A copula-based Markov chain model for serially dependent event times with a dependent terminal event

Abstract

Copula modeling for serial dependence has been extensively discussed in a time series context. However, fitting copula-based Markov models for serially dependent survival data is challenging due to the complex censoring mechanisms. The purpose of this paper is to develop likelihood-based methods for fitting a copula-based Markov chain model to serially dependent event times that are dependently censored by a terminal event, such as death. We propose a novel copula-based Markov chain model for describing serial dependence in recurrent event times. We also apply another copula model for handling dependent censoring. Due to the complex likelihood function with the two copulas, we propose a two-stage estimation method under Weibull distributions for fitting the survival data. The asymptotic normality of the proposed estimator is established through the theory of estimating functions. We propose a jackknife method for interval estimates, which is shown to be asymptotically consistent. To select suitable copulas for a given dataset, we propose a model selection method according to the 2nd stage likelihood. We conduct simulation studies to assess the performance of the proposed methods. For illustration, we analyze survival data from colorectal cancer patients. We implement the proposed methods in our original R package “Copula.Markov.survival” that is made available in CRAN (https://cran.r-project.org/).

Appendices

Appendix 1: partial derivatives

The 1st-stage transformed log-likelihood is

$$\tilde{\ell }_{i}^{(1)} (\,\tilde{\lambda },\;\tilde{\nu }_{2} \,) = \delta_{i}^{*} (\,\tilde{\lambda } + \tilde{\nu }_{2} \,) + \delta_{i}^{*} \log (\,T_{i}^{*} \,)\{ \,\exp (\,\tilde{\nu }_{2} \,) - 1\,\} - \exp (\,\tilde{\lambda }\,)T_{i}^{{*\exp (\,\tilde{\nu }_{2} \,)}} .$$

Its partial derivatives are

$$\begin{aligned} \partial \tilde{\ell }_{i}^{(1)} (\,\tilde{\lambda },\;\tilde{\nu }_{2} \,)/\partial \tilde{\lambda }& = \delta_{i}^{*} - \exp (\,\tilde{\lambda }\,)T_{i}^{{*\exp (\,\tilde{\nu }_{2} \,)}} \hfill \\ \partial \tilde{\ell }_{i}^{(1)} (\,\tilde{\lambda },\;\tilde{\nu }_{2} \,)/\partial \tilde{\nu }_{2} &= \delta_{i}^{*} + \delta_{i}^{*} \exp (\,\tilde{\nu }_{2} \,)\log (\,T_{i}^{*} \,) - T_{i}^{{*\exp (\,\tilde{\nu }_{2} \,)}} \exp (\,\tilde{\lambda }\,)\log (\,T_{i}^{*} \,). \hfill \\ \end{aligned}$$

Define notations $w_{i} = \exp (\tilde{\lambda })T_{i}^{{*\exp (\,\tilde{\nu }_{2} \,)}}$, $u_{ij} = \exp (\tilde{r})T_{ij}^{{\exp (\tilde{\nu }_{1} )}}$,

$$\tilde{D}^{[\,2,\square ]} (s,t) = - \frac{\partial }{\partial s}\tilde{D}^{[\,1,\cdot ]} (s,t)\;{\text{and}}\,\tilde{D}^{[\,\cdot ,2]} (s,t) = - \frac{\partial }{\partial t}\tilde{D}^{[\,\cdot ,1]} (s,t)$$

The 2nd stage transformed log-likelihood is

$$\begin{aligned} \tilde{\ell }_{i} (\,\tilde{r},\;\tilde{\nu }_{1} ,\;\tilde{\theta },\;\tilde{\alpha },\;\tilde{\lambda },\;\tilde{\nu }_{2} \,) & = \delta_{i}^{*} \{ {\tilde{\lambda } + \tilde{\nu }_{2} + (\exp (\tilde{\nu }_{2} ) - 1)\log (\,T_{i}^{*} \,)} \} + \delta_{i1} \{ {\tilde{r} + \tilde{\nu }_{1} + (\exp (\tilde{\nu }_{1} ) - 1)\log (\,T_{i1} \,)\,} \} \\ & \quad + \delta_{i}^{*} \delta_{i1} \log \tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,) + \delta_{i}^{*} (\,1 - \delta_{i1} \,)\log \tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,) \\ & \quad + (\,1 - \delta_{i}^{*} \,)\delta_{i1} \log \tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,) + (\,1 - \delta_{i}^{*} \,)(\,1 - \delta_{i1} \,)\log \tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,) \\ & \quad + \sum\limits_{j = 2}^{{n_{i} }} {\left[ {\,u_{i(j - 1)} + \delta_{ij} \left\{ {\tilde{r} + \tilde{\nu }_{1} + (\exp (\tilde{\nu }_{1} ) - 1)\log (\,T_{ij} \,)} \right\}} \right.} \\ & \quad + \left. {\delta_{ij} \log D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,) + (\,1 - \delta_{ij} \,)\log D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)} \right]. \\ \end{aligned}$$

