Appendix 1: Some derivatives used in the estimation method
The marginal likelihood is
$$\begin{aligned} l({\varTheta })\propto \prod _{i=1}^n(\sigma ^2)^{-n_i/2}(\sigma _{\alpha }^2)^{-1/2}\int _{-\infty }^{+\infty } e^{\lambda (\alpha _i)}d\alpha _i. \end{aligned}$$
We approximate \(\lambda (\alpha _i)\) at \(\hat{\alpha }_i\) by Taylor expansion and \(\lambda (\alpha _i)\approx \lambda (\hat{\alpha }_i)+\frac{1}{2}(\alpha _i-\hat{\alpha }_i)^2\lambda ^{(2)}(\hat{\alpha }_i)\) since \(\lambda ^{(1)}(\hat{\alpha }_i)=0\). Thus
$$\begin{aligned} l({\varTheta })\approx & {} \prod _{i=1}^n(\sigma ^2)^{-n_i/2}(\sigma _{\alpha }^2)^{-1/2}e^{\lambda (\hat{\alpha }_i)}\int _{-\infty }^{+\infty } e^{-\frac{(\alpha _i-\hat{\alpha }_i)^2}{2(-\lambda ^{(2)}(\hat{\alpha }_i))^{-1}}}d\alpha _i\\= & {} \prod _{i=1}^n(\sigma ^2)^{-n_i/2}(\sigma _{\alpha }^2)^{-1/2}e^{\lambda (\hat{\alpha }_i)} \sqrt{2\pi |\lambda ^{(2)}(\hat{\alpha }_i)|^{-1}} \end{aligned}$$
and the first order and second-order Laplace-approximated marginal log-likelihood
$$\begin{aligned} \widetilde{LL}({\varTheta })= & {} -\frac{\sum _{i=1}^nn_i}{2}\log {\sigma ^2}-\frac{n}{2}\log {\sigma _{\alpha }^2}+\sum _{i=1}^n\lambda (\hat{\alpha }_i),\\ LL({\varTheta })= & {} -\frac{\sum _{i=1}^nn_i}{2}\log {\sigma ^2}-\frac{n}{2}\log {\sigma _{\alpha }^2}+\sum _{i=1}^n\lambda (\hat{\alpha }_i) -\frac{1}{2}\sum _{i=1}^n\log |\lambda ^{(2)}(\hat{\alpha }_i)|. \end{aligned}$$
Denote \(\lambda _{PJPL}(\alpha )=\sum _{i=1}^n\lambda _{PJPL}(\alpha _i), \alpha =(\alpha _1, \ldots , \alpha _n)^T\), then
$$\begin{aligned} \frac{\partial \lambda _{PJPL}\left( \alpha \right) }{\partial \alpha _i}= & {} -\frac{1}{2\sigma ^2}\sum _{j=1}^{n_i}\frac{\partial d\left( y_{ij}; \mu _{ij}\right) }{\partial \alpha _i}-\frac{\alpha _i}{\sigma ^2_{\alpha }}+\delta _i\phi -\sum _{l=1}^i\delta _l\frac{e^{Z_i^T\gamma +\phi \alpha _i}\phi }{\sum _{k\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}},\\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha \right) }{\partial \alpha _i^2}= & {} -\frac{1}{2\sigma ^2}\sum _{j=1}^{n_i}\frac{\partial ^2 d\left( y_{ij}; \mu _{ij}\right) }{\partial \alpha _i^2}-\frac{1}{\sigma ^2_{\alpha }}\\&\quad -\,\sum _{l=1}^i\delta _l\frac{e^{Z_i^T\gamma +\phi \alpha _i}}{\sum _{k\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\left[ 1-\frac{e^{Z_i^T\gamma +\phi \alpha _i}}{\sum _{k\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\right] \phi ^2,\\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha \right) }{\partial \alpha _i\partial \alpha _m}= & {} e^{Z_i^T\gamma +\phi \alpha _i}\phi \sum _{l=1}^i\delta _l\frac{e^{Z_m^T\gamma +\phi \alpha _m}\phi }{\left( \sum _{k\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}I_{\left( t_m\in R\left( t_l\right) \right) },\\&\quad \, i, m=1, \ldots , n, i\ne m, \\ \frac{\partial \lambda _{PJPL}\left( \alpha \right) }{\partial \beta }= & {} -\frac{1}{2\sigma ^2}\sum _{i=1}^n\sum _{j=1}^{n_i}\frac{\partial d\left( y_{ij}; \mu _{ij}\right) }{\partial \beta },\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha \right) }{\partial \beta \partial \beta ^T}\\ \end{aligned}$$
