Estimating the hazard functions of two alternating recurrent events in the presence of covariates

Abstract

The motivation for this paper is a cystic fibrosis data which records a patient’s times to relapse and times to cure under several recurrences of the disease. The idea is to study the impact of covariates on the hazard rates of two alternately occurring events. The dependence between the times to the two events over the different cycles is modeled through an autoregressive-type setup. The partial likelihood function is then derived and the estimators obtained. The estimators are shown to be consistent and asymptotically normal. The technique is applied to study the motivating data. A simulation study is also conducted to corroborate the results.

Appendix

Here we show that the maximum likelihood estimators are consistent and asymptotically normal (as before we restrict ourselves to $(\varvec{\theta },\varvec{\mu })$ and the results for $(\varvec{\eta },\varvec{\gamma })$ follow similarly).

Let $N_{ji}(t) = I \{X_{ji} \le t , \delta _{1ji}=1 , \delta ^*_{1ji}=1\}$, where $N_{ji}(1)$ is a.s. finite.

Observe that, $\lambda _{Eji}(t)=Q_{ji}(t)\lambda _{E0j}(t)r_{Eji}(t)$, so that the log-likelihood evaluated at time t is

$$\begin{aligned} C(\varvec{\theta ,\mu } ; t)= & {} \sum _{j=1}^{m}\sum _{i=1}^{n} \int _{0}^t \left[ \varvec{\theta _j'z_{ji}}(u) + \varvec{\mu }' \varvec{y}_{ji}(u) \right] ~\text {d}N_{ji}(u)\\&- \,\sum _{j=1}^{m} \sum _{i=1}^{n} \int _{0}^t \log ~\left[ ~\sum _{l=1}^{n} Q_{jl}(u) e^{\{\varvec{\theta _j'Z_{jl}}(u) + \varvec{\mu }' \varvec{y}_{jl}(u)\}}\right] ~ \text {d}{N_{ji}}(u). \end{aligned}$$

Let $(\varvec{\theta _0},\varvec{\mu _0})$ be the true value of $(\varvec{\theta },\varvec{\mu })$. Then define

$$\begin{aligned} C^*(\varvec{\theta },\varvec{\mu },t)= & {} C(\varvec{\theta },\varvec{\mu },t)-C(\varvec{\theta }_0,\varvec{\mu }_0,t)\\= & {} \sum _{j=1}^{m}\sum _{i=1}^{n} \int _{0}^t [ (\varvec{\theta _j}-\varvec{\theta _{0j}})'\varvec{z}_{ji}(u) + (\varvec{\mu }-\varvec{\mu }_0)' \varvec{y}_{ji}(u) ]~\text {d}N_{ji}(u) \\&-\,\sum _{j=1}^{m} \sum _{i=1}^{n} \int _{0}^t \log ~\left[ ~\frac{\sum _{l=1}^{n} Q_{jl}(u) e^{\{\varvec{\theta }_j'\varvec{Z}_{jl}(u) + \varvec{\mu }' \varvec{y}_{jl}(u)\}}}{\sum _{l=1}^{n} Q_{jl}(u) e^{\{\varvec{\theta }_{0j}'\varvec{Z}_{jl}(u) + \varvec{\mu }_0' \varvec{y}_{jl}(u)\}}}\right] ~ \text {d}{N_{ji}}(u). \end{aligned}$$

Let $S^{(0)}_j ({\varvec{\theta }}_j,\varvec{\mu },t)=n_m^{-1}\sum _{l=1}^{n} Q_{jl}(t) r_{Ejl}$,

Then the first and second order derivatives of $S^{(0)}_j ({\varvec{\theta }}_j,\varvec{\mu },t)$ are

