Joint modeling of generalized scale-change models for recurrent event and failure time data

Abstract

Recurrent event and failure time data arise frequently in many clinical and observational studies. In this article, we propose a joint modeling of generalized scale-change models for the recurrent event process and the failure time, and allow the two processes to be correlated through a shared frailty. The proposed joint model is flexible in that it requires neither the Poisson assumption for the recurrent event process nor a parametric assumption on the frailty distribution. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. Simulation studies are conducted to evaluate the finite sample performances of the proposed method. An application to a medical cost study of chronic heart failure patients is provided.

Appendices

Appendix

In order to prove Theorem 1, we study the asymptotic distribution of $n^{1/2}(\hat{\theta } - \theta _0)$, $n^{1/2}(\hat{\xi } - \xi _0),$ $n^{1/2}\{\widehat{\Lambda }_0(t) - \Lambda _0(t) \}$ and $n^{1/2}\{\widehat{H}_0(t) - H_0(t) \}$.

1. Asymptotic Linearity of $\widehat{\Lambda }_0(t; \alpha )$

By the uniform strong law of large numbers (Pollard 1990), we obtain that $Q_n(t; \alpha ) \rightarrow Q(t; \alpha )$ and $R_n(t; \alpha ) \rightarrow R(t; \alpha )$ almost surely uniformly in $\alpha$ and $t \in [0, \tau ].$ In addition, the functional central limit theorem (Pollard 1990) implies that $||Q_n(t; \alpha ) - Q(t; \alpha )|| = O_p(n^{-1/2})$ and $||R_n(t; \theta ) - R(t; \theta )|| = O_p(n^{-1/2})$, where $|| \cdot ||$ denotes the supremum norm. Then using the asymptotic properties of the product-integral (Gill and Johansen 1990), we have

$$\begin{aligned} || -\log \widehat{\Lambda }_0(t; \alpha ) - \int _t^\tau R^{-1}_n(u; \alpha )dQ_n(u; \alpha ) || = o_p(n^{-1/2}). \end{aligned}$$

(13)

Let $\Lambda _0(t; \alpha ) =\exp \{ -\int ^\tau _t {R^{-1}(u; \alpha )} {dQ(u; \alpha )}\}$ with $\Lambda _0(t; \alpha _0) \equiv \Lambda _0(t).$ It follows from the functional delta method that uniformly in $\alpha$ and $t \in [0, \tau ],$

$$\begin{aligned}&\int ^\tau _t \frac{dQ_n(u;\alpha )}{R_n(u;\alpha )} - \int ^\tau _t \frac{dQ(u;\alpha )}{R(u;\alpha )}\nonumber \\&\quad = \int ^\tau _t \frac{d\{Q_n(u;\alpha ) - Q(u;\alpha )\}}{R(u;\alpha )} - \int ^\tau _t \frac{[R_n(u;\alpha ) -R(u;\alpha ) ]dQ(u;\alpha )}{R(u;\alpha )^2} + o_p(n^{-1/2}) \nonumber \\&\quad = \int ^\tau _t \frac{dQ_n(u;\alpha )}{R(u;\alpha )} - \int ^\tau _t \frac{R_n(u;\alpha ) dQ(u;\alpha )}{R(u;\alpha )^2} + o_p(n^{-1/2}) \nonumber \\&\quad = -\sum ^n_{i=1} \phi _i(t; \alpha ) + o_p(n^{-1/2}), \end{aligned}$$

(14)

where

$$\begin{aligned} \phi _i(t; \alpha ) = \sum ^{m_i}_{j=1} \int ^\tau _t \frac{ I\{t^*_{ij}(\alpha ) \le u \le T^*_i(\alpha )\} dQ(u;\alpha )}{R(u;\alpha )^2} -\sum ^{m_i}_{j=1} \int ^\tau _t \frac{d I\{t^*_{ij}(\alpha ) \le u \}}{R(u;\alpha )}, \end{aligned}$$

with $E\phi _i(t; \alpha ) = 0.$ Then by (13) and (14), uniformly in $\alpha$ and $t \in [0, \tau ],$

$$\begin{aligned} n^{1/2} \{\widehat{\Lambda }_0(t;\alpha ) - \Lambda _0(t; \alpha )\} = n^{-1/2} \Lambda _0(t; \alpha ) \sum ^n_{i=1} \phi _i(t; \alpha ) + o_p(1), \end{aligned}$$

(15)

which gives an asymptotic i.i.d. representation of $n^{1/2} \{\widehat{\Lambda }_n(t;\alpha ) - \Lambda _0(t; \alpha )\}.$

We next show the asymptotic linearity of $n^{1/2} \{\widehat{\Lambda }_0(t; \alpha ) - \widehat{\Lambda }_0(t; \alpha _0)\}.$ Note that