$$\begin{aligned}= & {} -\frac{1}{2\sigma ^2}\sum _{i=1}^n\sum _{j=1}^{n_i}\frac{\partial ^2 d\left( y_{ij}; \mu _{ij}\right) }{\partial \beta \partial \beta ^T},\\ \frac{\partial \lambda _{PJPL}\left( \alpha \right) }{\partial \gamma }= & {} \sum _{i=1}^n\delta _i\left( Z_i^T-\frac{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}Z_k^T}{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\right) ,\\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha \right) }{\partial \gamma \partial \gamma ^T}= & {} -\sum _{i=1}^n\delta _i\left( \frac{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}Z_kZ_k^T}{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\right. \\&\left. -\frac{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}Z_k\cdot \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}Z_k^T}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}\right) , \end{aligned}$$
$$\begin{aligned} \frac{\partial LL_{PJPL}}{\partial \sigma ^2}= & {} -\frac{\sum _{i=1}^n n_i}{2}\frac{1}{\sigma ^2}-\frac{1}{2}\sum _{i=1}^n\frac{1}{\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }\frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2}\\&+\sum _{i=1}^n\frac{\partial \lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2},\\ \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2\partial \sigma ^2}= & {} \frac{\sum _{i=1}^n n_i}{2\sigma ^4}-\frac{1}{2}\sum _{i=1}^n\frac{1}{\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }\left( \frac{\partial ^2\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2\partial \sigma ^2}\right. \\&\left. -\frac{1}{\lambda ^{\left( 2\right) }\left( \alpha _i\right) }\left( \frac{\partial \lambda ^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2}\right) ^2\right) +\,\sum _{i=1}^n\frac{\partial ^2\lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2\partial \sigma ^2},\\ \frac{\partial LL_{PJPL}}{\partial \sigma _{\alpha }^2}= & {} -\frac{n}{2\sigma ^2_{\alpha }}+\sum _{i=1}^n\frac{\partial \lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }} -\frac{1}{2}\sum _{i=1}^n\frac{1}{\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }\frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }},\\ \frac{\partial ^2 LL_{PJPL}}{\partial \sigma _{\alpha }^2\partial \sigma _{\alpha }^2}= & {} \frac{n}{2\sigma ^4_{\alpha }} -\frac{1}{2}\sum _{i=1}^n\frac{1}{\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }\left( \frac{\partial ^2\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma _{\alpha }^2\partial \sigma _{\alpha }^2} -\frac{1}{\lambda ^{\left( 2\right) }\left( \alpha _i\right) }\left( \frac{\partial \lambda ^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }}\right) ^2\right) \\ \end{aligned}$$
$$\begin{aligned}&+\,\sum _{i=1}^n\frac{\partial ^2\lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }\partial \sigma ^2_{\alpha }},\\ \frac{\partial LL_{PJPL}}{\partial \phi }= & {} \sum _{i=1}^n\frac{\partial \lambda _{PJPL}\left( \alpha _i\right) }{\partial \phi } -\frac{1}{2}\sum _{i=1}^n\frac{1}{\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }\frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \phi },\\ \frac{\partial ^2 LL_{PJPL}}{\partial \phi \partial \phi }= & {} -\frac{1}{2}\sum _{i=1}^n\frac{1}{\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }\left( \frac{\partial ^2\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \phi \partial \phi } -\frac{1}{\lambda ^{\left( 2\right) }\left( \alpha _i\right) }\left( \frac{\partial \lambda ^{\left( 2\right) }\left( \alpha _i\right) }{\partial \phi }\right) ^2\right) \\&+\,\sum _{i=1}^n\frac{\partial ^2\lambda _{PJPL}\left( \alpha _i\right) }{\partial \phi \partial \phi },\\ \frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \beta }= & {} \frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}}\frac{\partial \mu _{ij}}{\partial \beta },\ \frac{\partial ^2 d\left( y_{ij};\mu _{ij}\right) }{\partial \beta \partial \beta ^T}=\frac{\partial ^2d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}\partial \mu _{ij}}\frac{\partial \mu _{ij}}{\partial \beta }\frac{\partial \mu _{ij}}{\partial \beta ^T}\\ \end{aligned}$$
