $\kappa$ exists, define $N(\epsilon ,\mathcal {L}_n, L_1(P_n))=\infty$. Also for the proof, we need first to establish the following two lemmas, whose proofs are similar to those for Lemmas 1 and 2 in Zhou et al. (2016).

Lemma 1

Assume that the regularity conditions (A.1)-(A.3) given above hold. Then, we have that the covering number of the class $\mathcal {L}_n =\{l(\xi ,O):\xi \in \Psi _n\}$ satisfies

$$\begin{aligned} N\Big (\epsilon ,\mathcal {L}_n, L_1(P_n)\Big )\le KM_n^{2(m+1)}\epsilon ^{-(p_{\eta }+2(m+1))} \end{aligned}$$

for a constant K, where $m=o(n^q)$ with $q\in (0, 1)$ is the degree of Bernstein polynomials, and $M_n = O(n^a)$ with $a>0$ controls the size of the sieve space $\Psi _n$.

Lemma 2

Assume that the regularity conditions (A.1)–(A.3) given above hold. Then, we have that

$$\begin{aligned} \sup _{\xi \in \Psi _n}|P_nl(\xi ,O)-Pl(\xi ,O)|\rightarrow 0 \end{aligned}$$

almost surely.

Now, we are ready to prove Theorems 1–3.

Proof of Theorem 1

We first prove the strong consistency of ${\hat{\xi }}_n$. Let $l(\xi ,O)$ denote the log-likelihood function based on a given single observation O and consider the class of functions $\mathcal {L}_n =\{l(\xi ,O ):\xi \in \Psi _n\}$. By Lemma 1, the covering number of $\mathcal {L}_n$ satisfies

$$\begin{aligned} N\Big (\epsilon ,\mathcal {L}_n, L_1(P_n)\Big )\le KM_n^{2(m+1)}\epsilon ^{-(p_{\eta }+2m+2)}. \end{aligned}$$

Furthermore, by Lemma 2, we have

$$\begin{aligned} \begin{aligned} \sup _{\xi \in \Psi _n}|P_nl(\xi ,O)-Pl(\xi ,O)|\rightarrow 0 \ \text {almost surely}. \end{aligned} \end{aligned}$$

(5)

Note that $Pl(\xi ,O)=P\{pl(\xi ,O)\}=Pl(\xi ,O)$ and $\xi _0$ maximizes $Pl(\xi ,O)$. Let $M(\xi ,O)=-l(\xi ,O)$, and define $K_{\epsilon }=\{\xi :d(\xi ,\xi _0)\ge \epsilon ,\xi \in \Psi _n\}$ for $\epsilon >0$ and

$$\begin{aligned} \zeta _{1n}=\sup _{\xi \in \Psi _n}|P_nM(\xi ,O)-PM(\xi ,O)|,\zeta _{2n}=P_nM(\xi _0,O)-PM(\xi _0,O). \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned} \inf _{K_{\epsilon }}PM(\xi ,O)=&\inf _{K_{\epsilon }}\Big \{PM(\xi ,O)-P_nM(\xi ,O)+P_nM(\xi ,O)\Big \}\\&\le \zeta _{1n}+\inf _{K_{\epsilon }}P_nM(\xi ,O). \end{aligned} \end{aligned}$$

(6)

If ${\hat{\xi }}_n\in K_{\epsilon }$, then we have

$$\begin{aligned} \begin{aligned} \inf _{K_{\epsilon }}P_nM(\xi ,O)=P_nM({\hat{\xi }}_n,O)\le P_nM(\xi _0,O)=\zeta _{2n}+PM(\xi _0,O). \end{aligned} \end{aligned}$$

(7)

