On semiparametric transformation model with LTRC data: pseudo likelihood approach

Abstract

When the distribution of the truncation time is known up to a finite-dimensional parameter vector, many researches have been conducted with the objective to improve the efficiency of estimation for nonparametric or semiparametric model with left-truncated and right-censored (LTRC) data. When the distribution of truncation times is unspecified, one approach is to use the conditional maximum likelihood estimators (cMLE) (Chen and Shen in Lifetime Data Anal https://doi.org/10.1007/s10985-016-9385-9, 2017). Although the cMLE has nice asymptotic properties, it is not efficient since the conditional likelihood function does not incorporate information on the distribution of truncation time. In this article, we aim to develop a more efficient estimator by considering the full likelihood function. Following Turnbull (J R Stat Soc B 38:290–295, 1976) and Qin et al. (J Am Stat Assoc 106:1434–1449, 2011), we treat the unobserved (left-truncated) subpopulation as missing data and propose a two-stage approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first stage, the distribution of left truncation time is estimated by the inverse-probability-weighted (IPW) estimator (Wang in J Am Stat Assoc 86:130–143, 1991). In the second stage, we obtain the pseudo complete-data likelihood function by replacing the distribution of truncation time with the IPW estimator in the full likelihood. We propose an expectation–maximization algorithm for obtaining the PMLE and establish the consistency of the PMLE. Simulation results show that the PMLE outperforms the cMLE in terms of mean squared error. The PMLE can also be used to analyze the length-biased data, where the truncation time is uniformly distributed. We demonstrate that the PMLE works more robust against the support assumption of truncation time for length-biased data compared with the MLE proposed by Qin et al. (2011). We apply our proposed method to the channing house data. While the PMLE is quite appealing under specific cases with independent censoring and time-invariant covariates, its applicability, as shown in simulation study, can be rather restricted for more general settings.

Appendices

Appendix 1: Proof of Theorem 1

Our proofs follow essentially the same steps as Zeng and Lin (2006) (also see Murphy 1994, 1995; Parner 1998).

By condition (C2), there exists a constant M such that $\sup _{\beta \in {{\mathcal {B}}}} |\beta ^T Z^{*}|\le M$ with probability one. Hence, the $i^{th}$ term in (3) satisfies

$$\begin{aligned}&\int _{0}^{\tau _c} \Big (\log \{g(R(X_i\wedge t)e^{\beta ^T Z_i})\}+\log \bigr \{R\{t\}e^{\beta ^T Z_i}\bigl \}\Big ) dN_i(t) \\&\qquad -G(R(X_i\wedge \tau _c)\beta ^T Z_i)-\log \alpha (R,\beta ,K;Z_i) \\&\quad \le G(R(\tau _c\wedge C_i)e^{M} \biggl [{{\log \bigl \{R(X_i\wedge \tau _c)e^{M}\sup _{y\le R(X_i)e^{M}}g(y)\bigr \}}\over {G(R(X_i\wedge \tau _c)e^{-M})}}-1\biggr ]\\&\qquad -\log \alpha (R,\beta ,K;Z_i). \end{aligned}$$

Under condition (C3), this quantity diverges to $-\infty$ if $R\{X_i\}$ tends to $\infty$ for some $X_i$. Hence, the jump sizes of R must be finite. Since $\zeta _n$ maximizes the likelihood function $l_n({\hat{\zeta }}_n,{\hat{K}}_n)$, the following inequality holds $l_n({\hat{\zeta }}_n,{\hat{K}}_n)-l_n({\tilde{\zeta }}_n,{\hat{K}}_n)\ge 0$, where ${\tilde{\zeta }}_n=({\tilde{R}}_n,{\hat{\beta }}_n)$, ${\tilde{R}}_n(t)={\hat{R}}_n(t)/\xi _n$, $\xi _n={\hat{R}}_n(\tau _c)$. Since ${\hat{K}}_n$ is a consistent estimator of $K_0$ (Wang 1991), $l_n({\hat{\zeta }}_n,K_0)-l_n({\tilde{\zeta }}_n,K_0)$ is asymptotically nonnegative. Using the approach of Zeng and Lin (2006), we first show that ${\hat{{\varLambda }}}_n(\tau )$ is bounded almost surely by contradiction. From (3) and $n^{-1}[l_n({\hat{\zeta }}_n,K_0)-l_n({\tilde{\zeta }}_n,K_0)]\ge 0$, we obtain

