Empirical likelihood inference for the panel count data with informative observation process

Abstract

Panel count data refer to interval-censored recurrent event data. Each study subject can only be observed at discrete time points, leading to knowledge about the total number of events occurring between observations. The observation times can be also different among subjects and carry important information about the underlying recurrent process. In this paper, an empirical likelihood (EL) method for panel count data with informative observation times is proposed. Based on the influence function, we formulate an empirical likelihood ratio for the vector of regression coefficients, and the Wilks’ theorem is established. Simulation studies are carried out to compare the performance of empirical likelihood with normal approximation methods. Finally, the EL method is compared with existing approaches, utilizing an illustrative example drawn from a bladder cancer study.

Appendix: Proofs of theorems

We assume the following regularity conditions throughout the paper, which are similar to Li et al. (2010).

• (C.1) The covariate vector X(t) and the weight function W(t) are bounded for all $t \in [0,\tau ]$.

• (C.2) The function g is twice continuously differentiable.

• (C.3) The function W(t) converges uniformly to a deterministic function w(t) for all $t \in [0, \tau ]$ a.s.

• (C.4) There exists a $\tau >0$ such that $Pr(C_i \ge \tau ) > 0, i = 1, \cdots , n.$

• (C.5) The matrix $\Sigma$ is positive definite for all $t \in [0,\tau ]$.

• (C.6) $A_*=E\left[ \int _0^{\tau } \{ Z(t) - {\bar{z}}(t;\gamma _0)\}^{\otimes 2} \Delta _i(t) e^{\gamma _0' Z(t)} \lambda _0(t) dt \right]$ is positive definite, where E is the expectation.

Lemma A.1

Assume that the conditions (C.1)–(C.6) hold. If $\beta _0= (\beta _{10}', \beta _{20}')'$ are the true values of the parameters, then

$$\begin{aligned} \frac{1}{\sqrt{n}} \sum _{i=1}^n U_{ni} (\beta _0;{\hat{\gamma }}) \overset{{\mathfrak {D}}}{\rightarrow } N(0, \Sigma ). \end{aligned}$$

Proof

Note that

$$\begin{aligned}{} & {} \sum _{i=1}^n d{\hat{M}}_i(t; \beta _0,{\hat{\gamma }}) \\{} & {} \quad = \sum _{i=1}^n \left[ Y_i(t) \Delta _i(t)dO_i(t)- g\{{{\hat{\mu }}_0(t; \beta _0,{\hat{\gamma }})}e^{\beta _0'X_i(t)}\}\Delta _i(t)e^{{\hat{\gamma }}'Z_i(t)}d{\hat{\Lambda }}_0(t;{\hat{\gamma }})\right] =0, \end{aligned}$$

and

$$\begin{aligned} \sum _{i=1}^n d{\hat{M}}^{*}_i(t;{\hat{\gamma }})= & {} \sum _{i=1}^n \left[ \Delta _i(t)dO_i(t)- \Delta _i(t)e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }})\right] \\= & {} \sum _{i=1}^n \left[ \Delta _i(t)dO_i(t)- \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} \frac{\sum _{i=1}^n \Delta _i(t) dO_i(t)}{\sum _{i=1}^n \Delta _i(t)e^{{\hat{\gamma }}'Z_i(t)}}\right] \\= & {} \sum _{i=1}^n \Delta _i(t) dO_i(t)- \sum _{i=1}^n \Delta _i(t)e^{{\hat{\gamma }}'Z_i(t)} \frac{\sum _{i=1}^n \Delta _i(t) dO_i(t)}{\sum _{i=1}^n \Delta _i(t)e^{{\hat{\gamma }}'Z_i(t)}}\\= & {} \sum _{i=1}^n \Delta _i(t) dO_i(t)-\sum _{i=1}^n \Delta _i(t) dO_i(t)\\= & {} 0. \end{aligned}$$

Since ${\hat{\gamma }}$ is obtained by solving $V(\gamma )=0$ at $\gamma ={\hat{\gamma }}$, we have (cf. pp. 148–149, Therneau and Grambsch (1990))

$$\begin{aligned} 0= & {} V({\hat{\gamma }})\\= & {} \sum _{i=1}^{n}\int _0^{\tau }\{Z_i(t)-{\bar{Z}}(t;{\hat{\gamma }})\}\Delta _i(t)dO_i(t)\\= & {} \sum _{i=1}^n \int _0^{\tau } \{ Z_i(t) - {\bar{Z}}(t;{\hat{\gamma }}) \} d{\hat{M}}_i^*(t;{\hat{\gamma }}). \end{aligned}$$

