Statistical inference for Cox model under case-cohort design with subgroup survival information

Abstract

With the explosive growth of data, it is a challenge to infer the quantity of interest by combining the existing different research data about the same topic. In the case-cohort setting, our aim is to improve the efficiency of parameter estimation for Cox model by using subgroup information in the aggregate data. So we put forward the generalized moment method (GMM) to use the auxiliary survival information at some critical time points. However, the auxiliary information is likely obtained from other studies or populations, two extended GMM estimators are proposed to account for multiplicative and additive inconsistencies. We establish the consistency and asymptotic normality of the proposed estimators. In addition, the uniform consistency and asymptotic normality of Breslow estimator are also presented. From the asymptotic normality, we show that the proposed approaches are more efficient than the traditional weighted estimating equation method. In particular, if the number of subgroups is equal to one, the asymptotic variance-covariances of the GMM estimators are identical with the weighted score estimate. Some simulation studies and a real data study demonstrate the proposed methods and theories. In the numerical studies, our approaches are even better than the full cohort estimator and the extended GMM methods are robust.

Appendix

For convenience, we use $\mathbf {0}$ to represent a matrix of 0 with proper dimension.

The proofs of the three lemmas are similar to Shang and Wang (2017).

Proof of Theorem 1

Define $U_w(\varvec{\beta })=(U_w^{(1)}(\varvec{\beta }),\ldots ,U_w^{(p)}(\varvec{\beta }))^\mathrm{T}$ and ${\varvec{Z}}_i=(Z_i^{(1)},\ldots ,Z_i^{(p)})^\mathrm{T},\ i=1,\ldots ,n$, where $U_w^{(l)}(\varvec{\beta })$ and $Z_i^{(l)}$ are the lth element of $U_w(\varvec{\beta })$ and ${\varvec{Z}}_i$, respectively. Let ${\varvec{S}}_w^{(m,l)}(t,\varvec{\beta })=\frac{1}{n}\sum _{i=1}^n w_i Y_i(t) Z_i^{(l)}{\varvec{Z}}_i^{\otimes m}\exp (\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i)$ with $m=0,1,2$, $l=1,\dots ,p$, ${\varvec{a}}^{\otimes 0}=1$ and ${\varvec{a}}^{\otimes 1}={\varvec{a}}$. The second order partial derivatives of $\Psi _n(\varvec{\theta })$ and $\phi ({\varvec{Z}},\varvec{\theta })$ are established as follows. For $k=1,\ldots ,K$ and $l=1,\ldots ,p$,