$$\begin{aligned}&n^{-1}\sum _{i=1}^{n}\int _{0}^{\tau _c} \log \left\{ \xi _n g(\xi _n{\tilde{R}}_n(X_i\wedge t)e^{{\hat{\beta }}_n^T Z_i})\right\} dN_i(t) \\&\qquad -\,n^{-1}\sum _{i=1}^{n}G\left\{ \xi _n{\tilde{R}}_n(X_i\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_i}\right\} -\,n^{-1}\sum _{i=1}^{n}\log \left\{ \alpha ({\hat{R}}_n,{\hat{\beta }}_n,K_0;Z_i)\right\} \\&\quad \ge n^{-1}\sum _{i=1}^{n}\int _{0}^{\tau _c} \log \left\{ g({\tilde{R}}_n(X_i\wedge t)e^{{\hat{\beta }}_n^T Z_i})\right\} dN_i(t) -n^{-1}\sum _{i=1}^{n}G\left\{ {\tilde{R}}_n(X_i\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_i}\right\} \\&\qquad -\,n^{-1}\sum _{i=1}^{n}\log \left\{ \alpha ({\tilde{R}}_n,{\hat{\beta }}_n,K_0;Z_i)\right\} . \end{aligned}$$

Note that the right-hand side is bounded from below by

$$\begin{aligned} \log \min _{y\le e^{M}}g(y)\{n^{-1}\sum _{i=1}^{n}N_i(\tau _c)\} -G(e^{M}) > -\infty . \end{aligned}$$

The left-hand side is bounded from above by

$$\begin{aligned}&n^{-1}\sum _{i=1}^{n}\int _{0}^{\tau _c\wedge C_i}dN_i(t) \log \xi _n\sup _{y\le \xi _n e^{M}}g(y)- n^{-1}\sum _{i=1}I_{[X_i\ge \tau _c]} G(e^{-M}\xi _n) \\&\quad -\,n^{-1}\sum _{i=1}^{n}\log \{\alpha ({\hat{R}}_n,{\hat{\beta }}_n,K_0;Z_i)\}. \end{aligned}$$

Under condition (C3), $\log \xi _n\sup _{y\le \xi _n e^{M}}g(y)\le \epsilon G(\xi _n e^{M})$ for any $\epsilon$ when n is large enough. It follows that if we choose $\epsilon$ such that $\epsilon E[N_i(\tau _c)]\le P(X_i\ge \tau _c)/2$, the left-hand side diverges to $-\infty$ when $\xi _n\rightarrow \infty$. This is a contradiction. Thus, ${\hat{R}}_n$ is bounded on $[0,\tau _c]$ with probability one. By the Helly’s selection theorem, along a subsequence, we assume that ${\hat{\zeta }}_n$ converges to $\zeta ^{*}=(R^{*},\beta ^{*})$.

Next, by differentiating $l_n(R,\beta ,K_0)$ with respect to $R\{X_i\}$ and setting the derivative to be zero,we obtain $[n{\hat{R}}_n\{X_i\}]^{-1}=\phi _n(X_i;{\hat{\beta }}_n,{\hat{R}}_n,K_0),$ where