$$\begin{aligned}&+\,\frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}}\frac{\partial ^2\mu _{ij}}{\partial \beta \partial \beta ^T},\\ \frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \alpha _i}= & {} \frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}}\mu _{ij}\left( 1-\mu _{ij}\right) ,\ \frac{\partial ^2 d\left( y_{ij};\mu _{ij}\right) }{\partial \alpha _i^2}\\= & {} \frac{\partial ^2d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}^2}\left( \frac{\partial \mu _{ij}}{\partial \alpha _i}\right) ^2+\,\frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}}\frac{\partial ^2\mu _{ij}}{\alpha _i^2}, \end{aligned}$$
$$\begin{aligned} \frac{\partial d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}}= & {} -2\frac{d\left( y_{ij};\mu _{ij}\right) }{y_{ij}-\mu _{ij}}-2d\left( y_{ij};\mu _{ij}\right) \frac{1-2\mu _{ij}}{\mu _{ij}\left( 1-\mu _{ij}\right) },\ \frac{\partial \mu _{ij}}{\partial \beta }\\= & {} \mu _{ij}\left( 1-\mu _{ij}\right) X_{ij},\\ \frac{\partial ^2d\left( y_{ij};\mu _{ij}\right) }{\partial \mu _{ij}^2}= & {} -2\frac{\partial d\left( y_{ij};\mu _{ij}\right) /\partial \mu _{ij}}{y_{ij}-\mu _{ij}}-\frac{2d\left( y_{ij};\mu _{ij}\right) }{\left( y_{ij}-\mu _{ij}\right) ^2}\\&-\,\frac{2\left[ \partial d\left( y_{ij};\mu _{ij}\right) /\partial \mu _{ij}\left( 1-2\mu _{ij}\right) -2d\left( y_{ij};\mu _{ij}\right) \right] }{\mu _{ij}\left( 1-\mu _{ij}\right) }\\&+\,\frac{2d\left( y_{ij};\mu _{ij}\right) \left( 1-2\mu _{ij}\right) ^2}{\mu _{ij}^2\left( 1-\mu _{ij}\right) ^2},\\ \frac{\partial ^2\mu _{ij}}{\partial \beta \partial \beta ^T}= & {} \left( 1-2\mu _{ij}\right) \left( 1-\mu _{ij}\right) \mu _{ij}X_{ij}X_{ij}^T,\\ \end{aligned}$$
$$\begin{aligned} \frac{\partial \lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2}= & {} \frac{1}{2}\sum _{j=1}^{n_i}\frac{d\left( y_{ij};\mu _{ij}\right) }{\sigma ^4},\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2\partial \sigma ^2}=-\frac{\sum _{j=1}^{n_i}d\left( y_{ij};\mu _{ij}\right) }{\sigma ^6},\\ \frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2}= & {} \frac{1}{2\sigma ^4}\sum _{j=1}^{n_i}\frac{\partial ^2d\left( y_{ij};\mu _{ij}\right) }{\partial \alpha _i^2},\ \frac{\partial ^2\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2\partial \sigma ^2}= -\frac{1}{\sigma ^6}\sum _{j=1}^{n_i}\frac{\partial ^2d\left( y_{ij};\mu _{ij}\right) }{\partial \alpha _i^2},\\ \frac{\partial \lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }}= & {} \frac{\alpha _i^2}{2\sigma ^4_{\alpha }},\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }\partial \sigma ^2_{\alpha }}=-\frac{\alpha _i^2}{\sigma ^6_{\alpha }},\ \frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }}\\ \end{aligned}$$