Its partial derivatives are

$$\begin{aligned} \frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{r}}} & = \delta_{i1} - \left\{ {\delta_{i}^{*} \delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,2,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}} + \delta_{i}^{*} (\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}} \right. \\ & \quad + \left. {(\,1 - \delta_{i}^{*} \,)\delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,2,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}} + (\,1 - \delta_{i}^{*} \,)(\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,)}}} \right\}u_{i1} \\ & \quad + \sum\limits_{j = 2}^{{n_{i} }} {\left[ {\,u_{i(j - 1)} + \delta_{ij} - \left\{ {\delta_{ij} \frac{{D_{{\tilde{\theta }}}^{[\,2,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}} + (\,1 - \delta_{ij} \,)\frac{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}} \right\}u_{ij} } \right.} \\ & \quad - \left. {\left\{ {\delta_{ij} \frac{{D_{{\tilde{\theta }}}^{[\,1,\;2\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}} + (\,1 - \delta_{ij} \,)\frac{{D_{{\tilde{\theta }}}^{[\,0,\;2\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}} \right\}u_{i(j - 1)} \,} \right] \\ \end{aligned}$$

$$\begin{aligned} \frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\lambda }}} = \delta_{i}^{*} - \left\{ {\delta_{i}^{*} \delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;2\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}} + \delta_{i}^{*} (\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;2\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}} \right. \hfill \\ & \quad \left. { + (\,1 - \delta_{i}^{*} \,)\delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}} + (\,1 - \delta_{i}^{*} \,)(\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,)}}} \right\}w_{i} \hfill \\ \end{aligned}$$

$$\begin{aligned} \frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\nu }_{1} }} & = \delta_{i1} \{ \,1 + \exp (\tilde{\nu }_{1} )\log (\,T_{i1} \,)\,\} \\ & \quad - \left\{ {\delta_{i}^{*} \delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,2,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}} + \delta_{i}^{*} (\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}} \right. \\ & \quad + \left. {(\,1 - \delta_{i}^{*} \,)\delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,2,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}} + (\,1 - \delta_{i}^{*} \,)(\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,)}}} \right\}u_{i1} \exp (\tilde{\nu }_{1} )\log T_{i1} \\ & \quad + \sum\nolimits_{j = 2}^{{n_{i} }} {\left[ {\,u_{i(j - 1)} \exp (\tilde{\nu }_{1} )\log T_{i(\,j - 1\,)} + \delta_{ij} \{ 1 + \exp (\tilde{\nu }_{1} )\log (\,T_{ij} \,)\,\} } \right.} \\ & \quad - \left\{ {\delta_{ij} \frac{{D_{{\tilde{\theta }}}^{[\,2,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}} - (\,1 - \delta_{ij} \,)\frac{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}} \right\}u_{ij} \exp (\tilde{\nu }_{1} )\log T_{ij} \\ & \quad - \left. {\left\{ {\delta_{ij} \frac{{D_{{\tilde{\theta }}}^{[\,1,\;2\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}} + (\,1 - \delta_{ij} \,)\frac{{D_{{\tilde{\theta }}}^{[\,0,\;2\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}} \right\}u_{i(j - 1)} \exp (\tilde{\nu }_{1} )\log T_{i(\,j - 1\,)} \,} \right] \\ \end{aligned}$$