$$\begin{aligned}&\phi _n(X_i;{\hat{R}}_n,{\hat{\beta }}_n,K_0) =n^{-1}\sum _{k=1}^{n}g({\hat{R}}_n(X_k\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_k}) e^{{\hat{\beta }}_n^T Z_k}I_{[X_k\ge X_i]} \\&\quad -\,n^{-1}\sum _{k=1}^{n}{{\int _{0}^{\infty }\exp \{-G({\hat{R}}_n(s\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_k})\} g({\hat{R}}_n(s\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_k}) e^{{\hat{\beta }}_n^T Z_k}I_{[s\ge X_i]}k_0(s)ds} \over {\alpha ({\hat{R}}_n,{\hat{\beta }}_n,K_0;Z_k)}} \\&\quad -\,n^{-1}\sum _{k=1}^{n}\int _{0}^{\tau _c} {{I_{[X_k\wedge t\ge X_i]}e^{{\hat{\beta }}_n^T Z_k} \dot{g}({\hat{R}}_n(X_k\wedge t)e^{{\hat{\beta }}_n^T Z_k})} \over {g({\hat{R}}_n(X_k\wedge t)e^{{\hat{\beta }}_n^T Z_k})}}dN_k(t), \end{aligned}$$

where $\dot{g}(x)=dg(x)/dx$. It follows that

$$\begin{aligned} {\hat{R}}_n(t)=\int _{0}^{t}{{n^{-1}\sum _{i=1}^{n}dN_i(s)} \over {\phi _n(s;{\hat{R}}_n,{\hat{\beta }}_n,K_0)}}. \end{aligned}$$

By the Glivenko–Cantelli theorem, $\phi _n(s;{\hat{R}}_n,{\hat{\beta }}_n,K_0)$ uniformly converges to a continuously differentiable function $\phi (s;R^{*},\beta ^{*},K_0)$. Similar to the arguments of Zeng and Lin (2006), it follows that when n is large enough $|{\hat{\phi }}_n(t;{\hat{R}}_n,{\hat{\beta }}_n,K_0)|>\epsilon$ for some $\epsilon _0$. Let

$$\begin{aligned} {\hat{R}}_n^{0}(t) =\int _{0}^{t}{{n^{-1}\sum _{i=1}^{n}dN_i(s)}\over {\phi _n(t;R_0,\beta _0,K_0)}}. \end{aligned}$$

By the Glivenko–Cantelli theorem, ${\hat{R}}_n^{0}(t)$ converges uniformly to ${\varLambda }_0$ almost surely. By the lower bound of $|\phi _n(t)|$, ${\hat{R}}_n(t)$ is absolutely continuous respect to ${\hat{R}}_n^{0}(t)$ and $d{\hat{R}}_n/d{\hat{R}}_n^{0}$ converges to a bounded measurable function, i.e., $R^{*}(t)=\int _{0}^{t}b(s)dR_0(s)$. Thus, $R^{*}$ is absolutely continuous with derivative $r^{*}(t)$ and $b(t)=r^{*}(t)/r_0(t)$. Since $l_n(R,\beta ,K_0)$ is maximized at $({\hat{R}}_n,{\hat{\beta }}_n)$, we have

$$\begin{aligned}&n^{-1}\sum _{i=1}^{n}\biggl [\int _{0}^{\tau _c} \log {{{\hat{R}}_n(t)}\over {{\hat{R}}_n^{0}(t)}}dN_i(t)- G\{{\hat{R}}_n(X_i\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_i}\} +G\{{\hat{R}}_n^{0}(X_i\wedge \tau _c)e^{{\hat{\beta }}_n^T Z_i}\} \\&\quad +\int _{0}^{\tau _c}\log \{g({\hat{R}}_n(X_i\wedge t)e^{{\hat{\beta }}_n^T Z_i})\}dN_i(t) -\int _{0}^{\tau _c}\log \{g({\hat{R}}_n^{0}(X_i\wedge t)e^{{\hat{\beta }}_n^T Z_i})\}dN_i(t) \\&\quad -\log \{\alpha ({\hat{R}}_n,{\hat{\beta }}_n,K_0;Z_i)\}+ \log \{\alpha ({\hat{R}}_n^{0},{\hat{\beta }}_n,K_0;Z_i)\}\biggr ]\ge 0. \end{aligned}$$