$$\begin{aligned}&S^{(1)}_{j{\varvec{\theta }_j}} ({\varvec{\theta }}_j,\varvec{\mu },t)=n_m^{-1}\sum _{l=1}^{n} Q_{jl}(t) r_{Ejl}{\varvec{z}}_{jl}\\&S^{(1)}_{j\varvec{\mu }} ({\varvec{\theta }}_j,\varvec{\mu },t)= n_m^{-1}\sum _{l=1}^{n} Q_{jl}(t) r_{Ejl}~{\varvec{y}}_{jl},\\&S^{(2)}_{j{\varvec{\theta }_j}} ({\varvec{\theta }}_j,\varvec{\mu },t)=n_m^{-1}\sum _{l=1}^{n}Q_{jl}(t) r_{Ejl}{\varvec{z}}_{jl}{\varvec{z}}_{jl}',\\&S^{(2)}_{j\varvec{\mu }} ({\varvec{\theta }}_j,\varvec{\mu },t)=n_m^{-1}\sum _{l=1}^{n}Q_{jl}(t) r_{Ejl}{\varvec{y}}_{jl}{\varvec{y}}_{jl}',\\&S^{(2)}_{j{\varvec{\theta }_j\mu }} ({\varvec{\theta }}_j,\varvec{\mu },t)=n_m^{-1}\sum _{l=1}^{n}Q_{jl}(t) r_{Ejl}{\varvec{z}}_{jl}{\varvec{y}}_{jl}'. \end{aligned}$$

Also let $E_{j{\varvec{\theta }_j}}({\varvec{\theta }_j},\varvec{\mu },t)=\frac{S^{(1)}_{j{\varvec{\theta }_j}}}{S^{(0)}_j}, \quad \quad E_{j\varvec{\mu }}({\varvec{\theta }_j},\varvec{\mu },t)=\frac{S^{(1)}_{j\varvec{\mu }}}{S^{(0)}_{j}}$,

$$\begin{aligned} V_{j{\varvec{\theta }_j}}({\varvec{\theta }_j},\varvec{\mu },t)=\frac{S^{(2)}_{j{\varvec{\theta }_j}}}{S^{(0)}_j}-E_{j{\varvec{\theta }_j}} E'_{j{\varvec{\theta }_j}}, \quad V_{j\varvec{\mu }}({\varvec{\theta }_j},\varvec{\mu },t)=\frac{S^{(2)}_{j\varvec{\mu }}}{S^{(0)}_{j}}-E_{j\varvec{\mu }} E'_{j\varvec{\mu }}, \end{aligned}$$

and $\quad V_{j{\varvec{\theta }_j}\mu }({\varvec{\theta }_j},\varvec{\mu },t)=\frac{S^{(2)}_{j{\varvec{\theta }_j}\mu }}{S^{(0)}_j}-E_{j{\varvec{\theta }_j}} E'_{j{\varvec{\mu }}}$.

Next suppose the following conditions hold :

• A1. $\int _{0}^{1} \lambda _{E0j}(t)\text {d}t <\infty$.

• A2. There exists a neighborhood $\Theta _j$ of $\theta _j$ and $s^{(0)}_j$, $s^{(k)}_{j{\varvec{\theta }_j}}$ for $k=1,2$, defined on $\Theta _j\times [0,1]$, such that $sup_{t\epsilon [0,1],\theta _j\epsilon \Theta _j} \parallel S^{(0)}_j(t) - s^{(0)}_j(t) \parallel {\mathop {\rightarrow }\limits ^{p}} 0,$ and $sup_{t\epsilon [0,1],\theta _j\epsilon \Theta _j} \parallel S^{(k)}_{j{\varvec{\theta }_j}}(t) - s^{(k)}_{j{\varvec{\theta }_j}}(t) \parallel {\mathop {\rightarrow }\limits ^{p}} 0,$. For all $t\epsilon [0,1],\theta _j\epsilon \Theta _j$, with $s^{(1)}_{j{\varvec{\theta }_j}}=\frac{\delta }{\delta \theta _j}s^{(0)}_j$ and $s^{(2)}_{j{\varvec{\theta }_j}}=\frac{\delta ^2}{\delta ^2\theta _j}s^{(0)}_j$, $s^{(0)}_j(t), s^{(1)}_{j{\varvec{\theta }_j}}(t)$ and $s^{(2 )}_{j{\varvec{\theta }_j}}(t)$ are continuous functions of $\theta _j\epsilon \Theta _j$, uniformly in $t\epsilon [0,1], s^{(0)}_j, s^{(1)}_{j{\varvec{\theta }_j}}$ and $s^{(2)}_{j{\varvec{\theta }_j}}$ are bounded on $\Theta _j\times [0,1]$, $s^{(0)}_j$ is bounded away from zero on $\Theta _j\times [0,1]$ and the matrix $\Sigma _{j\theta _j}$ is positive definite, where $\Sigma _{j\theta _j}=\int _{0}^{1} v_{j\theta _j}(\theta _{0j},\mu _0,t)~s_j^{(0)}(\theta _{0j},\mu _0,t)~\lambda _{E0j}(t)\text {d}t$.