$$\begin{aligned}&\int ^\tau _t \frac{dQ_n(t; \alpha )}{R_n(t; \alpha )} - \int ^\tau _t \frac{dQ_n(t; \alpha _0)}{R_n(t; \alpha _0)} \nonumber \\&\quad = \int ^\tau _t \frac{dQ(t; \alpha )}{R(t; \alpha )} - \int ^\tau _t \frac{dQ(t; \alpha _0)}{R(t; \alpha _0)} \nonumber \\&\qquad + \int ^\tau _t \Big \{ \frac{1}{R_n(u;\alpha )} - \frac{1}{R(u;\alpha )} -\frac{1}{R_n(u;\alpha _0)} + \frac{1}{R(u;\alpha _0)} \Big \} dQ_n(u; \alpha ) \nonumber \\&\qquad + \int ^\tau _t \Big \{ \frac{1}{R(u;\alpha )} - \frac{1}{R(u;\alpha _0)} \Big \} d\{Q_n(u;\alpha ) - Q(u;\alpha )\} \nonumber \\&\qquad + \int ^\tau _t \Big \{ \frac{1}{R_n(u;\alpha _0)} - \frac{1}{R(u;\alpha _0)} \Big \} d\{Q_n(u;\alpha ) - Q(u;\alpha _0)\} \nonumber \\&\qquad + \int ^\tau _t \frac{1}{R_(u;\alpha _0)} d\{Q_n(u;\alpha ) - Q(u;\alpha ) -Q_n(u;\alpha _0) + Q(u;\alpha _0)\} \nonumber \\&\quad := I_1(t; \alpha ) + I_2(t; \alpha ) + I_3(t; \alpha ) + I_4(t; \alpha ) + I_ 5(t; \alpha ). \end{aligned}$$

(16)

For any positive sequence $\varepsilon _n \rightarrow 0$ and $||\alpha - \alpha _0|| \le \varepsilon _n$, following similar arguments as in the proof of Theorem 1 of Ying (1993), we have that uniformly in $\{\alpha : ||\alpha - \alpha _0|| \le \varepsilon _n \}$ and $u \in [0, \tau ]$,

$$\begin{aligned} \frac{1}{R_n(u;\alpha )} - \frac{1}{R(u;\alpha )} -\frac{1}{R_n(u;\alpha _0)} + \frac{1}{R(u;\alpha _0)} = o_p(n^{-1/2}), \end{aligned}$$

which implies that $I_2(t; \alpha ) = o_p(n^{-1/2})$ uniformly in $\alpha$ and $t \in [0, \tau ]$. In addition, using a similar argument as in Lemma 3 of Ying (1993), we obtain that $I_3(t; \alpha ) + I_4(t; \alpha ) = o_p(n^{-1/2})$ uniformly in $\alpha$ and $t \in [0, \tau ]$. Note that uniformly in $\{\alpha : ||\alpha - \alpha _0|| \le \varepsilon _n \}$ and $t \in [0, \tau ],$

$$\begin{aligned} Q_n(u;\alpha ) - Q(u;\alpha )-Q_n(u;\alpha _0) + Q(u; \alpha _0) = o_p(n^{-1/2}). \end{aligned}$$

Then by integration by parts, $I_ 5(t; \alpha ) = o_p(n^{-1/2})$ uniformly in $\{\alpha : ||\alpha - \alpha _0|| \le \varepsilon _n \}$ and $t \in [0, \tau ]$. Thus, $I_2(t; \alpha ) + I_3(t; \alpha ) + I_4(t; \alpha ) + I_ 5(t; \alpha ) = o_p(n^{-1/2})$ uniformly in $\{\alpha : ||\alpha - \alpha _0|| \le \varepsilon _n \}$ and $u \in [0, \tau ]$. Furthermore, in view of the definitions of $Q(t; \alpha )$ and $R(t; \alpha )$, a straightforward calculation yields that uniformly in $\alpha$ and $t \in [0, \tau ],$

$$\begin{aligned} I_1(t; \alpha ) =&\int ^\tau _t \frac{dQ(u; \alpha ) - E[\nu _i e^{X_i^\prime (\eta _0 - \alpha _0) } I\{T^*_i(\alpha ) \ge u\}]d\Lambda _0(u) }{R(u; \alpha )} \\&\quad + \int ^\tau _t \frac{E[\nu _i e^{X_i^\prime (\eta _0 - \alpha _0) } I\{T^*_i(\alpha ) \ge u\}] d\Lambda _0(u)}{R(u; \alpha )} \\&-\int ^\tau _t \frac{E[\nu _i e^{X_i^\prime (\eta _0 - \alpha _0) } I\{T^*_i(\alpha _0) \ge u\}] d\Lambda _0(u)}{R(u; \theta _0)}\\ =&\int ^\tau _t \frac{E[\nu _i X_i^\prime e^{X_i^\prime (\eta _0 - \alpha _0)} I\{T^*_i(\alpha _0) \ge u\}]}{R(u; \theta _0)} d\{\lambda _0(u)u\} (\alpha _0 - \alpha ) \\&\quad + \int ^\tau _t \frac{\partial \{E[\nu _i e^{X_i^\prime (\eta _0 - \alpha _0) } I\{T^*_i(\alpha ) \ge u\}] R(u; \alpha )^{-1}\}}{\partial \alpha ^\prime } \Big |_{\alpha = \alpha _0} d\Lambda _0(u) (\alpha - \alpha _0)\\&\quad + o_p(||\alpha - \alpha _0||). \end{aligned}$$

Hence it follows from (13) and (16) that uniformly in $\{\alpha : ||\alpha - \alpha _0|| \le \varepsilon _n \}$ and $t \in [0, \tau ],$

$$\begin{aligned} \widehat{\Lambda }_0(t; \alpha ) - \widehat{\Lambda }_0(t; \alpha _0) = \Lambda _0(t) \varphi (t)^\prime (\alpha - \alpha _0) + o_p(n^{-1/2}+||\alpha -\alpha _0|| ), \end{aligned}$$