Define $\delta _{\epsilon }=\inf _{K_{\epsilon }}PM(\xi ,O)-PM(\xi _0,O)$. Under Condition (A.4), we have $\delta _{\epsilon } > 0$. It follows from (6) and (7) that

$$\begin{aligned} \inf _{K_{\epsilon }}PM(\xi ,O)\le \zeta _{1n}+\zeta _{2n}+PM(\xi _0,O)=\zeta _n+PM(\xi _0,O), \end{aligned}$$

with $\zeta _n = \zeta _{1n} + \zeta _{2n}$, and hence, $\zeta _n\ge \delta _{\epsilon }$. This gives $\{{\hat{\xi }}_n\in K_{\epsilon }\}\subseteq \{\zeta _n\ge \delta _{\epsilon }\}$, and by (5) and the strong law of large numbers, we have both $\zeta _{1n}\rightarrow 0$ and $\zeta _{2n}\rightarrow 0$ almost surely. Therefore, $\bigcup _{k=1}^{\infty }\bigcap _{n=k}^{\infty }\{{\hat{\xi }}_n\in K_{\epsilon }\}\subseteq \bigcup _{k=1}^{\infty }\bigcap _{n=k}^{\infty }\{\zeta _n\ge \delta _{\epsilon }\}$, which proves that $d({\hat{\xi }}_n,\xi _0)\rightarrow 0$ almost surely. $\square$

Proof of Theorem 2

We will show the convergence rate of ${\hat{\xi }}_n$ by using Theorem 3.4.1 of Van and Wellner (1996). First note from Theorem 1.6.2 of Lorentz (1986) that exists Bernstein polynomials $\Lambda _{tn0}$ and $\Lambda _{wn0}$ such that $\Vert \Lambda _{tn0}-\Lambda _{t0}\Vert _{\infty }=O(m^{-v/2})$ and $\Vert \Lambda _{wn0}-\Lambda _{w0}\Vert _{\infty }=O(m^{-v/2})$. Then, we have $d(\xi _{n0}-\xi _0)=O(n^{-vq/2})$. For any $s>0$, define the class of functions $\mathcal {F}_s=\{l(\xi ,O)-l(\xi _{n0},O):\xi \in \Psi _n,s/2<d(\xi -\xi _{n0})\le s\}$. One can easily show that $P\{l(\xi _0,O)-l(\xi _{n0},O)\}\le Kd^2(\xi _0,\xi _{n0})\le Kn^{-vq}$. Hence, under Condition (A.4), we have for large n, $P\{l(\xi ,O)-l(\xi _{n0},O)\}=P\{l(\xi ,O)-l(\xi _0,O)\}+P\{l(\xi _0,O)-l(\xi _{n0},O)\}\le -Ks^2+Kn^{-vq}=-Ks^2$, for any $l(\xi ,O)-l(\xi _{n0},O)\in \mathcal {F}_s$.

Following the calculations in Shen and Wong (1994), we can establish that for $0<\epsilon <s$, $\log N_{[]}(\epsilon , \mathcal {F}_s, L_2(P))\le KN\log (s/\epsilon )$ with $N = 2(m+1)$. Moreover, some algebraic manipulations yield that $P\{(l(\xi ,O)-l(\xi _{n0},O)\}^2\le Ks^2$ for any $l(\xi ,O)-l(\xi _{n0},O)\in \mathcal {F}_s$. It is easy to see that $\mathcal {F}_s$ is uniformly bounded. Therefore, by Lemma 3.4.2 of Van and Wellner (1996), we obtain

$$\begin{aligned} E_{P}\left\| n^{1 / 2}\left( P_{n}-P\right) \right\| _{\mathcal {F}_{s}} \le K J_{[]}\left( s, \mathcal {F}_{s}, L_{2}(P)\right) \left\{ 1+\frac{J_{[]}\left( s, \mathcal {F}_{s}, L_{2}(P)\right) }{s^{2} n^{1 / 2}}\right\} , \end{aligned}$$

where $J_{[]}\left( s, \mathcal {F}_{s}, L_{2}(P)\right) =\int _0^s\{1+\log N_{[]}(\epsilon , \mathcal {F}_s, L_2(P))\}^{1/2}d\epsilon \le KN^{1/2}s$. This yields $\phi _n(s)=N^{1/2}s+N/n^{1/2}$. It is easy to see that $\phi _n(s)/s$ is decreasing in s, and $v_n^2\phi _n(1/v_n)=v_nN^{1/2}+v_n^2N/n^{1/2}\le Kn^{1/2}$, where $v_n=N^{-1/2}n^{1/2}=n^{(1-q)/2}$.