$$\begin{aligned}= & {} \frac{1}{\sigma ^4_{\alpha }},\ \frac{\partial ^2\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2_{\alpha }\partial \sigma ^2_{\alpha }}=-\frac{2}{\sigma ^6_{\alpha }},\\ \frac{\partial \lambda _{PJPL}\left( \alpha _i\right) }{\partial \phi }= & {} \delta _i\left( \alpha _i-\frac{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k}{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\right) ,\\ \frac{\partial ^2\lambda _{PJPL}\left( \alpha _i\right) }{\partial \phi ^2}= & {} -\delta _i\left( \frac{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k^2}{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}-\left( \frac{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k}{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\right) ^2\right) ,\\ \end{aligned}$$
$$\begin{aligned} \frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \phi }= & {} -\delta _i\left[ \frac{e^{Z_i^T\gamma +\phi \alpha _i}\left( \phi ^2\alpha _i+2\phi \right) }{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}- \frac{e^{Z_i^T\gamma +\phi \alpha _i}\phi ^2\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}\right. \\&\quad -\,\frac{e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\left( 2\phi ^2\alpha _i+2\phi \right) }{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}\\&\quad \left. +\,\frac{2e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\phi ^2\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^3}\right] , \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2\lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \phi ^2}=\,-\delta _i&\left[ \frac{e^{Z_i^T\gamma +\phi \alpha _i}\phi ^2\left( \alpha _i^2\phi ^2+4\phi \alpha _i+2\right) }{\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\right. \\&\quad \left. -\, \frac{e^{Z_i^T\gamma +\phi \alpha _i}\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k\left( 2\alpha _i\phi ^2+4\phi \right) }{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}\right. \\ \end{aligned}$$
$$\begin{aligned}&\quad -\,\frac{e^{Z_i^T\gamma +\phi \alpha _i}\phi ^2\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k^2}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}\\&\quad \left. +\,2\frac{e^{Z_i^T\gamma +\phi \alpha _i}\phi ^2\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k\right) ^2}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^3}\right. \\&\quad \left. -\,4\frac{e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\alpha _i\left( \phi ^2\alpha _i+\phi \right) }{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}-2\frac{e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\left( 2\phi \alpha _i+1\right) }{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2}\right. \\ \end{aligned}$$
$$\begin{aligned}&\quad +\,4\frac{e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k^2\left( \phi ^2\alpha _i+2\phi \right) }{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^3}\\&\quad \left. +\,4\frac{e^{\left( 2Z_i^T\gamma +\phi \alpha _i\right) }\alpha _i\phi ^2\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^3}\right. \\&\quad +\,2\frac{e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\phi ^2\sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k^2}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^3}\\ \end{aligned}$$
$$\begin{aligned}&\quad \left. -\,6\frac{e^{2\left( Z_i^T\gamma +\phi \alpha _i\right) }\phi ^2\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\alpha _k\right) ^2}{\left( \sum _{k\in R\left( t_i\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^4}\right] .\\ \end{aligned}$$
The MLEs of \({\varTheta }'\) can be achieved by Newton-Raphson algorithm through approximated PJPL log-likelihoods.