$$\frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\theta }}} = \sum\limits_{j = 2}^{{n_{i} }} {\left[ {\delta_{ij} \frac{{\frac{\partial }{{\partial \tilde{\theta }}}D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,1,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}} + (\,1 - \delta_{ij} \,)\frac{{\frac{\partial }{{\partial \tilde{\theta }}}D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}{{D_{{\tilde{\theta }}}^{[\,0,\;1\,]} (\,u_{ij} ,\;u_{i(j - 1)} \,)}}} \right]}$$

$$\begin{aligned} \frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\alpha }}} & = \delta_{i}^{*} \delta_{i1} \frac{{\frac{\partial }{{\partial \tilde{\alpha }}}\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}} + \delta_{i}^{*} (\,1 - \delta_{i1} \,)\frac{{\frac{\partial }{{\partial \tilde{\alpha }}}\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}} \\ & \quad + (\,1 - \delta_{i}^{*} \,)\delta_{i1} \frac{{\frac{\partial }{{\partial \tilde{\alpha }}}\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}} + (\,1 - \delta_{i}^{*} \,)(\,1 - \delta_{i1} \,)\frac{{\frac{\partial }{{\partial \tilde{\alpha }}}\tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,)}} \\ \end{aligned}$$

$$\begin{aligned} \frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\nu }_{2} }} & = \delta_{i}^{*} \left\{ {1 + \exp (\tilde{\nu }_{2} )\log (\,T_{i}^{*} \,)} \right. \\ & \quad + \left\{ {\delta_{i}^{*} \delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;2\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}} \right. + \delta_{i}^{*} (\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;2\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}} \\ & \quad - \left. {(\,1 - \delta_{i}^{*} \,)\delta_{i1} \frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}}^{[\,1,\;0\,]} (\,u_{i1} ,\;w_{i} \,)}} - (\,1 - \delta_{i}^{*} \,)(\,1 - \delta_{i1} \,)\frac{{\tilde{D}_{{\tilde{\alpha }}}^{[\,0,\;1\,]} (\,u_{i1} ,\;w_{i} \,)}}{{\tilde{D}_{{\tilde{\alpha }}} (\,u_{i1} ,\;w_{i} \,)}}} \right\}w_{i} \exp (\tilde{\nu }_{2} )\log T_{i}^{*} . \\ \end{aligned}$$

Appendix 2: sketch of the proof of Theorem 1

We first define

$${\mathbf{g}}_{i} (\,\tilde{\varTheta }\,) = \left[ {\begin{array}{*{20}c} {\frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{r}}}} & {\frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\nu }_{1} }}} & {\frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\theta }}}} & {\frac{{\partial \tilde{\ell }_{i} }}{{\partial \tilde{\alpha }}}} & {\frac{{\partial \tilde{\ell }_{i}^{(1)} }}{{\partial \tilde{\lambda }}}} & {\frac{{\partial \tilde{\ell }_{i}^{(1)} }}{{\partial \tilde{\nu }_{2} }}} \\ \end{array} } \right]^{\text{T}} .$$

By a Taylor series expansion, the estimating function ${\mathbf{g}}(\,\tilde{\varTheta }\,) = \sum\nolimits_{i = 1}^{N} {{\mathbf{g}}_{i} (\,\tilde{\varTheta }\,)}$ can be expanded around the true parameter value $\tilde{\varTheta }_{0} = (\,\tilde{r}_{0} ,\;\tilde{\nu }_{1\;0} ,\;\tilde{\theta }_{0} ,\;\tilde{\alpha }_{0} ,\;\tilde{\lambda }_{0} ,\;\tilde{\nu }_{2\;0} \,)$, such that

$${\mathbf{g}}(\,\tilde{\varTheta }\,) = {\mathbf{g}}(\,\tilde{\varTheta }_{0} \,) + \left. {\frac{{\partial {\mathbf{g}}(\,\tilde{\varTheta }\,)}}{{\partial \tilde{\varTheta }}}} \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} \left( {\tilde{\varTheta } - \tilde{\varTheta }_{0} } \right)^{\text{T}} + \overbrace {{O\left( {\left\| {\tilde{\varTheta } - \tilde{\varTheta }_{0} } \right\|^{2} } \right)}}^{\text{The 2nd - order and remainder terms}}.$$