We take the limits on both sides. Then, by the Glivenko–Cantelli theorem and the fact that ${\hat{R}}_n\{t\}/{\hat{R}}_n^{0}\{t\}$ converges uniformly to $r^{*}(t)/r_0(t)$, the Kullback–Leibler information between the density indexed by $(R^{*},\beta ^{*})$ and $(R_0,\beta _0)$ is negative. Thus, with probability one

$$\begin{aligned}&\int _{0}^{\tau _c}\log \{I_{[X\ge t]}\lambda ^{*}(t) e^{{\beta ^{*}}^T Z} g(R^{*}(X\wedge t) e^{{\beta ^{*}}^T Z}\}dN(t)-G\{R^{*}(X\wedge \tau _c) e^{{\beta ^{*}}^T Z}\} \\&\qquad -\log \{\alpha (R^{*},\beta ^{*},K_0;Z)\} \\&\quad =\int _{0}^{\tau _c}\log \{I_{[X\ge t]}\lambda _0(t) e^{\beta _0^{T} Z} g(R_0(X\wedge t)e^{\beta _0^{T}Z}\}dN(t)-G\{R_0(X\wedge \tau _c)e^{\beta _0^{T}Z}\} \\&\qquad -\log \{\alpha (R_0,\beta _0,K_0;Z)\}. \end{aligned}$$

This equality holds for the case $X\ge \tau _c$, $\delta _i=0$ and also holds for the case where $X\ge \tau _c$ and $N(t-)=1$ for $t\in [0,\tau _c]$, $N(\tau _c)=1$. The difference between the equalities from two cases entails that

$$\begin{aligned}\lambda ^{*}(t)e^{{\beta ^{*}}^T Z} g(R^{*}(X\wedge t) e^{{\beta ^{*}}^T Z}) =\lambda _0(t) e^{\beta _0^{T} Z} g(R_0(X\wedge t)e^{\beta _0^{T}Z}). \end{aligned}$$

Under (C2), it follows that $\beta ^{*}=\beta _0$ and $R^{*}=R_0$. Hence, ${\hat{\beta }}_n$ converges almost surely to $\beta _0$ and by (C1), ${\hat{R}}_n(t)$ converges uniformly in t for $t\in [0,\tau _c]$.

Appendix 2: Proof of the asymptotic distribution based on $l_n(\beta ,R,K_0)$

In additions conditions (C1)–(C3), we need the following condition:

(C4) Let ${\hat{R}}_n(\cdot ,\beta )$ be the maximizer of $l_n(R,\beta ,K_0)$ for given $\beta$. The information matrix $-\partial ^2E[l_n({\hat{R}}_n(\cdot ,\beta ),\beta ,K_0)]/\partial ^2\beta$ evaluated at true value $\beta _0$ is positive definite.

The proof of the asymptotic normality is similar to the work of Zeng and Lin (2006). We provide only a sketch of the proof. Let ${{\mathcal {P}}}_n$ denote the empirical measure determined by n i.i.d. observations and ${{\mathcal {P}}}_0$ denote its expectation. Furthermore, we define $l(\beta ,R,K_0)$ as the logarithm of the observed likelihood function from a single subject. Define the derivative of $l(R,\beta ,K_0)$ with respect to R as

$$\begin{aligned} l_{R}(R,\beta ,K_0)[{\varDelta } R] =\lim _{\epsilon \rightarrow 0}{{l_n(R+\epsilon {\varDelta } R,\beta ,K_0)-l_n(R,\beta ,K_0)}\over {\epsilon }} \end{aligned}$$

and also define

$$\begin{aligned} l_{RR}(R,\beta ,K_0)[{\varDelta }_1 R,{\varDelta }_2 R] =\lim _{\epsilon \rightarrow 0}{{l_{R}(R+\epsilon {\varDelta }_2 R,\beta ,K_0)[{\varDelta }_1 R]-l_R(R,\beta ,K_0)[{\varDelta }_1 R]}\over {\epsilon }}. \end{aligned}$$