• A3. There exists a neighborhood $\Theta _{\varvec{\mu }}$ of ${\varvec{\mu }}$ and $s^{(k)}_{j\varvec{\mu }}$ defined on $\Theta _{\varvec{\mu }} \times [0,1]$, such that $sup_{t\epsilon [0,1],{\varvec{\mu }}\epsilon \Theta _{\varvec{\mu }}} \parallel S^{(k)}_{j\varvec{\mu }}(t) - s^{(k)}_{j\varvec{\mu }}(t) \parallel {\mathop {\rightarrow }\limits ^{p}} 0$, for all $t\epsilon [0,1],{\varvec{\mu }}\epsilon \Theta _{\varvec{\mu }}$, with $s^{(1)}_{j\varvec{\mu }}=\frac{\delta }{\delta {\varvec{\mu }}}s^{(0)}_{j\varvec{\mu }}$ and $s^{(2)}_{j\varvec{\mu }}=\frac{\delta ^2}{\delta ^2{\varvec{\mu }}}s^{(0)}_{j\varvec{\mu }}$, $s^{(0)}_{j\varvec{\mu }}(t), s^{(1)}_{j\varvec{\mu }}(t)$ and $s^{(2)}_{j\varvec{\mu }}(t)$ are continuous functions of ${\varvec{\mu }}\epsilon \Theta _{\varvec{\mu }}$, uniformly in $t\epsilon [0,1], s^{(0)}_{j\varvec{\mu }}, s^{(1)}_{j\varvec{\mu }}$ and $s^{(2)}_{j\varvec{\mu }}$ are bounded on $\Theta _{\varvec{\mu }}\times [0,1]$, $s^{(0)}_{\varvec{\mu }}$ is bounded away from zero on $\Theta _{\varvec{\mu }}\times [0,1]$ and the matrix $\Sigma _{\varvec{\mu }}=\int _{0}^{1} \sum _{j=1}^{m}v_{j\varvec{\mu }}(\theta _{0j},\mu _0,t)~s_{j}^{(0)}(\theta _{0j},\mu _0,t)~\lambda _{E0j}(t)\text {d}t$ is positive definite.

• A4. For $s^{(2)}_{j{\varvec{\theta }_j\mu }}$ , defined on $\Theta _j\times [0,1]$ and $\Theta _\mu \times [0,1]$, such that $sup_{t\epsilon [0,1],\theta _j\epsilon \mu } \parallel S^{(2)}_{j{\varvec{\theta }_j\mu }}(t) - s^{(2)}_{j{\varvec{\theta }_j\mu }}(t) \parallel {\mathop {\rightarrow }\limits ^{p}} 0,$ for all $t\epsilon [0,1],\theta _j\epsilon \Theta _j,\theta _\mu \epsilon \Theta _\mu$, with $s^{(2)}_{j{\varvec{\theta }_j\mu }}=\frac{\delta ^2}{\delta ^2\theta _j\mu }s^{(0)}_j$, also, $s^{(2 )}_{j{\varvec{\theta }_j\mu }}(t)$ is a continuous function of $\theta _j\epsilon \Theta _j$ and $\theta _\mu \epsilon \Theta _\mu$, uniformly in $t\epsilon [0,1]$ and $s^{(2)}_{j{\varvec{\theta }_j\mu }}$ are bounded on $\Theta _j\times [0,1]$ and $\Theta _\mu \times [0,1]$ and the matrix $\Sigma _{j\theta _j\varvec{\mu }}=\int _{0}^{1} v_{j\theta _j\varvec{\mu }}(\theta _{0j},\mu _0,t)~s_{j}^{(0)}(\theta _{0j},\mu _0,t)~\lambda _{E0j}(t)\text {d}t$ exists.