$$\begin{aligned} \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}} & =\frac{1}{n}\sum _{i=1}^n w_i\exp \{-\gamma \exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i)}\} \\ & \{\gamma ^2I({\varvec{Z}}_i\in \Omega _k){\varvec{Z}}_i{\varvec{Z}}_i^\mathrm{T}\exp {(2\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i}) -\gamma I({\varvec{Z}}_i\in \Omega _k){\varvec{Z}}_iZ_i^\mathrm{T}\exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i})\}, \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\beta }\partial \gamma } = \, &\frac{\gamma }{n}\sum _{i=1}^n w_i\exp \{-\gamma \exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i)}\} I({\varvec{Z}}_i\in \Omega _k){\varvec{Z}}_i\exp (2\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i),\\ \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \gamma ^2} = \, &\frac{1}{n}\sum _{i=1}^n w_i\exp \{-\gamma \exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i)}\} I({\varvec{Z}}_i\in \Omega _k)\exp (2\varvec{\beta }^\mathrm{T}{\varvec{Z}}_i), \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} = \, &\left( \begin{array}{cc} \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}} &{} \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\beta }\partial \gamma }\\ \frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\beta }^\mathrm{T}\partial \gamma } &{}\frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \gamma ^2} \end{array}\right) ,\\ \small \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}} = \, &\exp \{-\gamma \exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}})}\} \{\gamma ^2I({\varvec{Z}}\in \Omega _k) {\varvec{Z}}{\varvec{Z}}^\mathrm{T}\exp {(2\varvec{\beta }^\mathrm{T}{\varvec{Z}}}) -\gamma I({\varvec{Z}}\in \Omega _k){\varvec{Z}}{\varvec{Z}}^\mathrm{T}\exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}}})\},\\ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\beta }\partial \gamma } =\, &\gamma \exp \{-\gamma \exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}})}\} I({\varvec{Z}}\in \Omega _k) {\varvec{Z}}\exp (2\varvec{\beta }^\mathrm{T}{\varvec{Z}}),\\ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \gamma ^2} =\, &\exp \{-\gamma \exp {(\varvec{\beta }^\mathrm{T}{\varvec{Z}})}\}I({\varvec{Z}}\in \Omega _k)\exp (2\varvec{\beta }^\mathrm{T}{\varvec{Z}}),\\ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} =\, &\left( \begin{array}{cc} \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}} &{} \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\beta }\partial \gamma }\\ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\beta }^\mathrm{T}\partial \gamma } &{}\frac{\partial ^2 \phi ^{(k)}(Z,\varvec{\theta })}{\partial \gamma ^2} \end{array}\right) ,\\ \frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}} =\, &\frac{1}{n} \sum _{i=1}^n w_i\int _0^\tau \left\{ \frac{{\varvec{S}}_w^{(1)}(t,\varvec{\beta }){\varvec{S}}_w^{(1,l)}(t,\varvec{\beta })^\mathrm{T} +{\varvec{S}}_w^{(1,l)}(t,\varvec{\beta }) {\varvec{S}}_w^{(1)}(t,\varvec{\beta })^\mathrm{T} +S_w^{(0,l)}(t,\varvec{\beta }){\varvec{S}}_w^{(2)}(t,\varvec{\beta }) }{S_w^{(0)}(t,\varvec{\beta })^2} \right. \\&\left. -\frac{{\varvec{S}}_w^{(2,l)}(t,\varvec{\beta })}{ S_w^{(0)}(t,\varvec{\beta })}-\frac{2 S_w^{(0,l)}(t,\varvec{\beta }) {\varvec{S}}_w^{(1)}(t,\beta ){\varvec{S}}_w^{(1)}(t,\varvec{\beta })^\mathrm{T}}{S_w^{(0)}(t,\varvec{\beta })^3}\right\} \mathrm{d}N_i(t), \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 U_w^{(l)}(\varvec{\beta }))}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} =\, &\left( \begin{array}{cc} \frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}}&{} \mathbf {0}\\ \mathbf {0} &{}0 \end{array}\right) . \end{aligned}$$

(I) Applying the strong law of large numbers, the law of large numbers and the double expectation, we obtain that

$$\begin{aligned}&\frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} \, {\mathop {\rightarrow }\limits \, ^{a.s.}}E\left\{ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}\right\} ,\ k=1,\ldots ,K,\\&\Psi _n(\varvec{\theta }) \, {\mathop {\rightarrow }\limits \, ^{a.s}}E\left\{ \Psi _n(\varvec{\theta })\right\} ,\\&\frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}} \, {\mathop {\rightarrow }\limits \, ^{p}}E\left\{ \frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}}\right\} ,\\&{\varvec{S}}_w^{(m,l)}(t,\varvec{\beta }_0) \, {\mathop {\rightarrow }\limits \, ^{p}}E\left\{ Y(t)Z^{(l)}{\varvec{Z}}^{\otimes m}\exp (\varvec{\beta }_0^\mathrm{T}{\varvec{Z}})\right\} ,\ m=0,1,2;\ l=1,\ldots ,p. \end{aligned}$$

From the conditions of C1 and C2, we know $E\left\{ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} <\infty$ and $\frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}|_{\varvec{\theta }=\varvec{\theta }_0}<\infty , k=1,\ldots ,K$, as n is greater than a large enough positive number. Since Y(t) is less than one,

$$\begin{aligned} E\{\Vert Y(t)Z^{(l)}{\varvec{Z}}^{\otimes m}\exp (\varvec{\beta }_0^\mathrm{T}{\varvec{Z}})\Vert \}< E\{\Vert Z^{(l)}{\varvec{Z}}^{\otimes m}\exp (\varvec{\beta }_0^\mathrm{T}{\varvec{Z}})\Vert \}. \end{aligned}$$

Following condition C3, we get $E\{S_w^{(m,l)}(t,\varvec{\beta }_0)\}<\infty$. Thus $E\left\{ \frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}}|_{\varvec{\beta }=\varvec{\beta }_0}\right\} <\infty$ and $\frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\beta }\partial \varvec{\beta }^\mathrm{T}}|_{\varvec{\beta }=\varvec{\beta }_0}<\infty ,l=1,\ldots ,p$ in probability, as n is large enough. From the continuity of $\frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}$ and $\frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}$, we conclude that $E\left\{ \frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}\right\}$ and $E\left\{ \frac{\partial ^2 \phi ^{(k)}({\varvec{Z}},\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}\right\}$ are finite, $\frac{\partial ^2 \Phi _{ w}^{(k)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}$ and $\frac{\partial ^2 U_w^{(l)}(\varvec{\beta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}$ are finite in probability, as n is greater than a large enough positive number and $\varvec{\theta }$ lies in the neighborhood of $\varvec{\theta }_0$.