(17)

where

$$\begin{aligned} \varphi (t) =&\int ^\tau _t \frac{E[\nu _i X_i^\prime e^{X_i^\prime (\eta _0 - \alpha _0)} I\{T^*_i(\alpha _0) \ge u\}]}{R(u; \theta _0)} d\{\lambda _0(u)u\} \\&- \int ^\tau _t \frac{\partial \{E[\nu _i e^{X_i^\prime (\eta _0 - \alpha _0) } I\{T^*_i(\alpha ) \ge u\}] R(u; \alpha )^{-1}\}}{\partial \alpha ^\prime } \Big |_{\alpha = \alpha _0} d\Lambda _0(u). \end{aligned}$$

2. Asymptotic properties of $\hat{\theta }$ and $\widehat{\Lambda }_0(t)$

We first show the asymptotic normality of $S_n(\theta _0)$. Note that by (6) and (7),

$$\begin{aligned} S(\theta _0) =&\frac{1}{n} \sum ^n_{i=1} X^*_i(\theta _0)m_i e^{X_i^\prime (\alpha _0-\eta _0)} \big [ \widehat{\Lambda }_0^{-1} \{T^*_i(\alpha _0)\} - \Lambda _0^{-1} (T^*_i(\alpha _0))\big ] \nonumber \\&\quad + \frac{1}{n} \sum ^n_{i=1} X^*_i(\theta _0) \big [m_ie^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0)) - \mu _{\nu }\big ] \nonumber \\&- \frac{1}{n} \sum ^n_{i=1} X^*_i(\theta _0) \big [ \widehat{\mu }_\nu (\theta _0) - \mu _{\nu } \big ]. \end{aligned}$$

(18)

In view of (15), the first term on the right-hand side of (18) equals

$$\begin{aligned}&- \frac{1}{n}\sum ^n_{i=1} X^*_i(\theta _0) m_i e^{X_i^\prime (\alpha _0-\eta _0)} \frac{n^{-1} \sum ^n_{j=1} \phi _j (T^*_i(\alpha _0); \alpha _0)}{ \Lambda _0 (T^*_i(\alpha _0))} + o_p(n^{-1/2}) \\&\quad = - \frac{1}{n} \sum ^n_{i=1} \int {x^* m e^{x^\prime (\alpha _0-\eta _0)} \phi _i (y; \alpha _0)} {\Lambda _0^{-1}(t^*)} d V_1(x, m, t) + o_p(n^{-1/2}), \end{aligned}$$

where $V_1(x, m, t)$ is the joint probability measure of $(X_i, m_i, T_i)$ with $x^*=(x, \Phi (x; \theta _0)^\prime )^\prime$ and $t^*=te^{x'\alpha _0}.$ In a similar manner, we have

$$\begin{aligned} \widehat{\mu }_\nu (\theta _0) - \mu _{\nu } =&\frac{1}{n} \sum ^n_{i = 1} m_i e^{ X_i^\prime (\alpha _0-\eta _0)} \widehat{\Lambda }_0^{-1}\{T^*_i(\alpha _0)\} -\frac{1}{n} \sum ^n_{i = 1} m_i e^{ X_i^\prime (\alpha _0-\eta _0)} {\Lambda }_0^{-1}(T^*_i(\alpha _0)) \\&\quad + \frac{1}{n} \sum ^n_{i = 1} m_i e^{ X_i^\prime (\alpha _0-\eta _0)} {\Lambda }_0^{-1}(T^*_i(\alpha _0)) -\mu _\nu \\ =&\frac{1}{n} \sum ^n_{i=1}\Big [- \int { m e^{x^\prime (\alpha _0-\eta _0)} \phi _i (y; \alpha _0) {\Lambda _0^{-1} (y)} d V(x, x^*, m, y)} \\&+ m_i e^{ X_i^\prime (\alpha _0-\eta _0)} {\Lambda }_0^{-1}(T^*_i(\alpha _0)) -\mu _\nu \Big ]+ o_p(n^{-1/2}). \end{aligned}$$

Thus, by combining the above results, we obtain

$$\begin{aligned} S (\theta _0) = \frac{1}{n} \sum ^n_{i=1} \Upsilon _i + o_p(n^{-1/2}), \end{aligned}$$

(19)

which implies that $n^{1/2}S (\theta _0)$ is asymptotically normal, where

$$\begin{aligned} \Upsilon _i =&\int {\big [E\{ X^*_i(\theta _0)\} - x^*\big ] m e^{x^\prime (\alpha _0-\eta _0)} \phi _i (y; \alpha _0) \Lambda _0^{-1} (y) d V_1(x, m, y) } \\&\quad + \big [ X^*_i - E\{ X^*_i(\theta _0)\}\big ] \big \{m_ie^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0)) - \mu _{\nu }\big \}. \end{aligned}$$

Next, we show the consistency of $\hat{\theta }.$ Let $S(\theta )=(S_{1}(\theta )^\prime , S_{2}(\theta )^\prime )^\prime ,$ where $S_{1}(\theta )$ is the vector consisting of the first p components of $S(\theta )$. Write

$$\begin{aligned} S_{1}(\theta ) - S_{1}(\theta _0) = \{S_{1}(\alpha , \eta ) - S_{1}(\alpha _0, \eta )\} + \{S_{1}(\alpha _0, \eta ) - S_{1}(\alpha _0, \eta _0)\}. \end{aligned}$$