Let $l_{\beta }(R,\beta ,K_0)$ denote the score vector of $l(R,\beta ,K_0)$ for $\beta$ and $l_{\beta \beta }(R,\beta ,K_0)$ the Hessian matrix of $l(R,\beta ,K_0)$. Define $\psi (t;R,\beta )=\dot{g}\{R(X\wedge t)e^{\beta ^T Z}\}/g\{R(X\wedge t)e^{\beta ^T Z}\}.$ We choose $\epsilon _0$ small enough and define a map $U_n :=(U_{1n},U_{2n})$ from ${{\mathcal {S}}}=\{(R,\beta ): ||R-R_0||_{l^{\infty }[0,\tau _c]}<\epsilon _0,|\beta -\beta _0| <\epsilon _0 \}\subset {{\mathcal {R}}}_p\times l^{\infty }({{\mathcal {D}}})$ to $l^{\infty }({{\mathcal {D}}}) \times {{\mathcal {R}}}_p$ as follows: for any $q(t)\in {{\mathcal {D}}}$,

$$\begin{aligned}&U_{1n}(R,\beta )[q] ={d\over {d\epsilon }}{{\mathcal {P}}}_n\bigl \{l_n(R(t)+\epsilon \int _{0}^{t}q(s)dR(s),\beta ,K_0) \bigr \}\bigl |_{\epsilon =0} \\&\quad ={{\mathcal {P}}}_n\biggl \{\int _{0}^{\tau _c}\psi (t;R,\beta ) \int _{0}^{t}I_{[X\ge s]}q(s)e^{\beta ^T Z}dR(s)dN(t) \\&\qquad +\int _{0}^{\tau _c}q(t)dN(t)-g\{R(X\wedge \tau _c)e^{\beta ^T Z}\} \int _{0}^{\tau _c}I_{[X\ge t]}q(t)e^{\beta ^T Z}dR(t) \\&\qquad +\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R(v\wedge \tau _c)\exp (\beta ^T Z)\bigr \}\bigr ) g\{R(v\wedge \tau _c)e^{\beta ^T Z}\} \\&\qquad \times \int _{0}^{\tau _c}I_{[v\ge t]}q(t)e^{\beta ^T Z}dR(t) \bigr ]k_0(v)dv \\&\qquad \times \{\alpha (R,\beta ,K_0;Z)\}^{-1}\biggr \} \end{aligned}$$

and

$$\begin{aligned}&U_{2n}(R,\beta )=\nabla _{\beta }{{\mathcal {P}}}_n\{l(R,\beta ,K_0)\} \\&\quad ={{\mathcal {P}}}_n\biggl \{\int _{0}^{\tau _c}\psi (t;R,\beta ) R(X\wedge t)e^{\beta ^T Z}ZdN(t)+ZN(\tau _c)\\&\qquad -g\{R(X\wedge \tau _c)e^{\beta ^T Z}\}R(X\wedge \tau _c)e^{\beta ^T Z}Z \\&\qquad +\int _{0}^{\infty }[\exp \bigl (-G\bigl \{R(v\wedge \tau _c)\exp (\beta ^T Z)\bigr \}\bigr ) g\{R(v\wedge \tau _c)e^{\beta ^T Z}\}R(v\wedge \tau _c)e^{\beta ^T Z}Z ]k_0(v)dv \\&\qquad \times \{\alpha (R,\beta ,K_0;Z)\}^{-1}\biggr \}. \end{aligned}$$