Some simple algebra leads to

$$\begin{aligned} \sum _{i=1}^n U_{ni}(\beta _0;{\hat{\gamma }})= & {} \sum _{i=1}^n\Bigg [\int _0^{\tau }W(t)\{X_i(t)-{\hat{E}}_X(t;\beta _0,{\hat{\gamma }})\}d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }})\\{} & {} -\int _0^{\tau }\frac{W(t){\hat{R}}(t;\beta _0,{\hat{\gamma }})}{S^{(0)}(t,{\hat{\gamma }})}d{\hat{M}}_i^{*}(t;{\hat{\gamma }})\\{} & {} - {\hat{P}}(\beta _0,{\hat{\gamma }}) {\hat{D}}^{-1} \int _0^{\tau } \{ Z_i(t) - {\bar{Z}}(t;{\hat{\gamma }}) \} d{\hat{M}}_i^*(t;{\hat{\gamma }}) \Bigg ] \\= & {} \sum _{i=1}^n \int _0^{\tau }W(t)\{X_i(t)-{\hat{E}}_X(t;\beta _0,{\hat{\gamma }})\}d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }})\\{} & {} - \int _0^{\tau }\frac{W(t){\hat{R}}(t;\beta _0,{\hat{\gamma }})}{{S}^{(0)}(t,{\hat{\gamma }})} \sum _{i=1}^n d{\hat{M}}_i^{*}(t;{\hat{\gamma }})\\{} & {} - {\hat{P}}(\beta _0,{\hat{\gamma }}) {\hat{D}}^{-1} \sum _{i=1}^n \int _0^{\tau } \{ Z_i(t) - {\bar{Z}}(t;{\hat{\gamma }}) \} d{\hat{M}}_i^*(t;{\hat{\gamma }})\\= & {} \sum _{i=1}^n \int _0^{\tau }W(t) X_i(t) d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }})\\{} & {} - \sum _{i=1}^n \int _0^{\tau }W(t) {\hat{E}}_X(t;\beta _0,{\hat{\gamma }}) d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }}) \\= & {} \sum _{i=1}^n \int _0^{\tau }W(t) X_i(t) d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }})\\{} & {} - \int _0^{\tau }W(t) {\hat{E}}_X(t;\beta _0,{\hat{\gamma }}) \sum _{i=1}^n d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }}) \\= & {} \sum _{i=1}^n \int _0^{\tau }W(t) X_i(t) d{\hat{M}}_i(t;\beta _0,{\hat{\gamma }})\\= & {} \sum _{i=1}^n \int _0^{\tau } W(t) X_i(t)\\{} & {} \Big [ Y_i(t) \Delta _i(t) dO_i(t) - g\{{\hat{\mu }}_0(t; \beta _0,{\hat{\gamma }}) e^{\beta _0'X_i(t)}\} \Delta _i(t)\\{} & {} e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }})\Big ]\\= & {} U(\beta _0, {\hat{\gamma }}). \end{aligned}$$

As shown in the Appendix of Li et al. (2010) that $n^{-1/2} U(\beta _{0}; {\hat{\gamma }})$ converges to a zero mean Gaussian distribution with covariance matrix $\Sigma$, we have

$$\begin{aligned} \frac{1}{\sqrt{n}} \sum _{i=1}^n U_{ni} (\beta _0;{\hat{\gamma }})=\frac{1}{\sqrt{n}} U(\beta _0;{\hat{\gamma }}) \overset{{\mathfrak {D}}}{\rightarrow } N(0, \Sigma ). \end{aligned}$$

$\square$

Lemma A.2

Assume that the conditions (C.1)-(C.6) hold. If $\beta _0= (\beta _{10}', \beta _{20}')'$ are the true values of the parameters, then

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n U_{ni} (\beta _0;{\hat{\gamma }}) U_{ni}' (\beta _0;{\hat{\gamma }}) \overset{p}{\rightarrow }\ \Sigma . \end{aligned}$$