Define the overall weighted loss function $l(\varvec{\theta })= E\{\Psi _n(\varvec{\theta })^\mathrm{T}\} \varvec{\Sigma }^{-1} E\{\Psi _n(\varvec{\theta })\}/2$. Thus

$$\begin{aligned} \frac{\partial l_n(\varvec{\theta })}{\partial \varvec{\theta }} =\, &\frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}^\mathrm{T} \widehat{\varvec{\Sigma }}(\widehat{\varvec{\theta }}_n^1)^{-1} \Psi _n(\varvec{\theta }),\\ \frac{\partial ^2 l_n(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} =\, &\frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}^\mathrm{T}\widehat{\varvec{\Sigma }} (\widehat{\varvec{\theta }}_n^1)^{-1}\frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }} +\sum _{j=1}^{K+p}\left\{ \widehat{\varvec{\Sigma }}(\widehat{\varvec{\theta }}_n^1)^{-1} \Psi _n(\varvec{\theta })\right\} ^{(j)} \frac{\partial ^2 \Psi _n^{(j)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}},\\ \frac{\partial l(\varvec{\theta })}{\partial \varvec{\theta }} =\, &E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}\right\} ^\mathrm{T} \varvec{\Sigma }^{-1} E\left\{ \Psi _n(\varvec{\theta })\right\} ,\\ \frac{\partial ^2 l(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} =\, &E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}\right\} ^\mathrm{T} \varvec{\Sigma }^{-1}E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}\right\} \\&+\sum _{j=1}^{K+p}[\varvec{\Sigma }^{-1} E\left\{ \Psi _n(\varvec{\theta })\right\} ]^{(j)}E\left\{ \frac{\partial ^2 \Psi _n^{(j)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}\right\} , \end{aligned}$$

where $\{\widehat{\varvec{\Sigma }}(\widehat{\varvec{\theta }}_n^1)^{-1} \Psi _n(\varvec{\theta })\}^{(j)}$ and $[\varvec{\Sigma }^{-1} E\{\Psi _n(\varvec{\theta })\}]^{(j)}$ are the jth elements of $\widehat{\varvec{\Sigma }}(\widehat{\varvec{\theta }}_n^1)^{-1} \Psi _n(\varvec{\theta })$ and $\varvec{\Sigma }^{-1} E\{\Psi _n(\varvec{\theta })\}$, respectively.

Let $\mathscr {F}_t=\sigma \{N_i(s),Y_i(s+),{\varvec{Z}}_i;0\le s\le t,i=1,2,\ldots ,n\}$ denote a filtration. Define $M_i(t,\varvec{\beta })=N_i(t)-\int _0^t Y_i(u)\lambda _0(u)e^{\varvec{\beta }^\mathrm T{\varvec{Z}}_i}\mathrm{d} u,\ i=1,\ldots ,n$. Then $M_i(t,\varvec{\beta }_0),\ i=1,\ldots ,n$ are martingales on the time interval $[0,\tau ]$. Therefore,

$$\begin{aligned} E\{U_w(\varvec{\beta }_0)\}&=E\left[ \frac{1}{n}\sum _{i=1}^n w_i\int _0^\tau \left\{ {\varvec{Z}}_i- \frac{S_w^{(1)}(t,\varvec{\beta }_0)}{S_w^{(0)}(t,\varvec{\beta }_0)}\right\} \quad \mathrm{d}M_i(t,\varvec{\beta }_0)\right] \\&=\frac{1}{n}\sum _{i=1}^n E\left( E\left[ \int _0^\tau \left\{ {\varvec{Z}}_i- \frac{S_w^{(1)}(t,\varvec{\beta }_0)}{S_w^{(0)}(t,\varvec{\beta }_0)}\right\} \quad \mathrm{d}M_i(t,\varvec{\beta }_0)|\mathscr {F}_0\right] \right) \\&=\mathbf {0}. \end{aligned}$$