Similarly, we can define the limit version of $U_n$ as $U_0 :=(U_{10},U_{20})$ by replacing ${{\mathcal {P}}}_n$ by ${{\mathcal {P}}}_0$, i.e., $U_{10}(R,\beta )[q]=E_0[U_{1n}(R,\beta )[q]]$ and $U_{20}(R,\beta )=E_0[U_{2n}(R,\beta )]$. Clearly, $U_n({\hat{R}}_n,{\hat{\beta }}_n)$ is asymptotically equal to zero and $U_0(R_0,\beta _0)=0$. By conditions (C1) and (C2) and the Donsker theorem, we can show that $\sqrt{n}(U_n-U_0)({\hat{R}}_n,{\hat{\beta }}_n)-\sqrt{n}(U_n-U_0)(R_0,\beta _0)=o_p(1)$ in the metric space ${{\mathcal {R}}}^{p}\times l^{\infty }({{\mathcal {D}}})$. Since $\sqrt{n}(U_n-U_0)(R_0,\beta _0)$ is a sum of i.i.d. random quantities, by empirical theory it converges weakly to ${{\mathcal {W}}}=({{\mathcal {W}}}_1,{{\mathcal {W}}}_2)$, where ${{\mathcal {W}}}_1$ is a tight Gaussian process and ${{\mathcal {W}}}_2$ is a Gaussian random vector. Furthermore, the covariance matrix for ${{\mathcal {W}}}_2$ is ${\varSigma }_{22}=E_0[U_{2n}(R_0,\beta _0)^{\otimes 2}]$ and covariance between ${{\mathcal {W}}}_1(s)$ and ${{\mathcal {W}}}_1(t)$ is ${\varSigma }_{11}(s,t)=E_0[U_{1n}(R_0,\beta _0)[q_1]U_{1n}(R_0,\beta _0)[q_2]]$, where $q_1(\cdot )=I_{[\cdot \le s]}$ and $q_2(\cdot )=I_{[\cdot \le t]}$. By Theorem 3.3.1 of van der Vaart and Wellner (1996), it remains to show that $U_0$ is Fréchet-differentiable at $\zeta _0=(R_0,\beta _0)$ and the derivative is continuously invertible in the set ${{\mathcal {S}}}$. The Fréchet-differentiability can be verified directly. The derivative of $U_0$ maps ${{\mathcal {S}}}$ to $l^{\infty }({{\mathcal {D}}})\times {{\mathcal {R}}}^p$ and has the form

$$\begin{aligned} \begin{pmatrix} U_{11} &{} U_{12}\\ U_{21}&{} U_{22} \end{pmatrix} \begin{pmatrix} R-R_0\\ \beta -\beta _0 \end{pmatrix}\biggl [ \begin{pmatrix} b\\ q\end{pmatrix}\biggr ] =\begin{pmatrix}U_{11}(R-R_0)[q]+ U_{12}(\beta -\beta _0)[q]\\ U_{21}(R-R_0)^{T} b+U_{22}(\beta -\beta _0)^{T}b\end{pmatrix}, \end{aligned}$$

where $U_{11}(R-R_0)[q]=\int (-p(t)I+D)[q]d(R-R_0)$, $U_{12}(\beta -\beta _0)[q]=A[\int q dR_0](\beta -\beta _0)$, $U_{21}(R-R_0)=A^{*}[R-R_0]$, $U_{22}(\beta -\beta _0)=B(\beta -\beta _0)$, where $p(t) > 0$, I is identity operator, A and D are both linear operator, $A^{*}$ is the dual operator of A and B is a $p\times p$ matrix. Specifically,

$$\begin{aligned}&p(t)=E_0\bigl [I_{[X\ge t]}e^{\beta _0^T Z} g\{R_0(X\wedge t)e^{\beta _0^T Z}\}\bigr ] \\&\quad -\,E_0\biggl [\biggr \{\int _{0}^{\infty } \bigl [\exp \bigl (-G\bigl \{R_0(v)\exp (\beta _0^T Z_i)\bigr \}\bigr ) \bigl [I_{[v\ge t]}e^{\beta _0^T Z} g\{R_0(v)e^{\beta _0^T Z}\}\bigr ]k_0(v)dv\biggr \} \\&\quad \times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-1}\biggr ], \\&D[q]=-E_0\biggl [I_{[X\ge t]}e^{\beta _0^T Z} \dot{g}\{R_0(X\wedge \tau _c)e^{\beta _0^T Z}\}\int _{0}^{\tau _c}q(s)I_{[X\ge s]} e^{\beta _0^T Z}dR_0(s)\biggr ] \\&\quad +E_0\biggl [I_{[X\ge t]}e^{\beta _0^T Z} \int _{t}^{\tau _c}\psi ^{'}(s;R_0,\beta _0)\int _{0}^{s}q(u)I_{[X\ge u]} e^{\beta _0^T Z}dR_0(u)dN(s)\biggr ] \\&\quad -\,E_0\biggl [\biggr \{\int _{0}^{\infty } \bigl [\exp \bigl (-G\bigl \{R_0(v)\exp (\beta _0^T Z_i)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\} \\&\quad \times \, \int _{0}^{\infty }I_{[v\ge t]}q(t)e^{\beta _0^T Z}dR_0(t) \bigr ]k_0(v)dv\biggr \}^2\times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-2}\biggr ] \\&\quad -\,E_0\biggl [\biggr \{\int _{0}^{\infty } \bigl [\exp \bigl (-G\bigl \{R_0(v)\exp (\beta _0^T Z)\bigr \}\bigr ) \bigl \{g\{R_0(v)e^{\beta _0^T Z}\} \\&\quad \times \,\int _{0}^{\infty }I_{[v\ge t]}q(t)e^{\beta _0^T Z}dR_0(t)\bigr \}^2 \bigr ]k_0(v)dv\biggr \}\times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-1}\biggr ] \end{aligned}$$