Note that

$$\begin{aligned} S_{1}(\alpha , \eta ) - S_{1}(\alpha _0, \eta ) =&\frac{1}{n} \sum ^{n}_{i=1} X_i m_i \Big [ e^{X_i^\prime (\alpha -\eta )} \widehat{\Lambda }_0^{-1} \{T^*_i(\alpha )\} - e^{X_i^\prime (\alpha _0-\eta )} \widehat{\Lambda }_0^{-1} \{T^*_i(\alpha )\Big ]\nonumber \\&\quad + \frac{1}{n} \sum ^{n}_{i=1} X_i m_i e^{X_i^\prime (\alpha _0-\eta )} \Big [ \widehat{\Lambda }_0^{-1} \{T^*_i(\alpha )- \widehat{\Lambda }_0^{-1} \{T^*_i(\alpha _0)\}\Big ]\nonumber \\&-\frac{1}{n} \sum ^{n}_{i=1} X_i \big [ \widehat{\mu }_\nu (\alpha , \eta ) - \widehat{\mu }_\nu (\alpha _0, \eta ) \big ]. \end{aligned}$$

(20)

For any positive sequence $\varepsilon _n \rightarrow 0,$ by the Taylor expansion and the uniform strong law of large numbers, we have that for $||\theta - \theta _0|| \le \varepsilon _n,$ the first term on the right-hand side of (20) is

$$\begin{aligned} E\big [ X_i X_i^\prime m_i e^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0))\big ](\alpha -\alpha _0)+ o_p(||\alpha -\alpha _0||). \end{aligned}$$

In view of (17), by following the similar argument as the above, the second term on the right-hand side of (20) is

$$\begin{aligned}&-E\Big [ X_i m_i e^{X_i^\prime (\alpha _0-\eta _0)} \big \{ \frac{\varphi (T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))}+\frac{\lambda _0(T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))^2} T^*_i(\alpha _0)X_i\big \}^\prime \Big ] (\alpha - \alpha _0)\\&\quad + o_p(n^{-1/2}+||\alpha -\alpha _0||). \end{aligned}$$

Similarly, the third term on the right-hand side of (20) equals

$$\begin{aligned}&-E(X_i) E\big [ X_i^\prime m_i e^{X_i^\prime (\alpha - \alpha _0)} \Lambda _0^{-1}(T^*_i(\alpha _0))\big ] (\alpha _0-\eta _0)\\&+ E(X_i) E\Big [ m_i e^{X_i^\prime (\alpha _0-\eta _0)} \big \{\frac{ \varphi (T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))}+\frac{\lambda _0(T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))^2}T^*_i(\alpha _0)X_i \big \}^\prime \Big ] (\alpha - \alpha _0)\\&+ o_p(n^{-1/2}+||\alpha -\alpha _0||). \end{aligned}$$

Hence for any positive sequence $\varepsilon _n \rightarrow 0$ and $||\theta - \theta _0|| \le \varepsilon _n$,

$$\begin{aligned} S_{1}(\alpha , \eta ) - S_{1}(\alpha _0, \eta )=D_{11} (\alpha - \alpha _0)+ o_p(n^{-1/2}+||\alpha - \alpha _0||), \end{aligned}$$

(21)

where

$$\begin{aligned} D_{11}=&E\big [ \{X_i-E(X_i)\} X_i^\prime m_i e^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0))\big ]\\&-E\Big [\{ X_i-E(X_i)\} m_i e^{X_i^\prime (\alpha _0-\eta _0)} \big \{ \frac{\varphi (T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))}+\frac{\lambda _0(T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))^2} T^*_i(\alpha _0)X_i\big \}^\prime \Big ]. \end{aligned}$$

Likewise, we have

$$\begin{aligned} S_{1}(\alpha _0, \eta ) - S_{1}(\alpha _0, \eta _0)= D_{12} (\eta - \eta _0)+ o_p(n^{-1/2}+||\eta - \eta _0||), \end{aligned}$$

(22)

where

$$\begin{aligned} D_{12}=&-E\big [\{X_i-E(X_i)\} X_i^\prime m_i e^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0))\big ]. \end{aligned}$$

Using (21) and (22), we have that for any sequence $\varepsilon _n \rightarrow 0$ and $||\theta - \theta _0|| \le \varepsilon _n$,

$$\begin{aligned} S_{1}(\theta ) - S_{1}(\theta _0)= (D_{11}, D_{12} )(\theta - \theta _0)+ o_p(n^{-1/2}+||\theta - \theta _0||). \end{aligned}$$

In a similar manner, we obtain that for any sequence $\varepsilon _n \rightarrow 0$ and $||\theta - \theta _0|| \le \varepsilon _n$,

$$\begin{aligned} S_{2}(\theta ) - S_{2}(\theta _0) = (D_{21}, D_{22}) (\theta - \theta _0) + o_p(n^{-1/2}+||\theta - \theta _0||), \end{aligned}$$

where

$$\begin{aligned} D_{21}=&-E\big [\{\Phi (X_i; \theta _0)-E(\Phi (X_i; \theta _0))\} X_i^\prime m_i e^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0))\big ]\\&-E\Big [\{\Phi (X_i; \theta _0)-E(\Phi (X_i; \theta _0))\} m_i e^{X_i^\prime (\alpha _0-\eta _0)}\\&\times \big \{ \frac{\varphi (T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))}+\frac{\lambda _0(T^*_i(\alpha _0))}{\Lambda _0(T^*_i(\alpha _0))^2} T^*_i(\alpha _0)X_i\big \}^\prime \Big ],\\ D_{22}=&-E\big [\{\Phi (X_i; \theta _0)-E(\Phi (X_i; \theta _0))\} X_i^\prime m_i e^{X_i^\prime (\alpha _0-\eta _0)} \Lambda _0^{-1}(T^*_i(\alpha _0))\big ]. \end{aligned}$$

Therefore,

$$\begin{aligned} S(\theta ) - S(\theta _0) = {D} (\theta - \theta _0) + o_p(1 + n^{1/2}||\theta - \theta _0||), \end{aligned}$$