Thus $E\{\Psi _n(\varvec{\theta }_0)\}=\mathbf {0}$. Since $\Psi _n(\varvec{\theta })$ is continuous, we obtain that $|\Psi _n(\varvec{\theta })|$ is less than an arbitrarily small value in probability if $\varvec{\theta }$ lies in the neighborhood of $\varvec{\theta }_0$. From Lemma 2, we get that $\frac{\partial ^2 l_n(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}$ is a positive definite matrix in probability. Then $l_n(\varvec{\theta })$ is convex function and $\widehat{\varvec{\theta }}_n$ is the unique minimum point of $l_n(\varvec{\theta })$ in the neighborhood of $\varvec{\theta }_0$. Since $E\left\{ \frac{\partial ^2 \Psi _n^{(j)}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}\right\}$ is bounded and $E\{\Psi _n(\varvec{\theta }_0)\}=0$, we show that

$$\begin{aligned} \frac{\partial ^2 l(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} |_{\varvec{\theta }=\varvec{\theta }_0}= E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} ^\mathrm{T} \varvec{\Sigma }^{-1}E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} . \end{aligned}$$

By Lemma 2 and the positive definite property of $\varvec{\Sigma }$, we know that $\frac{\partial ^2 l(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}|_{\varvec{\theta }=\varvec{\theta }_0}$ is a positive definite matrix. Therefore, $l(\varvec{\theta })$ is a convex function in the neighborhood of $\varvec{\theta }_0$. Due to $l(\varvec{\theta }_0)=0$ and $l(\varvec{\theta })\ge 0$, we obtain that $\varvec{\theta }_0$ is the unique minimum point of $l(\varvec{\theta })$ in the neighborhood of $\varvec{\theta }_0$. From Lemma 8.3.2 of Fleming and Harrington (1991), we conclude that $\widehat{\varvec{\theta }}_n$ coverage to $\varvec{\theta }_0$ in probability.

(II) By the Newton-Raphson method, we obtain that

$$\begin{aligned} \sqrt{n} (\widehat{\varvec{\theta }}_n-\varvec{\theta }_0)&=-\left\{ \frac{\partial ^2 l_n(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} |_{\varvec{\theta }=\tilde{\varvec{\theta }}_n}\right\} ^{-1}\sqrt{n}\frac{\partial l_n(\varvec{\theta })}{\partial \varvec{\theta }} |_{\varvec{\theta }=\varvec{\theta }_0}\\&=-\left\{ \frac{\partial ^2 l_n(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} |_{\varvec{\theta }=\tilde{\varvec{\theta }}_n}\right\} ^{-1}\frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}^\mathrm{T} |_{\varvec{\theta }=\varvec{\theta }_0}\widehat{\varvec{\Sigma }}(\widehat{\varvec{\theta }}_n^1)^{-1}\sqrt{n}\Psi _n(\varvec{\theta }_0), \end{aligned}$$

where $\tilde{\varvec{\theta }}_n$ is a point on the line of $\widehat{\varvec{\theta }}_n$ and $\varvec{\theta }_0$. From the law of large numbers and the continuous mapping theorem, we show that

$$\begin{aligned} \frac{\partial ^2 l_n(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}} |_{\varvec{\theta }=\tilde{\varvec{\theta }}_n} \, {\mathop {\rightarrow }\limits \, ^{p}}\frac{\partial ^2 l(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\mathrm{T}}|_{\varvec{\theta }= \varvec{\theta }_0}. \end{aligned}$$

By Lemma 1, the law of large numbers and the Slutsky Lemma,

$$\begin{aligned} \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }} |_{\varvec{\theta }=\varvec{\theta }_0} ^\mathrm{T}\widehat{\varvec{\Sigma }} (\widehat{\varvec{\theta }}_n^1)^{-1} \, {\mathop {\rightarrow }\limits \, ^{p}} E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} ^\mathrm{T}\varvec{\Sigma }^{-1}. \end{aligned}$$

The weighted auxiliary estimating function can be rewrote as follow,

$$\begin{aligned} {\sqrt{n}} \Phi _{ w}(\varvec{\theta }_0) ={\sqrt{n}}\Phi _F(\varvec{\theta }_0) +\frac{1}{\sqrt{n}}\sum _{i=1}^n( w_i-1)\phi ({\varvec{Z}}_i,\varvec{\theta }_0), \end{aligned}$$

where $\Phi _{F}(\varvec{\theta })=(\Phi _{F}^{(1)}(\varvec{\theta }),\ldots ,\Phi _{F}^{(K)}(\varvec{\theta }))^\mathrm{T}.$