$$\begin{aligned}&A\left[ \int q dR_0\right] = E_0\biggl [\int _{0}^{\tau _c}\psi ^{'}(t;R_0,\beta _0) \int _{0}^{t}I_{[X\ge s]}e^{\beta _0^T Z}q(s)dR_0(s) R_0(X\wedge t)e^{\beta _0^T Z}Z\biggr ] \\&\quad +\,E_0\biggl [\int _{0}^{\tau _c}\psi (t;R_0,\beta _0)\int _{0}^{t}I_{[X\ge s]}e^{\beta _0^T Z}q(s)ZdR_0(s)\biggr ] \\&\quad -\,E_0\biggl [\int _{0}^{\tau _c}I_{[X\ge t]}e^{\beta _0^T Z}q(t)dR_0(t) R_0(X\wedge \tau _c)e^{\beta _0^T Z}Z\dot{g}(R_0(X\wedge \tau _c)e^{\beta _0^T Z})\biggr ] \\&\quad -\,E_0\biggl [\int _{0}^{\tau _c}I_{[X\ge t]}e^{\beta _0^T Z}q(t)ZdR_0(t) g(R_0(X\wedge \tau _c)e^{\beta _0^T Z})\biggr ] \\&\quad -\,E_0\biggl [\biggl \{\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R_0(v) \exp (\beta _0^T Z)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\}R_0(v)e^{\beta _0^T Z}Z \bigr ]k_0(v)dv\biggr \} \\&\quad \times \biggl \{\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R_0(v)\exp (\beta _0^T Z_i)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\}\int _{0}^{\infty }I_{[v\ge t]}q(t)e^{\beta _0^T Z}dR_0(t)\bigr ]k_0(v)dv\biggr \} \\&\quad \times \, \{\alpha (R_0,\beta _0,K_0;Z)\}^{-2}\biggr ] \\&\quad +\,E_0\biggl [\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R_0(v) \{\exp (\beta _0^T Z)\bigr \}\bigr ) \dot{g}\{R_0(v)e^{\beta _0^T Z}\}R_0(v)e^{\beta _0^T Z}Z \\&\quad \times \, \int _{0}^{\infty }I_{[v\ge t]}q(t)e^{\beta _0^T Z}dR_0(t)\bigr ]k_0(v)dv\times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-1}\biggr ] \\&\quad +E_0\biggr [\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R_0(v) \{\exp (\beta _0^T Z)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\}e^{\beta _0^T Z}Z \\&\quad \times \, \int _{0}^{\infty }I_{[v\ge t]}q(t)e^{\beta _0^T Z}dR_0(t)\bigr ] k_0(v)dv\times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-1}\biggr ] \\&\quad -\,E_0\biggl [\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R_0(v) \exp (\beta _0^T Z)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\} \int _{0}^{\infty }I_{[v\ge t]}q(t)e^{\beta _0^T Z}dR(t) \\&\quad \times \,g\{R_0(v)e^{\beta _0^T Z}\}R_0(v)e^{\beta _0^T Z}Z \bigr ]k_0(v)dv\times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-1}\biggr ]. \end{aligned}$$