(23)

where

$$\begin{aligned} {D} = \left( \begin{array}{cc} D_{11} &{} D_{12} \\ D_{21} &{} D_{22} \\ \end{array} \right) . \end{aligned}$$

Let $\mathcal {S}(\theta )$ be the limit of $n^{-1}S(\theta ).$ Note that $\mathcal {S}(\theta _0)=0$ and $\mathcal {S}(\theta )\ne 0$ for $\theta \ne \theta _0.$ Then following the argument used in Theorem 2 of Lin et al. (1998), we have that $\hat{\theta }$ is consistent. The above consistency proof is also suitable for the case when $\hat{\theta }$ is the zero-crossing of $S(\theta ).$

For the asymptotic normality of $\hat{\theta },$ it follows from (19) and (23) that

$$\begin{aligned} n^{1/2}(\hat{\theta } - \theta _0) = -{D}^{-1} n^{1/2}\sum ^n_{i=1} \Upsilon _i + o_p(1), \end{aligned}$$

(24)

which implies that $n^{1/2} (\hat{\theta } - \theta _0)$ is asymptotically normal with mean zero and covariance matrix ${D}^{-1}E\{\Upsilon _i \Upsilon _i^\prime \} ({D}^{-1})^\prime$.

To show the weak convergence of $\widehat{\Lambda }_0(t),$ it follows from (15), (17) and (24) that uniformly in $t \in [0, \tau ],$

$$\begin{aligned} \widehat{\Lambda }_0(t) - \Lambda _0(t) =&\{\widehat{\Lambda }_0(t; \hat{\alpha }) - \widehat{\Lambda }_0(t; \alpha _0) \} + \{ \widehat{\Lambda }_0(t; \alpha _0) - \Lambda _0(t) \}\nonumber \\ =&n^{-1}\sum ^n_{i=1} \Psi _i(t)+ o_p(n^{-1/2}), \end{aligned}$$

(25)

where $\Psi _i(t) =\Lambda _0(t) \big \{-\varphi (t)^\prime (I_p, 0_p) {D}^{-1}\Upsilon _i +\phi _i(t; \alpha _0) \big \},$ and $I_p$ and $0_p$ are the $p \times p$ identity matrix and zero matrix, respectively. Because $\Psi _i(t)$ $(i=1,...,n)$ are independent zero-mean random variables for each t, the multivariate central limit theorem implies that $n^{1/2} \{\widehat{\Lambda }_0(t) - \Lambda _0(t) \}$ converges in finite-dimensional distributions to a zero-mean Gaussian process. Since $\Psi _i(t)$ can be written as sums or products of monotone functions of t, and thus are manageable. Hence $\Psi _i(\cdot )$ is a tight process (van der Vaart and Wellner 1996). Therefore, $n^{1/2} \{\widehat{\Lambda }_0(t) - \Lambda _0(t) \}$ is tight and converges weakly to a zero-mean Gaussian process with covariance function $E\{\Psi _i(s)\Psi _i(t)\}$ at (s, t).

3. Asymptotic properties of $\hat{\xi }$ and $\widehat{H}_0(t)$

We first show the asymptotic normality of ${U}(\xi ).$ For $k=0$ and 1,  define

$$\begin{aligned} s^{(k)}(t) = E [ {W_i^*(t; \xi _0)}^k \nu _i e^{W_i^\prime (\gamma _0 - \beta _0)} I\{Y^*_i(\beta _0) \ge t\} ]. \end{aligned}$$

It follows from (15), (17) and (24) that uniformly in $t \in [0, \tau ],$ for $k=0$ and 1,

$$\begin{aligned}&\frac{1}{n} \sum ^n_{i=1}{W_i^*(t; \xi _0)}^k e^{W_i^\prime (\gamma _0 - \beta _0)} \widehat{\nu }_i I\{Y^*_i(\beta _0) \ge t\} - s^{(k)}(t) \nonumber \\&\quad = \frac{1}{n} \sum ^n_{i=1} {W_i^*(t; \xi _0)}^k e^{W_i^\prime (\gamma _0 - \beta _0)} \Big [\frac{I\{Y^*_i(\beta _0) \ge t\}m_i}{e^{ X_i^\prime (\hat{\alpha } - \hat{\eta })} \widehat{\Lambda }_0 \{T^*_i(\hat{\alpha }) \}} - \frac{I\{Y^*_i(\beta _0) \ge t\}m_i}{e^{ X_i^\prime ({\alpha _0} - {\eta _0})} \Lambda _0(T^*_i(\alpha _0))} \Big ] \nonumber \\&\qquad + \frac{1}{n} \sum ^n_{i=1} {W_i^*(t; \xi _0)}^k e^{W_i^\prime (\gamma _0 - \beta _0)} \frac{I\{Y^*_i(\beta _0) \ge t\}m_i}{e^{ X_i^\prime ({\alpha _0} - {\eta _0})} \Lambda _0(T^*_i(\alpha _0) )} - s^{(k)}(t; \xi _0) \nonumber \\&\quad = \frac{1}{n} \sum ^n_{i=1} \zeta _{ki}(t) + o_p(n^{-1/2}), \end{aligned}$$