For $i=1,\ldots ,n,$

$$\begin{aligned} \text{ cov }\left\{ \phi ({\varvec{Z}}_i,{\varvec{\theta }}_0), ( w_i-1)\phi ({\varvec{Z}}_i,{\varvec{\theta }}_0)\right\} =E\left\{ \phi ({\varvec{Z}}_i,{\varvec{\theta }}_0)^{\otimes 2} E( w_i-1|\xi _i,{\varvec{Z}}_i)\right\} =\mathbf {0}. \end{aligned}$$

Then $\text{ cov }\{\sqrt{n}\Phi _{ w}(\varvec{\theta }_0)\}=E\{\phi ({\varvec{Z}},\varvec{\theta }_0)\phi ({\varvec{Z}},\varvec{\theta }_0)^\mathrm{T}\}+(1-\alpha _0)\rho _0/\alpha _0E\{\phi ({\varvec{Z}},\varvec{\theta }_0)\phi ({\varvec{Z}},\varvec{\theta }_0)^\mathrm{T}\} =\phi _\phi .$

Because $\text{ cov }\{ U_w({\varvec{\beta }}_0),\Phi _{ w}({\varvec{\theta }}_0)\} =E[E\{ U_w({\varvec{\beta }}_0)|\mathscr {F}_0\}\Phi _{ w}({\varvec{\theta }}_0)^\mathrm T]=\mathbf {0},$ following the central limit theorem and the martingale central limit theorem, we get that

$$\begin{aligned} \sqrt{n}\Psi _n(\varvec{\theta }_0) \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N(\mathbf {0},\varvec{\Sigma }). \end{aligned}$$

By the Slutsky Lemma, we show that

$$\begin{aligned} \sqrt{n} (\widehat{\varvec{\theta }}_n-\varvec{\theta }_0) \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N (\mathbf {0},\left[ E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }} |_{\varvec{\theta }=\varvec{\theta }_0}\right\} ^\mathrm{T}\varvec{\Sigma }^{-1} E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} \right] ^{-1}), \end{aligned}$$

where

$$\begin{aligned}&E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} ^\mathrm{T} \varvec{\Sigma }^{-1}E\left\{ \frac{\partial \Psi _n(\varvec{\theta })}{\partial \varvec{\theta }}|_{\varvec{\theta }=\varvec{\theta }_0}\right\} \\&\quad =\left( \begin{array}{ll} \varvec{\Sigma }_1(\varvec{\Sigma }_1+ \varvec{\Sigma }_2)^{-1}\varvec{\Sigma }_1+\phi _{\varvec{\beta }}^\mathrm{T}\phi _\phi ^{-1}\phi _{\varvec{\beta }} &{} \phi _{\varvec{\beta }}^\mathrm{T}\phi _\phi ^{-1}\phi _\gamma \\ \phi _\gamma ^\mathrm{T}\phi _\phi ^{-1}\phi _{\varvec{\beta }} &{}\phi _\gamma ^\mathrm{T}\phi _\phi ^{-1}\phi _\gamma \end{array}\right) . \end{aligned}$$

From the proof of Lemma 2, $\phi _\gamma ^\mathrm{T}\phi _\phi ^{-1}\phi _\gamma >0$, then

$$\begin{aligned} \sqrt{n} (\widehat{\varvec{\beta }}_n-\varvec{\beta }_0) \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N(\mathbf {0},\{\varvec{\Sigma }_1(\varvec{\Sigma }_1+ \varvec{\Sigma }_2)^{-1}\varvec{\Sigma }_1+\varvec{\Sigma }_3\}^{-1}). \end{aligned}$$

By Lemma 3, we obtain that $\varvec{\Sigma }_3$ is a positive semi-definite matrix and equal to zero in case that $K=1$.