$$\begin{aligned}&B=E_0\biggl [\int _{0}^{\tau _c}\psi (t;R_0,\beta _0) R_0(X\wedge t)e^{\beta _0^T Z}Z Z^TdN(t)\\&\quad +\,\int _{0}^{\tau _c}\psi ^{'}(t;R_0,\beta _0) \{R_0(X\wedge t)e^{\beta _0^T Z}Z\}^{\otimes 2}dN(t)\biggr ] \\&\quad -\,E_0\biggl [g(R_0(X\wedge \tau _c)e^{\beta _0^T Z})R_0(X\wedge \tau _c)e^{\beta _0^T Z}Z Z^T\\&\quad +\,\dot{g}(R_0(X\wedge \tau _c)e^{\beta _0^T Z})\{R_0(X\wedge \tau _c)e^{\beta _0^T Z}Z\}^{\otimes 2}\biggr ] \\&\quad -E_0\biggl [\biggl \{\int _{0}^{\infty }\bigl [\exp \bigl (-G\bigl \{R_0(v)\exp (\beta _0^T Z)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\}R_0(v)e^{\beta _0^T Z}Z \bigr ]k_0(v)dv\biggr \}^{\otimes 2} \\&\quad \times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-2}\biggr ] \\&\quad +E_0\biggl [\int _{0}^{\infty }\biggl \{\exp \bigl (-G\bigl \{R_0(v) \{\exp (\beta ^T Z)\bigr \}\bigr ) \dot{g}\{R_0(v)e^{\beta _0^T Z}\}\{R_0(v)e^{\beta _0^T Z}Z\}^{\otimes 2} \\&\quad +\exp \bigl (-G\bigl \{R_0(v) \{\exp (\beta _0^T Z)\bigr \}\bigr ) g\{R_0(v)e^{\beta _0^T Z}\}R_0(v)e^{\beta _0^T Z}Z Z^T \\&\quad -\exp \bigl (-G\bigl \{R_0(v) \{\exp (\beta _0^T Z)\bigr \}\bigr ) \bigl \{g\{R_0(v)e^{\beta _0^T Z}\}R_0(v)e^{\beta _0^T Z}Z\bigr \}^{\otimes 2}\biggr \} k_0(v)dv \\&\quad \times \{\alpha (R_0,\beta _0,K_0;Z)\}^{-1}\biggr ] \end{aligned}$$

It suffices to show that $U_{22}$ and $L:=U_{11}-U_{12}U_{22}^{-1}U_{21}$ are continuously invertible. Note $-U_{22}$ is the information at $\beta _0$. By condition (C2), it follows that $U_{22}$ is continuously invertible. Furthermore, based on the following equality

$$\begin{aligned}&\begin{pmatrix} U_{11}(R-R_0)[q]+ U_{12}(\beta -\beta _0)[q]\\ U_{21}(R-R_0)^{T} b+U_{22}(\beta -\beta _0)^{T}b\end{pmatrix} \\&\quad =-{{\mathcal {P}}}\begin{pmatrix}l_{RR}(R_0,\beta _0,K_0)[(R-R_0),\int q dR_0]+ l_{R\beta }(R_0,\beta _0,K_0)\left[ \int q dR_0\right] ^T(\beta -\beta _0)\\ l_{\beta R}(R_0,\beta _0,K_0)[R-R_0]^{T} b+l_{\beta \beta }(R_0,\beta _0,K_0)(\beta -\beta _0)^{T}b\end{pmatrix}, \end{aligned}$$

and the arguments given in Zeng and Lin (2006), we can show that L is continuously invertible in the set ${{\mathcal {S}}}$. Denote $\dot{U}_{\zeta _0}$ as the Fréchet derivative of the map $U_0$ at $\zeta _0$. Thus, we have $\dot{U}_{\zeta _0}[\sqrt{n}(\zeta _n-\zeta _0)] =-\sqrt{n}(U_n-U_0)(R_0,\beta _0)+o_p(1)$. It follows that $\sqrt{n}[\zeta _n-\zeta _0]$ converges weakly to a mean zero Gaussian process $-\dot{U}^{-1}_{\zeta _0}({{\mathcal {W}}})$.