• A5. There exists $\epsilon >0$, such that,

  $$\begin{aligned} n_m^{-\frac{1}{2}}\hbox {sup}_{i,j,t}|(\varvec{z}_{ji}(t),\varvec{y}_{ji}(t))'|Q_{ji}(t)I\{\varvec{\theta }'_{0j}\varvec{z_{ji}}+\varvec{\mu '}\varvec{y}_{ji}>-\epsilon |(\varvec{z}'_{ji}(t),\varvec{y}'_{ji}(t))'|\}{\mathop {\rightarrow }\limits ^{p}} 0 . \end{aligned}$$

For $\lambda _{E00}(u) = \sum _{j=1}^m \lambda _{E0j}(u)$, define

$$\begin{aligned} A(\varvec{\theta },\varvec{\mu },t)= & {} \sum _{j=1}^{m}\sum _{i=1}^{n} \int _{0}^t [ (\varvec{\theta _j}-\varvec{\theta _{0j}})'\varvec{z}_{ji}(u) + (\varvec{\mu }-\varvec{\mu }_0)' \varvec{y}_{ji}(u) ]~\lambda _{Eji}(u)\text {d}u \\&- \,\sum _{j=1}^{m}\sum _{i=1}^{n} \int _{0}^{t}{\log } \frac{S_j^{(0)}(\varvec{\theta }_j,\varvec{\mu },u)}{S_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu _0},u)}\lambda _{Eji}(u)\text {d}u\\= & {} \int _{0}^{t}\left[ \sum _{j=1}^{m} (\varvec{\theta _j}-\varvec{\theta _{0j}})'S_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{0j},\varvec{\mu }_0,t)\lambda _{E0j}(u) \right. \\&\left. + \,(\varvec{\mu }-\varvec{\mu }_0)' S_{\varvec{\mu }}^{(1)}(\varvec{\theta }_0,\varvec{\mu }_0,t)\lambda _{E00}(u) \right. \\&\left. - \,\sum _{j=1}^{m} {\log } \frac{S_j^{(0)}(\varvec{\theta }_j,\varvec{\mu },u)}{S_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu _0},u)}S_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\lambda _{E0j}(u)\right] \text {d}u \end{aligned}$$

Next defining $M_{ji}(t) = N_{ji}(t) - \int _{0}^t Q_{ji}(u)\lambda _{Eji}(u)\text {d}u$ and $\bar{M}_j=\sum _{i=1}^{n}M_{ji}$, it is easy to see that $M_{ji}(t)$ are local martingales on the time interval [0, 1].

Thus for every $(\varvec{\theta },\varvec{\mu },t)$, $C^*(\varvec{\theta ,\mu } ; t) - A(\varvec{\theta },\varvec{\mu },t)$ is a local square integrable martingale with

$$\begin{aligned} B(\varvec{\theta },\varvec{\mu },t)= & {} <C^*(\varvec{\theta ,\mu } ; t) - A(\varvec{\theta },\varvec{\mu },t), C^*(\varvec{\theta ,\mu } ; t) - A(\varvec{\theta },\varvec{\mu },t)>\\= & {} \int _{0}^{t}\left[ \sum _{j=1}^{m}\sum _{i=1}^{n} (\varvec{\theta _j}-\varvec{\theta _{0j}})'\varvec{z}_{ji}(u) + (\varvec{\mu }-\varvec{\mu }_0)' \varvec{y}_{ji}(u) \right. \\&\left. - \,\sum _{j=1}^{m}\sum _{i=1}^{n} {\log } \frac{S_j^{(0)}(\varvec{\theta }_j,\varvec{\mu },u)}{S_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}S_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\right] ^2\lambda _{Eji}(u)\text {d}u. \end{aligned}$$

Under the conditions $A1-A3$,

$$\begin{aligned}&A(\varvec{\theta },\varvec{\mu },1) {\mathop {\rightarrow }\limits ^{p}} \int _{0}^{1}\left[ \sum _{j=1}^{m} (\varvec{\theta _j}-\varvec{\theta _{0j}})'s_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{0j},\varvec{\mu }_0,t)\lambda _{E0j}(u) \right. \nonumber \\&\qquad \left. + \,(\varvec{\mu }-\varvec{\mu }_0)' s_{\varvec{\mu }}^{(1)}(\varvec{\theta }_0,\varvec{\mu }_0,t)\lambda _{E00}(u)\right. \nonumber \\&\qquad \left. -\,\sum _{j=1}^{m} {\log } \frac{s_j^{(0)}(\varvec{\theta }_j,\varvec{\mu },u)}{s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu _0},u)}s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\lambda _{E0j}(u)\right] \text {d}u \end{aligned}$$

(6.1)

and $n_mB(\varvec{\theta },\varvec{\mu },t)$ converges in probability to some finite quantity. Hence $C^*(\varvec{\theta ,\mu } ; 1)$ converges in probability to the same limit as $A(\varvec{\theta },\varvec{\mu },1)$ (Lenglart 1977).