Finally, note that $P_n\{l({\hat{\xi }}_n,O)-l(\xi _{n0},O)\}\ge 0$ and $d({\hat{\xi }}_n,\xi _{n0})\le d({\hat{\xi }}_n,\xi _{0})+d(\xi _0,\xi _{n0})\rightarrow 0$ in probability. Thus by applying Theorem 3.4.1 of Van and Wellner (1996), we have $n^{(1-q)/2}d({\hat{\xi }}_n,\xi _{n0})= O_p(1)$. This together with $d(\xi _{n0},\xi _{0}) = O(n^{-vq/2})$ yields that $d({\hat{\xi }}_n,\xi _{0})=O_p(n^{-(1-q)/2}+n^{-vq/2})$ and the proof is completed.$\square$

Proof of Theorem 3

Now, we will prove the asymptotic normality of ${\hat{\eta }}_n$. Let V denote the linear span of $\Psi -\xi _0$ and define the Fisher inner product for $u,\tilde{u} \in V$ as $<u,\tilde{u}>=P\{\dot{l}(\xi _0,O)[u]\dot{l}(\xi _0,O)[\tilde{u}]\}$ and the Fisher norm for $u\in V$ as $||u||^2=<u,u>$, where

$$\begin{aligned} \dot{l}(\xi _0,O)[u]=\frac{dl(\xi _0+su,O)}{ds}\big |_{s=0} \end{aligned}$$

denotes the first-order directional derivative of $l(\xi ,O)$ at the direction $u\in V$ (evaluated at $\xi _0)$. Also, let $\bar{V}$ be the closed linear span of V under the Fisher norm. Then, $(\bar{V}, ||\cdot ||)$ is a Hilbert space. Furthermore, for a vector of $p_{\eta }$ dimension b with $||b||\le 1$ and for any $u\in V$, define a smooth function of $\xi$ as $h(\xi )=b^{T}\eta$ and

$$\begin{aligned} \dot{h}(\xi _0)[u]=\frac{dh(\xi _0+su)}{ds}\big |_{s=0} \end{aligned}$$

whenever the right-hand side limit is well defined. Then by the Riesz representation theorem, there exists $u^{*}\in \bar{V}$ such that $\dot{h}(\xi _0)[u]=<u,u^{*}>$ for all $u\in \bar{V}$ and $||u^{*}||=||\dot{h}(\xi _0)||$. Also, note that $h(\xi )-h(\xi _0) = \dot{h}(\xi _0)[\xi -\xi _0]$. It thus follows from the Cramér–Wold device that to prove the asymptotic normality for ${\hat{\eta }}_n$, i.e., $n^{1/2}({\hat{\eta }}_n-\eta _0)\rightarrow N(0, I^{-1}(\eta _0))$ in distribution, it suffices to show that

$$\begin{aligned} n^{1/2}<{\hat{\xi }}_n-\xi _0,u^{*}>\rightarrow _{d}N(0,b^{T}I^{-1}(\eta _0)b) \end{aligned}$$

since $b^{T}({\hat{\eta }}_n-\eta _0)=h({\hat{\xi }}_n)-h(\xi _0)=\dot{h}(\xi _0)[\xi -\xi _0]=<{\hat{\xi }}_n-\xi _0,u^{*}>$. In fact, the above holds since one can show that $n^{1/2}<{\hat{\xi }}_n-\xi _0,u^{*}>\rightarrow _{d}N(0,||u^{*}||^2)$ and $||u^{*}||^2=b^{T}I^{-1}(\eta _0)b$.