Proof

It can be shown that

$$\begin{aligned}{} & {} U_{ni}(\beta _0;{\hat{\gamma }}) = {U_{0i}}(\beta _0;\gamma _0)+ \int _0^{\tau } \{W(t) - w(t) \} X_i(t) Y_i(t) \Delta _i(t) dO_i(t) \hspace{150pt}\\{} & {} \quad +(-1) \int _0^{\tau } \Big ( W(t) {\hat{E}}_X(t;\beta _0,{\hat{\gamma }}) - w(t) e_x(t) \Big ) Y_i(t) \Delta _i(t) dO_i(t) \\{} & {} \quad +(-1) \Big ( \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }}) - \\{} & {} \quad \quad \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t) \Big ) \\{} & {} \quad + \Big ( \int _0^{\tau } W(t) {\hat{E}}_X(t;\beta _0,{\hat{\gamma }}) g\{{\hat{\mu }}_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }}) - \\{} & {} \quad \quad \int _0^{\tau } w(t) e_x(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t) \Big )\\{} & {} \quad + (-1) \int _0^{\tau } \Big ( \frac{W(t){\hat{R}}(t;\beta _0,{\hat{\gamma }})}{S^{(0)}(t;{\hat{\gamma }})} - \frac{w(t) {r}(t)}{{s}^{(0)}(t;\gamma _0)} \Big ) \Delta _i(t) dO_i(t) \\{} & {} + \int _0^{\tau } \Big [ {\hat{P}}(\beta _0,{\hat{\gamma }}) {\hat{D}}^{-1} \{ Z_i(t) - {\bar{Z}}(t;{\hat{\gamma }}) \} - P(\beta _0,\gamma _0)D^{-1} \{ Z_i(t) - {\bar{z}}(t;\gamma _0) \} \Big ] \Delta _i(t) dO_i(t)\\{} & {} \quad + \Big ( \int _0^{\tau } \Big [\frac{W(t){\hat{R}}(t;\beta _0,{\hat{\gamma }})}{S^{(0)}(t{;}{\hat{\gamma }})} - {\hat{P}}(\beta _0,{\hat{\gamma }}) {\hat{D}}^{-1} \{ Z_i(t) - {\bar{Z}}(t;{\hat{\gamma }}) \} \Big ] \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }}) - \\{} & {} \quad \quad \int _0^{\tau } \Big [\frac{w(t) {r}(t)}{{s}^{(0)}(t;\gamma _0)} - {P}(\beta _0,\gamma _0) {D}^{-1} \{ Z_i(t) - {\bar{z}}(t;\gamma _0) \} \Big ] \Delta _i(t) e^{{\gamma _0}'Z_i(t)} d{\Lambda }_0(t) \Big ) \\:= & {} {U_{0i}}(\beta _0;\gamma _0)+\epsilon _{i1}+\epsilon _{i2}+\epsilon _{i3}+\epsilon _{i4}+\epsilon _{i5}+\epsilon _{i6}+\epsilon _{i7}, \end{aligned}$$

where

$$\begin{aligned} {U_{0i}}(\beta _0;\gamma _0)= & {} \int _0^{\tau } w(t)\{X_i(t) - e_x(t)\} dM_i(t; \beta _0,\gamma _0) - \int _0^{\tau } \frac{w(t) {r}(t)}{{s}^{(0)}(t;\gamma _0)} {d}{M}^{*}_i(t; \gamma _0)\\{} & {} - P(\beta _0,\gamma _0)D^{-1} \int _0^{\tau } \{ Z_i(t) - {\bar{z}}(t;\gamma _0) \}{d}{M}^{*}_i(t; \gamma _0). \end{aligned}$$

Under conditions (C.1) and (C.3), $|| \epsilon _{i1}||=o_p(1), || \epsilon _{i2}||=o_p(1), || \epsilon _{i5}||=o_p(1)$ and $|| \epsilon _{i6}||=o_p(1)$ hold. Note that (cf. pp. 1103–1104, Andersen and Gill (1982))

$$\begin{aligned} {\hat{\Lambda }}_0(t;\gamma _0)-\Lambda _0(t)=\frac{1}{n} \sum _{i=1}^n \int _0^t \frac{dM_i^* (u)}{s^{(0)}({u};\gamma _0)}+o_p(n^{-1/2}). \end{aligned}$$