Now by the boundedness conditions in $A2-A3$, the first and second derivatives of the limiting function on the right-hand side of (5.1) can be obtained by taking partial derivatives under the integral sign. For $e_j(\varvec{\theta }_{j},\varvec{\mu },u)=\frac{s_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)}{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}$, the first derivative with respect to $\varvec{\theta }_j$ is

$$\begin{aligned}&\int _{0}^{1}\{s_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u) - \frac{s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}s_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)\}\lambda _{E0j}(u)\text {d}u \nonumber \\&\qquad \quad =\int _{0}^{1}\{e_j(\varvec{\theta }_{0j},\varvec{\mu }_0,u)~-~e_j(\varvec{\theta }_{j},\varvec{\mu },u)\}s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\lambda _{E0j}(u)\text {d}u \end{aligned}$$

(6.2)

while for $e_{j\varvec{\mu }}(\varvec{\theta }_{j},\varvec{\mu },u)=\frac{s_{j\varvec{\mu }}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)}{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}$, the first derivative with respect to $\varvec{\mu }$ is

$$\begin{aligned}&\int _{0}^{1}\sum _{j=1}^{m}\{s_{j\varvec{\mu }}^{(1)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u) - \frac{s_{j}^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}{s_{j}^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}s_{j\varvec{\mu }}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)\}\lambda _{E0j}(u)\text {d}u \nonumber \\&\qquad \quad =\int _{0}^{1}\sum _{j=1}^{m}\{e_{j\varvec{\mu }}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)~-~e_{j\varvec{\mu }}(\varvec{\theta }_{j},\varvec{\mu },u)\}s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\lambda _{E0j}(u)\text {d}u.\nonumber \\ \end{aligned}$$

(6.3)

Both (6.2) and (6.3) are zero at $(\varvec{\theta },\varvec{\mu })=(\varvec{\theta _0},\varvec{\mu _0})$ so that the first derivative of the limiting function is zero at $(\varvec{\theta _0},\varvec{\mu _0})$.

Next for $v_j(\varvec{\theta }_{j},\varvec{\mu },u)=\frac{s^{(2)}_{j{\varvec{\theta }_j}}}{s^{(0)}_j}-e_j e'_j$, taking second order derivative with respect to $\varvec{\theta }_j$ leads to

$$\begin{aligned}&\int _{0}^{1}\left\{ -s_{j{\varvec{\theta }_j}}^{(2)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\frac{s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}\right. \nonumber \\&\left. +\, \frac{s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}{{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}^2}s_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)\left( s_{j{\varvec{\theta }_j}}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)\right) ' \right\} \lambda _{E0j}(u)\text {d}u \nonumber \\&\qquad =-\,v_j(\varvec{\theta }_{j},\varvec{\mu },u)s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\lambda _{E0j}(u)du \quad \quad \quad \quad \end{aligned}$$

(6.4)

and with respect to $\varvec{\mu }$, for $v_{j\varvec{\mu }}(\varvec{\theta }_{j},\varvec{\mu },u)=\frac{s^{(2)}_{j\varvec{\mu }}}{s^{(0)}_j}-e_{j\varvec{\mu }} e'_{j\varvec{\mu }}$ gives

$$\begin{aligned}&\int _{0}^{1}\sum _{j=1}^{m}\left\{ -s_{j\varvec{\mu }}^{(2)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\frac{s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)} \right. \nonumber \\&\left. +\, \frac{s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)}{{s_j^{(0)}(\varvec{\theta }_{j},\varvec{\mu },u)}^2}s_{j\varvec{\mu }}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)\left( s_{j\varvec{\mu }}^{(1)}(\varvec{\theta }_{j},\varvec{\mu },u)\right) ' \right\} \lambda _{E0j}(u)\text {d}u \nonumber \\&=-\,\int _{0}^{1} \sum _{j=1}^{m} v_{j\varvec{\mu }}(\varvec{\theta }_{j},\varvec{\mu },u)s_j^{(0)}(\varvec{\theta }_{0j},\varvec{\mu }_0,u)\lambda _{E0j}(u)\text {d}u \end{aligned}$$