We first prove that $n^{1/2}<{\hat{\xi }}_n-\xi _0,u^{*}>\rightarrow _{d}N(0,||u^{*}||^2)$. Let $\delta _n=n^{-\min \{(1-q)/2,vq/2\}}$ denote the rate of convergence obtained in Theorem 2, and for any $\xi \in \Psi$ such that $d(\xi ,\xi _0)\le \delta _n$, define the first-order directional derivative of $l(\xi ,O)$ at the direction $u\in V$ as

$$\begin{aligned} \dot{l}(\xi ,O)[u]=\frac{dl(\xi +su,O)}{ds}\big |_{s=0} \end{aligned}$$

and the second-order directional derivative at the direction $u,\tilde{u} \in V$ as

$$\begin{aligned} {\ddot{l}}(\xi ,O)[u,\tilde{u}]=\frac{d^2l(\xi +su+\tilde{s}\tilde{u},O)}{d\tilde{s}ds} \big |_{s=0}\big |_{\tilde{s}=0}=\frac{d\dot{l}(\xi +\tilde{s}\tilde{u},O)[\tilde{u}]}{d\tilde{s}} \big |_{\tilde{s}=0}. \end{aligned}$$

Note that by Condition (A.3) and Theorem 1.6.2 of Lorentz (1986), there exists $\Pi _n u^{*}\in \Psi -\xi _0$ such that $||\Pi _n u^{*}-u^{*}||=O(n^{-qv})$. Furthermore, under the assumption $q>1/2v$, we have $\delta _n ||\Pi _n u^{*}-u^{*}||=o(n^{-1/2})$. Define $v[\xi -\xi _0,O]=l(\xi ,O)-l(\xi _0,O)-\dot{l}(\xi _0,O)[\xi -\xi _0]$ and let $\varepsilon _n$ be any positive sequence satisfying $\varepsilon _n=o(n^{-1/2})$. Then by the definition of ${\hat{\xi }}_n$, we have

$$\begin{aligned} \begin{aligned} 0 \le&P_{n}\left[ l\left( {\hat{\xi }}_{n}, O\right) -l\left( {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}, O\right) \right] \\ =&\pm \varepsilon _{n} P_{n} \dot{l}\left( \xi _{0}, O\right) \left[ \Pi _{n} u^{*}\right] +\left( P_{n}-P\right) \left\{ v\left[ {\hat{\xi }}_{n}-\xi _{0}, O\right] -v\left[ {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}-\xi _{0}, O\right] \right\} \\&+P\left\{ v\left[ {\hat{\xi }}_{n}-\xi _{0}, O\right] -v\left[ {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}-\xi _{0}, O\right] \right\} \\ =&\pm \varepsilon _{n} P_{n} \dot{l}\left( \xi _{0}, O\right) \left[ u^{*}\right] \pm \varepsilon _{n} P_{n} \dot{l}\left( \xi _{0}, O\right) \left[ \Pi _{n} u^{*}-u^{*}\right] +\left( P_{n}-P\right) \left\{ v\left[ {\hat{\xi }}_{n}-\xi _{0}, O\right] \right. \\&\left. -r\left[ {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}-\xi _{0}, O\right] \right\} +P\left\{ v\left[ {\hat{\xi }}_{n}-\xi _{0}, O\right] -v\left[ {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}-\xi _{0}, O\right] \right\} \\ =&\pm \varepsilon _{n} P_{n} \dot{l}\left( \xi _{0}, O\right) \left[ u^{*}\right] \pm I_{1}+I_{2}+I_{3}. \end{aligned} \end{aligned}$$