We plug-in the MLE in the above equation so that the left-side becomes zero. It follows that

$$N^{1/2} (\,\hat{\tilde{\varTheta }} - \tilde{\varTheta }_{0} \,)^{T} = \,\left\{ { - \frac{1}{N}\left. {\frac{{\partial {\mathbf{g}}(\,\tilde{\varTheta }\,)}}{{\partial \tilde{\varTheta }}}} \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} } \right\}^{ - 1} \,\frac{1}{{N^{1/2} }}{\mathbf{g}}(\,\tilde{\varTheta }_{0} \,) + O_{P} (N^{1/2} ||\,\hat{\tilde{\varTheta }} - \tilde{\varTheta }_{0} \,||^{2} )$$

(3)

Since $||\,\hat{\tilde{\varTheta }} - \tilde{\varTheta }_{0} \,|| = o_{P} (1)$, the term $O_{P} (N^{1/2} ||\,\tilde{\varTheta } - \tilde{\varTheta }_{0} \,||^{2} )$ is negligible relative to the left-side. The weak law of large number implies

$$\frac{1}{N}\left. {\frac{{\partial {\mathbf{g}}(\,\tilde{\varTheta }\,)}}{{\partial \tilde{\varTheta }}}} \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} \mathop \to \limits^{p} H(\,\tilde{\varTheta }_{0} \,).$$

Under some regularity conditions,

$$\begin{aligned} {\text{E}}\{ \,{\mathbf{g}}_{i} (\,\tilde{\varTheta }_{0} \,)\,\} & = \left. {\int {f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,)} {\mathbf{g}}_{i} (\,\tilde{\varTheta }\,){\text{d}}{\mathbf{x}}_{i} } \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} \\ & = \left. {\int {f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,)\frac{{\partial \log f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,)}}{{\partial \tilde{\varTheta }}}{\text{d}}{\mathbf{x}}_{i} } } \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} \\ & = \left. {\int {f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,)\frac{1}{{f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,)}}\frac{{\partial f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,)}}{{\partial \tilde{\varTheta }}}{\text{d}}{\mathbf{x}}_{i} } } \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} \\ & = \left. {\frac{{\partial \int {f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,){\text{d}}{\mathbf{x}}_{i} } }}{{\partial \tilde{\varTheta }}}} \right|_{{\tilde{\varTheta } = \tilde{\varTheta }_{0} }} = {\mathbf{0}}{ ,} \\ \end{aligned}$$

where $f(\,{\mathbf{x}}_{i} |\tilde{\varTheta }\,) = f(\,t_{i1} ,\; \ldots ,\;t_{{in_{i} }} ,\;\delta_{i1} ,\; \ldots ,\;\delta_{{in_{i} }} ,\;t_{i}^{*} ,\,\delta_{i}^{*} |\tilde{\varTheta }\,) = L_{i}$ is the density function for subject $i$.

Also, by the central limit theorem,

$$\frac{1}{{N^{1/2} }}{\mathbf{g}}(\,\tilde{\varTheta }_{0} \,)\mathop \to \limits^{d} N\left[ {{\text{E}}\{ \,{\mathbf{g}}_{i} (\,\tilde{\varTheta }_{0} \,)\,\} ,\;{\text{Var}}\{ \,{\mathbf{g}}_{i} (\,\tilde{\varTheta }_{0} \,)\,\} } \right] = N\left[ {{\mathbf{0}} ,\;{\text{E}}\{ \,{\mathbf{g}}_{i} (\,\tilde{\varTheta }_{0} \,){\mathbf{g}}_{i} (\,\tilde{\varTheta }_{0} \,)^{\text{T}} \,\} } \right] = N [\,{\mathbf{0}} ,\;M(\,\tilde{\varTheta }_{0} \,)\,],$$

Finally, by Slutsky’s theorem,

$$N^{1/2} ( {\hat{\tilde{\varTheta }} - \tilde{\varTheta }_{0} } )^{\text{T}} \mathop \to \limits^{d} N\left[ {{\mathbf{0}} ,\;\{ \,H(\,\tilde{\varTheta }_{0} \,)^{ - 1} \,\} M(\,\tilde{\varTheta }_{0} \,)\{ \,H(\,\tilde{\varTheta }_{0} \,)^{ - 1} \,\}^{\text{T}} } \right] = N\left[ {{\mathbf{0}} ,\;\tilde{V}(\,\tilde{\varTheta }_{0} \,)} \right].\;\square$$