(6.5)

The second derivative turns out to be negative in both (6.4) and (6.5) and hence for each $(\varvec{\theta ,\mu })$, $C^*(\varvec{\theta ,\mu } ; 1)$ converges to a concave function with a unique maximum at $(\varvec{\theta _0},\varvec{\mu _0})$. Hence it follows that since $(\varvec{\hat{\theta },\hat{\mu }})$ maximizes $C^*(\varvec{\theta ,\mu } ; 1)$, $(\varvec{\hat{\theta },\hat{\mu }}){\mathop {\rightarrow }\limits ^{p}}(\varvec{\theta _0},\varvec{\mu _0})$.

To prove asymptotic normality, define

$$\begin{aligned} U(\varvec{\theta },\varvec{\mu },t ) = \sum _{i=1}^{n} \int _{0}^{t} [\tilde{\mathbf{z}}_i(\varvec{\theta },\varvec{\mu },u) - \mathbf{r}(\varvec{\theta },\varvec{\mu },u)]\text {d}\bar{N}_i(u) \end{aligned}$$

and

            $V(\varvec{\theta ,\mu ,t}) = \left( \begin{array}{ll} V_{\varvec{\theta \theta }}(t) &{} V_{\varvec{\theta \mu }}(t) \\ V_{\varvec{\theta \mu }}(t)'&{} V_{\varvec{\mu \mu }}(t) \\ \end{array}\right) ,$ where $V_{\varvec{\theta \theta }}(t)^{mp \times mp} = diag((V_{\varvec{\theta _j\theta _j}}(t))), V_{\varvec{\theta }\mu }(t)^{mp \times 2} = (V'_{\varvec{\theta _1\mu }}(t)~\cdots ~V'_{\varvec{\theta _m\mu }}(t))'$

$$\begin{aligned} V_{\varvec{\theta _j\theta _j}}(t)= & {} \int _{0}^{t} \sum _{i=1}^{n} \left\{ (a_{Ej}^{-1}(u) \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl} {\varvec{z}}_{jil}{{\varvec{z}'}_{jil}} \right. \\&\left. - \,\left( a_{Ej}^{-2}(u) \left( \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}{\varvec{z}}_{jl}\right) \left( \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}{\varvec{z}}_{jl}\right) ' \right) \right\} \text {d}N_{ji}(u),\\ V_{\varvec{\theta _j\mu }}(t)= & {} \int _{0}^{t} \sum _{i=1}^{n} \left\{ \left( a_{Ej}^{-1}(u) \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}{\varvec{z}}_{jil}~{\varvec{y}}^{'}_{jil}\right) \right. \\&\left. - \,\left( a_{Ej}^{-2}(u)\left( \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}{\varvec{z}}_{jl}\right) \left( \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}~{\varvec{y}}_{jl}\right) '\right) \right\} \text {d}N_{ji}(u),\\ V_{\varvec{\mu \mu }}(t)= & {} \int _{0}^{t} \sum _{j=1}^m \sum _{i=1}^{n} \left\{ \left( a_{Ej}^{-1}(u) \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl} {\varvec{y}}_{jil}{\varvec{y}}^{'}_{jil}\right) \right. \\&\left. - \,\left( a_{Ej}^{-2}(u) \left( \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}~{\varvec{y}}_{jl}\right) \left( \sum _{l=1}^{n} Q_{jl}(u) r_{Ejl}~{\varvec{y}}_{jl}\right) '\right) \right\} \text {d}N_{ji}(u). \end{aligned}$$

Also let    $\Sigma = \left( \begin{array}{ll} \Sigma _{\varvec{\theta }} &{} \Sigma _{\varvec{\theta \mu }} \\ \Sigma _{\varvec{\theta \mu }}'&{} \Sigma _{\varvec{\mu }} \\ \end{array} \right) , \hbox {where} \quad \Sigma _{\varvec{\theta }}^{mp \times mp} = diag((\Sigma _{\varvec{j\theta _j}})), \Sigma _{\varvec{\theta }\mu }^{mp \times 2} = \left( \Sigma '_{\varvec{1\theta _1\mu }}~\cdots ~\Sigma '_{\varvec{m\theta _m\mu }}\right) '$.