We will investigate the asymptotic behavior of $I_1$, $I_2$ and $I_3$. For $I_1$, it follows from Conditions (A.1)-(A.3), Chebyshev inequality and $||\Pi _{n} u^{*}-u^{*}||=o(1)$ that $I_1 =\varepsilon _n \times o_p(n^{-1/2})$. For $I_2$, by the mean value theorem, we obtain that

$$\begin{aligned} \begin{aligned} I_2&=(P_n-P)\left\{ l({\hat{\xi }}_n,O)-l({\hat{\xi }}_n\pm \varepsilon _{n}\Pi _{n} u^{*},O)\pm \Pi \varepsilon _n\dot{l}(\xi _0,O)[\Pi _{n} u^{*}]\right\} \\&=\pm \varepsilon _n(P_n-P)\left\{ (\dot{l}({\tilde{\xi }},O)-\dot{l}({\tilde{\xi }}_0,O))[\Pi _{n} u^{*}]\right\} ,\\ \end{aligned} \end{aligned}$$

where ${\tilde{\xi }}$ lies between ${\hat{\xi }}_n$ and ${\hat{\xi }}_n\pm \varepsilon _n\Pi _{n} u^{*}$. By Theorem 2.8.3 of Van and Wellner (1996), we know that $\{\dot{l}(\xi ,O)[\Pi _{n} u^{*}]:||\xi -\xi _0||\le \delta _n\}$ is Donsker class. Therefore, by Theorem 2.11.23 of Van and Wellner (1996), we have $I_2 = \varepsilon _n\times o_p(n^{-1/2})$. For $I_3$, note that

$$\begin{aligned} \begin{aligned} P\left( v\left[ \xi -\xi _{0}, O\right] \right) =&P\left\{ l(\xi , O)-l\left( \xi _{0}, O\right) -\dot{l}\left( \xi _{0}, O\right) \left[ \xi -\xi _{0}\right] \right\} \\ =&2^{-1} P\left\{ \ddot{l}({\tilde{\xi }}, O)\left[ \xi -\xi _{0}, \xi -\xi _{0}\right] -\ddot{l}\left( \xi _{0}, O\right) \left[ \xi -\xi _{0}, \xi -\xi _{0}\right] \right\} \\&+2^{-1} P\left\{ \ddot{l}\left( \xi _{0}, O\right) \left[ \xi -\xi _{0}, \xi -\xi _{0}\right] \right\} \\ =&2^{-1} P\left\{ \ddot{l}\left( \xi _{0}, O\right) \left[ \xi -\xi _{0}, \xi -\xi _{0}\right] \right\} +\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) , \end{aligned} \end{aligned}$$

where ${\tilde{\xi }}$ lies between $\xi _0$ and $\xi$ and the last equation follows from Taylor expansion and Conditions (A.1)-(A.3). Therefore,

$$\begin{aligned} \begin{aligned} I_{3}&=-2^{-1}\left\{ \left\| {\hat{\xi }}_{n}-\xi _{0}\right\| ^{2}-\left\| {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}-\xi _{0}\right\| ^{2}\right\} +\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) \\&=\pm \varepsilon _{n}<{\hat{\xi }}_{n}-\xi _{0}, \Pi _{n} u^{*}>+2^{-1}\left\| \varepsilon _{n} \Pi _{n} u^{*}\right\| ^{2}+\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) \\&=\pm \varepsilon _{n}<{\hat{\xi }}_{n}-\xi _{0}, u^{*}>+2^{-1}\left\| \varepsilon _{n} \Pi _{n} u^{*}\right\| ^{2}+\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) \\&=\pm \varepsilon _{n}<{\hat{\xi }}_{n}-\xi _{0}, u^{*}>+\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) , \end{aligned} \end{aligned}$$

where the last equality holds due to the facts $\delta _n ||\Pi _n u^{*}-u^{*}||=o(n^{-1/2})$, Cauchy–Schwartz inequality and $||\Pi _nu^{*}||^2\rightarrow ||u^{*}||^2$. Combining the above facts, together with $P\dot{l}(\xi _0,O)[u^{*}]=0$, we can establish that