(26)

where

$$\begin{aligned} \zeta _{ki}(t) =&\int \frac{w^*(t)^k e^{w^\prime (\gamma _0 - \beta _0)} I\{y* \ge t \} m}{e^{ x'(\alpha _0-\eta _0)}\Lambda _0(t^*)} \big \{x'(I_p, -I_p)D^{-1} \Upsilon _i\\&\quad + \varphi (t)^\prime (I_p, 0_p)D^{-1} \Upsilon _i -\phi _i(t; \alpha _0)\big \} dV_2(x, w, m, t) \\&+ \frac{1}{n} \sum ^n_{i=1} {W_i^*(t; \xi )}^k e^{W_i^\prime (\gamma _0 - \beta _0)} \frac{I\{Y^*_i(\beta _0) \ge t\}m_i}{e^{ X_i^\prime ({\alpha _0} - {\eta _0})} \Lambda _0(T^*_i(\alpha _0) )} - s^{(k)}(t), \end{aligned}$$

and $V_2(x, w, m, t)$ denotes the joint probability measure of $(X_i, W_i, m_i, T_i)$ with $w^*(t)=(w^\prime , G(t, w; \xi _0)^\prime )^\prime ,$ $t^*=te^{x'\alpha _0}$ and $y^*=te^{w'\beta _0}.$ Let $\mathcal {U}(\xi )$ be the limit of $n^{-1} {U}(\xi ).$ Note that $\mathcal {U}(\xi _0)=0.$ Then the functional delta method yields

$$\begin{aligned} {U}(\xi _0) =&\frac{1}{n} \sum ^n_{i=1} \int _0^{\tau } W_i^*(t; \xi _0)d[ \Delta _i I\{Y^*_i(\beta _0) \le t] \} \nonumber \\&\quad - E\Big \{\int _0^{\tau } W_i^*(t; \xi _0)d[ \Delta _i I\{Y^*_i(\beta _0) \le t] \Big \} \nonumber \\&\quad - \int _0^\tau \frac{\sum ^n_{j=1} W_j^*(t; \xi _0) \widehat{\nu }_j e^{W_j^\prime (\gamma _0 - \beta _0)} I\{Y^*_j(\beta _0) \ge t \}}{\sum ^n_{j=1} \widehat{\nu }_j e^{W_j^*(t; \xi _0)^\prime (\gamma _0 - \beta _0)} I\{Y^*_j(\beta _0) \ge t \}}\nonumber \\&\quad \times d\left[ \frac{1}{n}\sum ^n_{i=1}\Delta _i I\{Y^*_i(\beta _0) \le u \}\right] \nonumber \\&\quad + \int _0^\tau \bar{w}(t) dE[\Delta _i I\{Y_i^*(\beta _0) \le t \}] \nonumber \\ =&\frac{1}{n} \sum ^n_{i=1} \Gamma _i + o_p(1), \end{aligned}$$

(27)

which implies that $n^{1/2}{U}(\xi _0)$ is asymptotically normal, where

$$\begin{aligned} \Gamma _i =&\int _0^{\tau } W_i^*(t; \xi _0)d[ \Delta _i I\{Y^*_i(\beta _0) \le t] \} - E\Big \{\int _0^{\tau } W_i^*(t; \xi _0)d[ \Delta _i I\{Y^*_i(\beta _0) \le t] \Big \} \\&\quad + \int _0^\tau \frac{\zeta _{0i}(t)\bar{w}(t) }{{s}^{(0)}(t)} dE[\Delta _i I\{Y_i^*(\beta _0) \le t \}] - \int _0^\tau \frac{\zeta _{1i}(t) }{{s}^{(0)}(t)} dE[\Delta _i I\{Y_i^*(\beta _0) \le t \}] \\&\quad - \int _0^\tau \bar{w}(t) d\big (\Delta _i I\{Y^*_i(\beta _0) \le t \} - E[\Delta _i I\{Y_i^*(\beta _0) \le t \}]\big ), \end{aligned}$$

and $\bar{w}(t)={s}^{(1)}(t)/{s}^{(0)}(t).$

Next, we show the consistency of $\hat{\xi }.$ Let ${U}(\xi )=( {U}_1(\xi )^\prime , {U}_2(\xi )^\prime )^\prime ,$ where ${U}_1(\xi )$ is the vector consisting of the first p components of ${U}(\xi )$. Write

$$\begin{aligned} {U}_{1}(\xi ) - {U}_{1}(\xi _0)&= \{{U}_{1}(\beta , \gamma ) - {U}_{1}(\beta , \gamma _0)\} + \{{U}_{1}(\beta , \gamma _0) - {U}_{1}(\beta _0, \gamma _0)\}. \end{aligned}$$

By similar arguments as in Chen and Jewell (2001) and Sun and Su (2008), we obtain that for any sequence $\varepsilon _n \rightarrow 0$ and $||\xi - \xi _0|| \le \varepsilon _n$,

$$\begin{aligned}&{U}_{1}(\beta , \gamma _0) - {U}_{1}(\beta _0, \gamma _0)= \tilde{D}_{11}(\beta -\beta _0)+o_p(n^{-1/2}+\Vert \beta -\beta _0\Vert ),\\&{U}_{1}(\beta , \gamma ) - {U}_{1}(\beta , \gamma _0)= \tilde{D}_{12}(\gamma -\gamma _0)+o_p(\Vert \gamma -\gamma _0\Vert ), \end{aligned}$$

where

$$\begin{aligned} \tilde{D}_{11}&=-\int _0^\tau E \big [ \{W_i -\bar{w}(t; \xi _0)\}^{\otimes 2} \nu _i e^{W_i^\prime (\gamma _0 -\beta _0)} I\{Y_i^*(\beta _0) \ge t\}\big ] d(h_0(t) t),\\ \tilde{D}_{12}&= -\int ^\tau _0 \big [s^{(2)}(t) - \bar{w}(t){s}^{(1)}(t)^\prime \big ] d H_0(t), \end{aligned}$$

and $s^{(2)}(t) = E[\nu _i e^{W_i^\prime (\gamma _0 - \beta _0)} I\{Y^*_i(\beta _0) \ge t\}W_i^{\otimes 2}]$ with $a^{\otimes 2}=a a^\prime$ for any vector a. Thus, we have that for any sequence $\varepsilon _n \rightarrow 0$ and $||\xi - \xi _0|| \le \varepsilon _n$,

$$\begin{aligned} {U}_{1}(\xi ) - {U}_{1}(\xi _0)=(\tilde{D}_{11}, \tilde{D}_{12}) (\xi -\xi _0)+o_p(n^{-1/2}+\Vert \xi -\xi _0\Vert ). \end{aligned}$$