We then need to show that for $(\varvec{\theta }^{*'},\varvec{\mu }^{*'})'{\mathop {\rightarrow }\limits ^{p}}(\varvec{\theta }_0^{'},\varvec{\mu }_0^{'})'$,

$$\begin{aligned} \sqrt{n_m} [(\varvec{\hat{\theta }'},\varvec{\hat{\mu }'})' - (\varvec{\theta '}_0,\varvec{\mu '}_0)']= & {} \left[ \frac{1}{n_m}V(\varvec{\theta }^{*'},\varvec{\mu }^{*'},1)\right] ^{-1}\frac{1}{\sqrt{n_m}}U(\varvec{\theta }_0,\varvec{\mu }_0,1) {\mathop {\rightarrow }\limits ^{d}} N(\mathbf{0}, \Sigma ^{-1}).\\ \text {Now} \quad \quad n_m^{-\frac{1}{2}}U(\varvec{\theta }_0,\varvec{\mu }_0,t )= & {} n_m^{-\frac{1}{2}}\sum _{i=1}^{n} \int _{0}^{t} \tilde{\mathbf{z}}_i(\varvec{\theta }_0,\varvec{\mu }_0,u) \text {d}\bar{N}_i(u) - n_m^{-\frac{1}{2}}\mathbf{r}(\varvec{\theta }_0,\varvec{\mu }_0,u)\text {d}\bar{N}(u)\\= & {} n_m^{-\frac{1}{2}}\sum _{i=1}^{n} \int _{0}^{t} \tilde{\mathbf{z}}_i(\varvec{\theta }_0,\varvec{\mu }_0,u)\text {d}\bar{M}_i(u) - n_m^{-\frac{1}{2}}{} \mathbf{r}(\varvec{\theta }_0,\varvec{\mu }_0,u)\text {d}\bar{M}(u) \\= & {} n_m^{-\frac{1}{2}}\sum _{i=1}^{n} \int _{0}^{t} n_m^{-\frac{1}{2}}(\tilde{\mathbf{z}}_i(\varvec{\theta }_0,\varvec{\mu }_0,u) - n_m^{-\frac{1}{2}}{} \mathbf{r}(\varvec{\theta }_0,\varvec{\mu }_0,u))\text {d}\bar{M_i}(u), \end{aligned}$$

where, $d\bar{M_i}(u)=\sum _{j=1}^{m_i}dM_{ji}(u), \tilde{\mathbf{z}}_i(\varvec{\theta }_0,\varvec{\mu }_0,u) = ({\varvec{z}'}_{1i}, \ldots , {\varvec{z}'}_{mi}, \sum _{j=1}^m {\varvec{y}}'_{ji})'$ and $\mathbf{r}(\varvec{\theta },\varvec{\mu },u) = (E_{1\theta _1}(u), \ldots , E_{m\theta _m}(u), \sum _{j=1}^m E_{j\mu }(u))'$.

By techniques similar to Andersen and Gill (1982), it can be shown that the conditions of Rebolledo’s (1980) CLT are satisfied. Taking $H_{il}(t)=n_m^{-\frac{1}{2}}(\tilde{\mathbf{z}}_l(t)-\mathbf{r}(\varvec{\theta }_0,\varvec{\mu }_0,t))_i$, an application of the CLT shows that $n_m^{-\frac{1}{2}}U(\varvec{\theta }_0,\varvec{\mu }_0,t)$ converges weakly to a continuous Gaussian process that has a covariance matrix $\Sigma$ when evaluated at time $t=1$. Hence, $n_m^{-\frac{1}{2}}U(\varvec{\theta }_0,\varvec{\mu }_0,1){\mathop {\rightarrow }\limits ^{d}} N(0,\Sigma )$.

Again, following arguments similar to Andersen and Gill (1982) and using Lenglart’s inequality, it can be shown that $||n_m^{-1}V(\varvec{\theta }^*,\varvec{\mu }^*,1) - \Sigma ||{\mathop {\rightarrow }\limits ^{p}} 0$, for any random $(\varvec{\theta }^{*},\varvec{\mu }^{*})$ such that $(\varvec{\theta }^{*'},\varvec{\mu }^{*'})'{\mathop {\rightarrow }\limits ^{p}}(\varvec{\theta }_0^{'},\varvec{\mu }_0^{'})'$ as $n_m\rightarrow \infty$.