Then under conditions (C.1)-(C.3) and by the consistency of ${\hat{\mu }}_0(t)$ to ${\mu }_0(t)$, we have

$$\begin{aligned} \epsilon _{i3}= & {} \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }}) \\{} & {} - \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t)\\= & {} \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\hat{\Lambda }}_0(t;{\hat{\gamma }})\\{} & {} - \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\Lambda }_0(t)\\{} & {} +\int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d{\Lambda }_0(t)\\{} & {} - \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t)\\= & {} \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{{\hat{\gamma }}'Z_i(t)} d[{\hat{\Lambda }}_0(t;{\hat{\gamma }})- {\Lambda }_0(t)]\\{} & {} + (1+o_p(1)) \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t)\\{} & {} - \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t)\\= & {} (1+o_p(1)) \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d[{\hat{\Lambda }}_0(t;{\hat{\gamma }})-{\Lambda }_0(t)]\\{} & {} + \int _0^{\tau } W(t) X_i(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t)\\{} & {} - \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t)+o_p(1)\\= & {} (1+o_p(1)) \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d[{\hat{\Lambda }}_0(t;{\hat{\gamma }})-{\Lambda }_0(t)]\\{} & {} + \int _0^{\tau } \Big ( W(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} - w(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Big ) \ X_i(t) \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t) +o_p(1)\\= & {} (1+o_p(1)) \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} d[{\hat{\Lambda }}_0(t;\gamma _0)-{\Lambda }_0(t)]\\{} & {} +(1+o_p(1)) \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)}d[{\hat{\Lambda }}_0(t;{\hat{\gamma }})-{\hat{\Lambda }}_0(t;\gamma _0)]\\{} & {} + \int _0^{\tau } \Big ( W(t) g\{{\hat{\mu }}_0(t)e^{\beta _0'X_i(t)}\} - w(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Big ) \ X_i(t) \Delta _i(t) e^{\gamma _0'Z_i(t)} d{\Lambda }_0(t) +o_p(1)\\= & {} (1+o_p(1)) \frac{1}{n} \int _0^{\tau } \frac{w(t) X_i(t) g\{{\mu }_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} \sum _{i=1}^n dM_i^* (t)}{s^{(0)}}\\{} & {} +(1+o_p(1)) \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)}d[{\hat{\Lambda }}_0(t;{\hat{\gamma }})-{\hat{\Lambda }}_0(t;\gamma _0)]\\{} & {} +o_p(1)\\:= & {} E_1+E_2+o_p(1). \end{aligned}$$

In the term $E_1,$ for each i, $w(t) X_i(t) g\{{\mu }_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} \Big / s^{(0)}(t; \gamma _0)$ is predictable and finite, and $M_i^* (t)$ is a martingale. Then

$$\begin{aligned} \int _0^{\tau } \frac{w(t) X_i(t) g\{{\mu }_0(t)e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} \sum _{i=1}^n dM_i^* (t)}{s^{(0)}(t; \gamma _0)} \end{aligned}$$

is a martingale integral and converges to zero in probability. Also, similar to p. 300 of Fleming and Harrington (1991), the Taylor series expansion of ${\hat{\Lambda }}_0(t;{\hat{\gamma }})-{\hat{\Lambda }}_0(t;\gamma _0)$ about $\gamma =\gamma _0$ in the term $E_2$ can be written as

$$\begin{aligned} H'(t; \gamma ^*) ({\hat{\gamma }} - \gamma _0), \end{aligned}$$

where $\gamma ^*$ is on the line segment between ${\hat{\gamma }}$ and $\gamma _0$ and H is defined as

$$\begin{aligned} H(t;\gamma )=- \sum _{i=1}^n \int _0^t \frac{S^{(1)}(u;\gamma ) \Delta _i(u) dO_i(u)}{n \{S^{(0)}(u;\gamma )\}^2 }. \end{aligned}$$

Following Lin et al. (2000), it can be shown that $H(t;\gamma _0)$ converges to the following function a.e.

$$\begin{aligned} h(t;\gamma _0)=-\int _0^t \frac{s^{{(1)}}(u; \gamma _0)}{s^{{(0)}}(u; \gamma _0)} \lambda _0(u) du \end{aligned}$$

uniformly in t and

$$\begin{aligned} {\hat{\gamma }}-\gamma _0 = n^{-1} A_*^{-1} \sum _{i=1}^n \int _0^{\tau } \{ Z_i(t)- {\bar{z}}(t;\gamma _0)\} dM_i^*(t) + o_p(n^{-1/2}). \end{aligned}$$

Therefore, ${\hat{\Lambda }}_0(t;{\hat{\gamma }})-{\hat{\Lambda }}_0(t;\gamma _0)$ in the term $E_2$ is tight and therefore equal to

$$\begin{aligned} n^{-1} h'(t;\gamma _0) A_*^{-1} \sum _{i=1}^n \int _0^{\tau } \{ Z_i(t)- {\bar{z}}(t;\gamma _0)\} dM_i^*(t) + o_p(n^{-1/2}). \end{aligned}$$