Let ${\tilde{s}}^{(1)}(t) =E[ \nu _i G(t, W_i; \xi ) e^{W_i^\prime (\gamma _0 - \beta _0)} I\{Y^*_i(\beta _0) \ge t\}],$ $\tilde{s}^{(2)}(t) =E[ \nu _i e^{W_i^\prime (\gamma _0 - \beta _0)} I\{Y^*_i(\beta _0) \ge t\} G(t, W_i; \xi ) W_i^\prime ],$ and $\tilde{w}(t)={\tilde{s}}^{(1)}(t)/s^{(0)}(t).$ In a similar manner, we get that for any sequence $\varepsilon _n \rightarrow 0$ and $||\xi - \xi _0|| \le \varepsilon _n$,

$$\begin{aligned} {U}_{2}(\xi ) -{U}_{2}(\xi _0)=(\tilde{D}_{21}, \tilde{D}_{22}) (\xi -\xi _0)+o_p(n^{-1/2}+\Vert \xi -\xi _0\Vert ), \end{aligned}$$

where

$$\begin{aligned} \tilde{D}_{21}&= -\int _0^\tau E \big [ \big \{G(t, W_i; \xi _0) -\tilde{w}(t) \big \}\{W_i-\bar{w}(t)\}^\prime \nu _i e^{W_i^\prime (\gamma _0 -\beta _0)} I\{Y_i^*(\beta _0) \ge t\}\big ]\nonumber \\&\quad \times d(h_0(t) t),\\ \tilde{D}_{22}&= -\int ^\tau _0 \big [\tilde{s}^{(2)}(t) - \tilde{w}(t) s^{(1)}(t)^\prime \big ] d H_0(t). \end{aligned}$$

Thus,

$$\begin{aligned} {U}(\xi ) - {U}(\xi _0) = \tilde{D} (\xi - \xi _0) + o_p(1 + n^{1/2}||\xi - \xi _0||), \end{aligned}$$

(28)

where

$$\begin{aligned} \tilde{D} = \left( \begin{array}{cc} \tilde{D}_{11} &{} \tilde{D}_{12} \\ \tilde{D}_{21} &{} \tilde{D}_{22} \\ \end{array} \right) . \end{aligned}$$

Using the consistency of $\hat{\theta }$ and the asymptotic linearity of $\widehat{\Lambda }_0(t; \alpha ),$ we have that $\mathcal {U}(\xi _0)=0,$ and $\mathcal {U}(\xi )\ne 0$ for $\xi \ne \xi _0.$ Thus, $\hat{\xi }$ is consistent (Lin et al. 1998). The above consistency proof is also suitable for the case when $\hat{\xi }$ is the zero-crossing of $U(\xi ).$

For the asymptotic normality of $\hat{\xi }$, it then follows from (27) and (28) that

$$\begin{aligned} n^{1/2}( \hat{\xi } - \xi _0) = -\tilde{D}^{-1} n^{1/2}\sum ^n_{i=1} \Gamma _i + o_p(1). \end{aligned}$$

(29)

This implies that $n^{1/2} (\hat{\xi } - \xi _0)$ is asymptotically normal with mean zero and covariance matrix $\tilde{D}^{-1}E\{\Gamma _i \Gamma _i^\prime \} (\tilde{D}^{-1})^\prime$.

To show the weak convergence of $\widehat{H}_0(t),$ write

$$\begin{aligned} \widehat{H}_0(t) - H_0(t)) =&\{\widehat{H}_0(t; \hat{\beta }, \hat{\gamma }) - \widehat{H}_0(t; \hat{\beta }, \gamma _0)\} + \{\widehat{H}_0(t; \hat{\beta }, \gamma _ 0) - \widehat{H}_0(t; {\beta }_0, \gamma _ 0)\}\\&+ \{\widehat{H}_0(t; {\beta }_0, \gamma _ 0) - H_0(t)\}. \end{aligned}$$

By the Taylor expansion and the uniform strong law of large numbers, we have that uniformly in $t \in [0, \tau ],$

$$\begin{aligned} \widehat{H}_0(t; \hat{\beta }, \hat{\gamma }) - \widehat{H}_0(t; \hat{\beta }, \gamma _0)&= - \int _0^t \frac{\bar{w}(u)^\prime }{s^{(0)}(u)} {dE[\Delta I\{Y^*(\beta _0) \le u \}]} (\hat{\gamma } - \gamma _0) + o_p(n^{-1/2}). \end{aligned}$$

By following a similar argument as in the proof of (17), it is seen that uniformly in $t \in [0, \tau ],$

$$\begin{aligned} \widehat{H}_0(t; \hat{\beta }, \gamma _ 0) -\widehat{H}_0(t; {\beta }_0, \gamma _ 0)&= - \int _0^t \bar{w}(u)^\prime d\{h_0(u)u\}({\hat{\beta }} - \beta _0) +o_p(n^{-1/2}). \end{aligned}$$