For the standard error of estimated parameters, we derive
$$\begin{aligned} \mathscr {B}_1=-\left( \begin{array}{llllllll} \frac{\partial ^2 \lambda _{PJPL}}{\partial \beta \partial \beta ^T}&{} \frac{\partial ^2 \lambda _{PJPL}}{\partial \beta \partial \gamma ^T} &{}\frac{\partial ^2 \lambda _{PJPL}}{\partial \beta \partial \alpha ^T}\\ \frac{\partial ^2 \lambda _{PJPL}}{\partial \gamma \partial \beta ^T}&{} \frac{\partial ^2 \lambda _{PJPL}}{\partial \gamma \partial \gamma ^T} &{}\frac{\partial ^2 \lambda _{PJPL}}{\partial \gamma \partial \alpha ^T}\\ \frac{\partial ^2 \lambda _{PJPL}}{\partial \alpha \partial \beta ^T}&{} \frac{\partial ^2 \lambda _{PJPL}}{\partial \alpha \partial \gamma ^T} &{}\frac{\partial ^2 \lambda _{PJPL}}{\partial \alpha \partial \alpha ^T}\\ \end{array}\right) ,\ \mathscr {B}_2=-\left( \begin{array}{llllllll} \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2\partial \sigma ^2}&{} \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2\partial \sigma ^2_{\alpha }} &{}\frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2\partial \phi }\\ \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2_{\alpha }\partial \sigma ^2}&{} \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2_{\alpha }\partial \sigma ^2_{\alpha }} &{}\frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2_{\alpha }\partial \phi }\\ \frac{\partial ^2 LL_{PJPL}}{\partial \phi \partial \sigma ^2}&{} \frac{\partial ^2 LL_{PJPL}}{\partial \phi \partial \sigma ^2_{\alpha }} &{}\frac{\partial ^2 LL_{PJPL}}{\partial \phi \partial \phi }\\ \end{array}\right) \end{aligned}$$
where
$$\begin{aligned} \frac{\partial ^2\lambda _{PJPL}}{\partial \beta \partial \gamma ^T}= & {} 0,\ \frac{\partial ^2\lambda _{PJPL}}{\partial \beta \partial \alpha ^T}=\left( \frac{\partial ^2\lambda _{PJPL}}{\partial \beta \partial \alpha _1},\ldots , \frac{\partial ^2\lambda _{PJPL}}{\partial \beta \partial \alpha _n}\right) ,\ \frac{\partial ^2\lambda _{PJPL}}{\partial \beta \partial \alpha _i}\\= & {} -\frac{1}{2\sigma ^2}\sum _{j=1}^{n_i}\frac{\partial ^2 d\left( y_{ij};\mu _{ij}\right) }{\partial \beta \partial \alpha _i},\\ \frac{\partial ^2 d\left( y_{ij};\mu _{ij}\right) }{\partial \beta \partial \alpha _i}= & {} \left[ -\,2\frac{\partial d\left( y_{ij};\mu _{ij}\right) /\partial \alpha _i}{y_{ij}-\mu _{ij}}\right. \\ \end{aligned}$$
$$\begin{aligned}&\left. -\,2\frac{d\left( y_{ij};\mu _{ij}\right) \partial \mu _{ij}/\partial \alpha _i}{\left( y_{ij}-\mu _{ij}\right) ^2} -2\frac{\left( 1-2\mu _{ij}\right) \partial d\left( y_{ij};\mu _{ij}\right) /\partial \alpha _i}{\mu _{ij}\left( 1-\mu _{ij}\right) }\right. \\&+\,4\frac{d\left( y_{ij};\mu _{ij}\right) \partial \mu _{ij}/\partial \alpha _i}{\mu _{ij}\left( 1-\mu _{ij}\right) }+2\frac{d\left( y_{ij};\mu _{ij}\right) \left( 1-2\mu _{ij}\right) \partial \mu _{ij}/\partial \alpha _i}{\mu _{ij}^2\left( 1-\mu _{ij}\right) }\\&\left. -\,2\frac{d\left( y_{ij};\mu _{ij}\right) \left( 1-2\mu _{ij}\right) \partial \mu _{ij}/\partial \alpha _i}{\mu _{ij}\left( 1-\mu _{ij}\right) ^2}\right] \\&\cdot \mu _{ij}\left( 1-\mu _{ij}\right) X_{ij} +\left[ -2\frac{d\left( y_{ij};\mu _{ij}\right) }{y_{ij}-\mu _{ij}}-2\frac{d\left( y_{ij};\mu _{ij}\right) \left( 1-2\mu _{ij}\right) }{\mu _{ij}\left( 1-\mu _{ij}\right) }\right] \\&\frac{\partial \mu _{ij}}{\partial \alpha _i} \left( 1-2\mu _{ij}\right) X_{ij},\\ \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2\lambda _{PJPL}}{\partial \gamma \partial \alpha ^T}= & {} \left( \frac{\partial ^2\lambda _{PJPL}}{\partial \gamma \partial \alpha _1},\ldots , \frac{\partial ^2\lambda _{PJPL}}{\partial \gamma \partial \alpha _n}\right) ,\\ \frac{\lambda _{PJPL}}{\partial \gamma \alpha _i}= & {} -\,e^{Z_i^T\gamma +\phi \alpha _i}Z_i\phi \sum _{l=1}^i\frac{\delta _l}{\sum _{K\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}}\\&+\,e^{Z_i^T\gamma +\phi \alpha _i}\phi \sum _{l=1}^i\frac{\delta _l\sum _{K\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}Z_k}{\left( \sum _{K\in R\left( t_l\right) }e^{Z_k^T\gamma +\phi \alpha _k}\right) ^2},\\ \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2\partial \sigma ^2_{\alpha }}= & {} \frac{1}{2\sigma ^4_{\alpha }}\sum _{i=1}^n\frac{1}{\left( \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) \right) ^2} \frac{\partial \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2},\ \frac{\partial ^2 LL_{PJPL}}{\partial \sigma ^2_{\alpha }\partial \phi }\\ \end{aligned}$$
$$\begin{aligned}= & {} \frac{1}{2\sigma ^4_{\alpha }}\sum _{i=1}^n\frac{1}{\left( \lambda _{PJPL}^{\left( 2\right) }\left( \alpha _i\right) \right) ^2} \frac{\partial \lambda ^{\left( 2\right) }_{PJPL}\left( \alpha _i\right) }{\partial \phi },\\ \frac{\partial ^2 LL_{PJPL}}{\partial \phi \partial \sigma ^2}= & {} \frac{1}{2}\sum _{i=1}^n\frac{1}{\left( \lambda ^{\left( 2\right) }\left( \alpha _i\right) \right) ^2}\frac{\partial \lambda ^{\left( 2\right) }\left( \alpha _i\right) }{\partial \sigma ^2}\frac{\partial \lambda ^{\left( 2\right) }\left( \alpha _i\right) }{\partial \phi }. \end{aligned}$$
Appendix 2: Details of data generation
To generate the longitudinal observation y following the simplex distribution (5) with mean \(\mu \) and dispersion \(\sigma ^2\), we use the following property of simplex distributed random variable, \(y=\frac{M}{1+M}\), where \(M=I_M M_1+(1-I_M)M_2\) and \(I_M\) is a binary random variable with probability \(\mu \). That means M is a mixture random variable with probability \(\mu \) being \(M_1\) and probability \(1-\mu \) being \(M_2\). Here \(M_1\) is a reciprocal of inverse-gaussian random variable with mean \(\frac{1-\mu }{\mu }\) and shape parameter \(\frac{1}{\sigma ^2\mu ^2}\), and \(M_2\) follows the inverse gaussian distribution with mean \(\frac{\mu }{1-\mu }\) and shape parameter \(\frac{1}{\sigma ^2(1-\mu )^2}\).
We firstly generate binary random variable \(I_M\) with probability \(\mu \). If \(I_M=1\), generate the inverse-gaussian random variable \(M_1'\) with mean \(\frac{1-\mu }{\mu }\) and shape parameter \(\frac{1}{\sigma ^2\mu ^2}\), then obtain \(M_1=\frac{1}{M_1'}\) and set \(M=M_1\). If \(I_M=0\), generate \(M_2\) from the inverse gaussian with mean \(\frac{\mu }{1-\mu }\) and shape parameter \(\frac{1}{\sigma ^2(1-\mu )^2}\), then set \(M=M_2\). Then we transform M and obtain \(y=\frac{M}{1+M}\).