Hence,

$$\begin{aligned} E_2= & {} (1+o_p(1)) \int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)}\\{} & {} [ h'(t;\gamma _0) A_*^{-1} \sum _{i=1}^n \frac{1}{n} \{ Z_i(t)- {\bar{z}}(t;\gamma _0)\} dM_i^*(t)] + o_p(n^{-1/2}), \end{aligned}$$

where $\int _0^{\tau } w(t) X_i(t) g\{{\mu }_0(t) e^{\beta _0'X_i(t)}\} \Delta _i(t) e^{\gamma _0'Z_i(t)} [h'(t;\gamma _0) A_*^{-1} \sum _{i=1}^n \frac{1}{n} \{ Z_i(t)- {\bar{z}}(t;\gamma _0)\} d M_i^*(t)]$ is a martingale integral and converges to zero in probability. Thus, $\epsilon _{i3} = o_p(1).$ Using similar arguments, it can be shown that $\epsilon _{i4}=o_p(1)$ and $\epsilon _{i7}=o_p(1).$ Therefore,

$$\begin{aligned} U_{ni}(\beta _{0};{\hat{\gamma }})={U_{0i}}(\beta _{0};\gamma _0)+o_p(1), \quad i= 1, 2, \cdots , n. \end{aligned}$$

(A.1)

Let ${\hat{Q}}_n = {n^{-1}} \sum _{i=1}^n U_{ni}(\beta _0;{\hat{\gamma }}) (U_{0i}(\beta _0;{\hat{\gamma }}))'$ and ${Q_n} = {n^{-1}} \sum _{i=1}^n {U_{0i}}(\beta _{0};\gamma _0) {U_{0i}}(\beta _{0};\gamma _0))'$. For any $c \in R^p$, the following decomposition holds:

$$\begin{aligned} c' ({\hat{Q}}_n - Q_n) c= & {} c' \Big ( \frac{1}{n} \sum _{i=1}^n U_{ni}(\beta _0;{\hat{\gamma }}) (U_{ni}(\beta _0;{\hat{\gamma }}))'\\{} & {} - \frac{1}{n} \sum _{i=1}^n {U_{0i}}(\beta _{0}; \gamma _0) ({U_{0i}}(\beta _{0}; \gamma _0))' \Big )c \\= & {} \frac{1}{n} \sum _{i=1}^n [ c' U_{ni}(\beta _0;{\hat{\gamma }})-c'{U_{0i}}(\beta _{0};\gamma _0) ]^2 \\{} & {} + \frac{2}{n} \sum _{i=1}^n [ c' {U_{0i}}(\beta _{0};\gamma _0)] [c'( U_{ni}(\beta _0;{\hat{\gamma }}) - {U_{0i}}(\beta _{0};\gamma _0)) ]\\:= & {} I_1+ 2 I_2. \end{aligned}$$

Both $I_1$ and $I_2$ are $o_p(1)$ by (A.1). As a result,

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n U_{ni}(\beta _0;{\hat{\gamma }}) U_{ni}' (\beta _0;{\hat{\gamma }})= & {} \frac{1}{n} \sum _{i=1}^n {U_{0i}}(\beta _{0};\gamma _0) {U_{0i}}'(\beta _{0};\gamma _0) + o_p(1)\\\overset{p}{\rightarrow } & {} E[{U_{0i}}(\beta _{0};\gamma _0) ({U_{0i}}(\beta _{0};\gamma _0))'] \\= & {} \Sigma . \end{aligned}$$

$\square$

Proof of Theorem 1

Following Owen (1988, 1990), we can prove the theorem similarly. From Lemma A.1, we have $\underset{n \rightarrow \infty }{ \text{ lim }} {\hat{Q}}_n = \Sigma .$ As $U_i(\beta _{0}; \gamma _0)$ are i.i.d. r.v. Then combining Owen (2001) and (A.1), we have

$$\begin{aligned} Z_n=\underset{1 \le i \le n}{\text{ max }} || U_{ni}(\beta _0;{\hat{\gamma }}) || = o_p(n^{1/2}). \end{aligned}$$

Combining Lemma A.1, Eq. (8) and Taylor expansion of $l(\beta _{0})$, one can prove Theorem 1. The details are omitted here. $\square$

Proof of Theorem 2

The proof is similar to arguments from Qin and Lawless (1994), Yu et al. (2011), Yang and Zhao (2012), Zhang and Zhao (2013), and Yu and Zhao (2019). By Lemma A.1, we complete the proof of Theorem 2. The details are omitted here. $\square$