Note that

$$\begin{aligned}&\widehat{H}_0(t; \xi _ 0) - H_0(t)\\&= \sum ^n_{i=1}\int _0^t \frac{ d M_i^*(u; \xi _0)}{ \sum ^n_{i=1} \nu _i e^{W_i^\prime (\gamma _0 - \beta _0)}I\{Y^*_i(\beta _0) \le u \} }\\&\quad + \int _0^t \Big [ \frac{ n s^{(0)}(u) d H_0(u)}{\sum ^n_{i=1} {\hat{\nu }}_i e^{W_i^\prime (\gamma _0 - \beta _0)}I\{Y^*_i(\beta _0) \le u \} } - \frac{n s^{(0)}(u) d H_0(u)}{ \sum ^n_{i=1} \nu _i e^{W_i^\prime (\gamma _0 - \beta _0)}I\{Y^*_i(\beta _0) \le u \} } \Big ] \\&\quad + \int _0^t \Big [ \frac{1 }{\sum ^n_{i=1} {\hat{\nu }}_i e^{W_i^\prime (\gamma _0 - \beta _0)}I\{Y^*_i(\beta _0) \le u \} } - \frac{1 }{\sum ^n_{i=1} \nu _i e^{W_i^\prime (\gamma _0 - \beta _0)}I\{Y^*_i(\beta _0) \le u \} } \Big ] \\&\quad \times \Big \{\frac{1}{n} \sum ^n_{i=1} d[\Delta _i I\{Y^*_i(\beta _0) \le u \}] - {s}^{(0)}(u, \xi _0)dH_0(u) \Big \}. \end{aligned}$$

Then by the argument used in the proofs of (26) and (27), we obtain that uniformly in $t \in [0, \tau ],$

$$\begin{aligned} \widehat{H}_0(t; \xi _ 0) - H_0(t) =&\frac{1}{n}\sum ^n_{i=1} \int _0^t \frac{ d M_i^*(u; \xi _0)}{ s^{(0)}(u) } - \frac{1}{n} \sum ^n_{i=1} \int _0^t \frac{\pi _{i}(u)}{s^{(0)}(u)} d H_0(u) + o_p(1), \end{aligned}$$

where

$$\begin{aligned} \pi _{i}(u)=&\int \frac{ e^{w^\prime (\gamma _0 - \beta _0)} I\{y \ge t \} m}{e^{ x'(\alpha _0-\eta _0)}\Lambda _0(t^*)} \big \{(I_p, -I_p)D^{-1} \Upsilon _i +\varphi (t)^\prime (I_p, 0_p)D^{-1} \Upsilon _i \\&-\phi _i(t; \alpha _0)\big \} dV_2(x, w, m, t) + \frac{1}{n} \sum ^n_{i=1} e^{W_i^\prime (\gamma _0 - \beta _0)} \frac{I\{Y^*_i(\beta _0) \ge t\}m_i}{e^{ X_i^\prime ({\alpha _0} - {\eta _0})} \Lambda _0(T^*_i(\alpha _0) )} \\&- \frac{1}{n} \sum ^n_{i=1} \nu _i e^{W_i^\prime (\gamma _0 - \beta _0)} I\{Y^*_i(\beta _0) \ge t\}. \end{aligned}$$

Thus, it follows from (29) that uniformly in $t \in [0, \tau ],$

$$\begin{aligned} \widehat{H}_0(t) - H_0(t) = \frac{1}{n} \sum ^n_{i=1} \psi _i(t) +o_p(n^{-1/2}), \end{aligned}$$

(30)

where

$$\begin{aligned} \psi _i(t)&= \int _0^t \frac{ d M_i^*(u; \xi _0)}{ s^{(0)}(u) }+ \int _0^t \frac{\bar{w}(u)^\prime }{s^{(0)}(u)} {dE[\Delta I\{Y^*(\beta _0) \le u \}]} (0_p, I_p) \tilde{D}^{-1} \Gamma _i\\&\quad + \int _0^t \bar{w}(u)^\prime d\{h_0(u)u\}(I_p, 0_p) \tilde{D}^{-1} \Gamma _i - \int _0^t \frac{\pi _{i}(u)}{s^{(0)}(u)} d H_0(u). \end{aligned}$$

Hence, $n^{1/2}\{\widehat{H}_0(t) - H_0(t)\}$ converges weakly to a zero-mean Gaussian process with covariance function $E\{\psi _i(s)\psi _i(t)\}$ at (s, t). Furthermore, it follows from (24) and (29) that $n^{1/2}(\hat{\theta } - \theta _0)$ and $n^{1/2}( \hat{\xi } - \xi _0)$ have asymptotically a joint normal distribution with mean zero and covariance matrix $E\{(\Upsilon _i^\prime ({D}^{-1})^\prime , \Gamma _i^\prime (\tilde{D}^{-1})^\prime )^\prime (\Upsilon _i^\prime ({D}^{-1})^\prime , \Gamma _i^\prime (\tilde{D}^{-1})^\prime ) \}.$ Also $n^{1/2}\{\widehat{\Lambda }_0(t) - \Lambda _0(t)\}$ and $n^{1/2}\{\widehat{H}_0(t) - H_0(t)\}$ jointly converge weakly to a zero-mean bivariate Gaussian process with covariance function $E(\Psi _i(s), \psi _i(s))^\prime (\Psi _i(t), \psi _i(t))\}$ at (s, t).