$$\begin{aligned} \begin{aligned} 0&\le P_{n}\left\{ l\left( {\hat{\xi }}_{n}, O\right) -l\left( {\hat{\xi }}_{n} \pm \varepsilon _{n} \Pi _{n} u^{*}, O\right) \right\} \\&=\pm \varepsilon _{n} P_{n} \dot{l}\left( \xi _{0}, O\right) \left[ u^{*}\right] \pm \varepsilon _{n}<{\hat{\xi }}_{n}-\xi _{0}, u^{*}>+\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) \\&=\pm \varepsilon _{n}\left( P_{n}-P\right) \{\dot{l}\left( \xi _{0}, O\right) \left[ u^{*}\right] \} \pm \varepsilon _{n}<{\hat{\xi }}_{n}-\xi _{0}, u^{*}>+\varepsilon _{n} \times o_{p}\left( n^{-1 / 2}\right) . \end{aligned} \end{aligned}$$

Therefore, we obtain $\pm n^{1/2}(P_{n}-P)\{{\dot{l}}(\xi _{0}, O)[u^{*}]\} \pm n^{1/2}<{\hat{\xi }}_{n}-\xi _{0}, u^{*}>+o_p(1)\ge 0$ and then $n^{1/2}<{\hat{\xi }}_{n}-\xi _{0}, u^{*}>=n^{1/2}\left( P_{n}-P\right) \left\{ {\dot{l}}\left( \xi _{0}, O\right) [u^{*}]\right\} +o_p(1)\rightarrow _{d}N(0,||u^{*}||^2)$ by the central limit theorem and $||u^{*}||^2=||\dot{l}\left( \xi _{0}, O\right) \left[ u^{*}\right] ||^2$.

Next we will prove that $||u^{*}||^2= b^{T}I^{-1}(\eta _0)b$. For each component $\eta _j$, $j=1,2,\ldots ,p_{\eta }$, we denote by $\phi ^{*}_j=(b^{*}_{1j}, b^{*}_{2j})$ the value of $\phi _j = (b_{1j}, b_{2j})$ minimizing

$$\begin{aligned} E \left\{ l_{\eta }\cdot e_j-l_{b_1}[b_{1j}]-l_{b_2}[b_{2j}]\right\} ^2, \end{aligned}$$

where $l_{\eta }$ is the score function for $\eta$, $l_{b_1}$ and $l_{b_2}$ are the score operator for $\Lambda _t$ and $\Lambda _w$, and $e_j$ is a $p_{\eta }$-dimensional vector of zeros except the j-th element equal to 1.

Define the j-th element of $l^{*}(\eta ,O)$ as $l_{\eta }\cdot e_j-l_{b_1}[b^{*}_{1j}]-l_{b_2}[b^{*}_{2j}]$, $j=1,2,\ldots p_{\eta }$, and $I(\eta )$ as $E(\{l^{*}(\eta ,O)\}^{\otimes 2})$. By Condition (A.5), the matrix $I(\eta _0)$ is positive definite. Furthermore, by following similar calculations in Chen et al. (2006), we obtain

$$\begin{aligned} ||u^{*}||^2=||\dot{h}(\xi _0)||^2=\sup _{u\in \bar{V}:||u||>0}\frac{|\dot{h}(\xi _0)[u]|^2}{||u||^2}=b^{T}[E(\{l^{*}(\eta _0,O)\}^{\otimes 2})]^{-1}b=b^{T}I^{-1}(\eta _0)b. \end{aligned}$$

Thus, we have shown that $n^{1/2}({\hat{\eta }}_n-\eta _0)\rightarrow N(0, I^{-1}(\eta _0))$ in distribution for the estimator ${\hat{\eta }}_n$.$\square$