The proofs of the asymptotic properties for ${\widehat{\Lambda }}_w(t,{\widehat{{\varvec{\beta }}}}_n)$, ${\widehat{\Lambda }}_w(t,{\widehat{{\varvec{\beta }}}}_n^w)$, ${\widehat{\Lambda }}_w(t,{\widehat{{\varvec{\beta }}}}_M)$ and ${\widehat{\Lambda }}_w(t,{\widehat{{\varvec{\beta }}}}_A)$ can be found in Kulich and Lin (2004). The asymptotic properties of ${\widehat{{\varvec{\beta }}}}_n^w$, ${\widehat{{\varvec{\beta }}}}_M$ and ${\widehat{{\varvec{\beta }}}}_A$ are similar with that of ${\widehat{{\varvec{\beta }}}}_n$. The consistency of ${\widehat{\eta }}_M$ and ${\widehat{\eta }}_A$ are similar with that of ${\widehat{{\varvec{\theta }}}}_n$. Hence we only illustrate the the asymptotic normality for ${\widehat{\eta }}_M$ and ${\widehat{\eta }}_A$

Proof of asymptotic normality for ${\widehat{\eta }}_M$: Following from the same discussion of the asymptotic normality for ${\widehat{{\varvec{\theta }}}}_M$,

$$\begin{aligned} \sqrt{n}({\widehat{\varphi }}_M-\varphi _0) \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N(0,\sigma _1^2). \end{aligned}$$

Because

$$\begin{aligned} \sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_M)-\Lambda _0(t_*)\}&=\frac{1}{\sqrt{n}}\sum _{i=1}^n w_i\int _0^{t_*} \frac{1}{S_w^{(0)}(t,{\widehat{{\varvec{\beta }}}}_M)}\mathrm{d}M_i(t,{\widehat{{\varvec{\beta }}}}_M)\\&=\frac{1}{\sqrt{n}}\sum _{i=1}^n w_i\int _0^{t_*} \frac{1}{S_w^{(0)}(t,\varvec{\beta }_0)}\mathrm{d}M_i(t,{\varvec{\beta }}_0)+o_p(1), \end{aligned}$$

then under the regularity conditions and C1-C4,

$$\begin{aligned} \sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_M)-\Lambda _0(t_*)\} \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N(0,\sigma _2^2). \end{aligned}$$

Since $\text{ cov }\left\{ \sqrt{n}\Psi _n(\varvec{\theta }_0),\frac{1}{\sqrt{n}}\sum _{i=1}^n w_i\int _0^{t_*} \frac{1}{S_w^{(0)}(t,\varvec{\beta }_0)}\mathrm{d}M_i(t,{\varvec{\beta }}_0)\right\} =\mathbf {0}$, then

$$\begin{aligned} \text{ cov }[\sqrt{n}({\widehat{\varphi }}_M-\varphi _0),\sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_M)-\Lambda _0(t_*)\}]=0. \end{aligned}$$

From $\sqrt{n}({\widehat{\eta }}_M-\eta _0)=\frac{{\widehat{\varphi }}_M\sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_M)-\Lambda _0(t_*)\}}{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_M)\Lambda _0(t_*)} +\frac{\sqrt{n}({\widehat{\varphi }}_M-\varphi _0)}{\Lambda _0(t_*)}$ and the Slutsky Lemma,

$$\begin{aligned} \sqrt{n} (\widehat{\eta }_M-\eta _0) \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N\left( 0,\frac{\sigma _1^2+\eta _0^2\sigma _2^2}{\Lambda _0(t_*)^2}\right) . \end{aligned}$$

Proof of asymptotic normality for ${\widehat{\eta }}_A$: Following from the same discussion of the proof for the asymptotic normality of ${\widehat{\eta }}_M$,

$$\begin{aligned} \sqrt{n}({\widehat{\varphi }}_A-\varphi _0) \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N(0,\sigma _1^2), \\ \sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_A)-\Lambda _0(t_*)\} \, {\mathop {\rightarrow }\limits \, ^{\mathscr {D}}} N(0,\sigma _2^2), \\ \text{ cov }[\sqrt{n}({\widehat{\varphi }}_A-\varphi _0),\sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_A)-\Lambda _0(t_*)\}]=0. \end{aligned}$$

From $\sqrt{n}({\widehat{\eta }}_A-\eta _0)=\sqrt{n} \{{\widehat{\Lambda }}_w(t_*,{\widehat{{\varvec{\beta }}}}_A)-\Lambda _0(t_*)\}/t_* +\sqrt{n}({\widehat{\varphi }}_A-\varphi _0)/t_*$ and the Slutsky Lemma,

$$\begin{aligned} \sqrt{n} (\widehat{\eta }_A-\eta _0) \, {\mathop {\rightarrow }\limits ^{\mathscr {D}}} \, N\left( 0,\frac{\sigma _1^2+\sigma _2^2}{t_*^2}\right) . \end{aligned}$$