Appendix 3: sketch of the proof of Theorem 2

Below, we give the outline for the proof of the consistency by showing the approximation

$$N^{ - 1} \tilde{V}(\,\tilde{\varTheta }_{0} \,) \approx \sum\limits_{k = 1}^{N} {( {\hat{\tilde{\varTheta }}^{(\, - k\,)} - \hat{\tilde{\varTheta }}} )^{\text{T}} ( {\hat{\tilde{\varTheta }}^{(\, - k\,)} - \hat{\tilde{\varTheta }}} )} .$$

We substitute $\hat{\tilde{\varTheta }}$ by $\hat{\tilde{\varTheta }}^{(\, - k\,)}$ in Eq. (3), such that

$$N^{1/2} ( {\hat{\tilde{\varTheta }}^{(\, - k\,)} - \tilde{\varTheta }_{0} } )^{T} \approx - H( {\tilde{\varTheta }_{0} } )^{ - 1} N^{ - 1/2} \sum\nolimits_{i \ne k} {{\mathbf{g}}_{i} ( {\tilde{\varTheta }_{0} } )}$$

It follows that

$$( {\hat{\tilde{\varTheta }}^{(\, - k\,)} - \tilde{\varTheta }_{0} })^{\text{T}} \approx - H(\,\tilde{\varTheta }_{0} \,)^{ - 1} \frac{1}{N}\sum\nolimits_{i \ne k} {{\mathbf{g}}_{i} (\,\tilde{\varTheta }_{0} \,)}$$

(4)

Also by Eq. (3), we have

$$N\{ \, - H(\,\tilde{\varTheta }_{0} \,)\,\} (\,\hat{\tilde{\varTheta }} - \tilde{\varTheta }_{0} \,)^{T} \approx {\mathbf{g}}(\,\tilde{\varTheta }_{0} \,)$$

(5)

After eliminating $\tilde{\varTheta }_{0}$ from Eqs. (4) and (5),

$$( {\hat{\tilde{\varTheta }}^{(\, - k\,)} - \hat{\tilde{\varTheta }}} )^{\text{T}} \approx - \frac{1}{N}H(\,\tilde{\varTheta }_{0} \,)^{ - 1} \,{\mathbf{g}}_{k} (\,\tilde{\varTheta }_{0} \,).$$

Finally, we verify the desired results:

$$\begin{aligned} &\sum\nolimits_{{k = 1}}^{N} {( {\hat{\tilde{\Theta }}^{{(\, - k\,)}} - \hat{\tilde{\Theta }}} )^{{\text{T}}} ( {\hat{\tilde{\Theta }}^{{(\, - k\,)}} - \hat{\tilde{\Theta }}} )} \\&\quad \approx \frac{1}{{N^{2} }}H(\,\tilde{\Theta }_{0} \,)^{{ - 1}} \sum\nolimits_{{k = 1}}^{N} {{\mathbf{g}}_{k} (\,\tilde{\Theta }_{0} \,){\mathbf{g}}_{k} (\,\tilde{\Theta }_{0} \,)^{{\text{T}}} } \{ {H(\,\tilde{\Theta }_{0} \,)^{{ - 1}} } \}^{{\text{T}}} \\&\quad \approx \frac{1}{N}H(\,\tilde{\Theta }_{0} \,)^{{ - 1}} E\{ {{\mathbf{g}}_{i} (\,\tilde{\Theta }_{0} \,){\mathbf{g}}_{i} (\,\tilde{\Theta }_{0} \,)^{{\text{T}}} } \}\{ {H(\,\tilde{\Theta }_{0} \,)^{{ - 1}} } \}^{{\text{T}}} \\&\quad \approx \frac{1}{N}H(\,\tilde{\Theta }_{0} \,)^{{ - 1}} {M(\,\tilde{\Theta }_{0} \,) \{ \,H(\,\tilde{\Theta }_{0} \,)^{{ - 1}} } \}^{{\text{T}}} = \frac{1}{N}\tilde{V}(\,\tilde{\Theta }_{0} \,).\; \end{aligned}$$

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