Appendix A: First- and second-order derivatives of the Q-function
1.1 Destructive Bernoulli cure rate model
$$\begin{aligned} Q({\varvec{\theta }},\varvec{\pi }^{(k)})\propto & {} \displaystyle \sum _{I_1} \log \left( \frac{\eta _ip_if(t_i)}{1+\eta _i}\right) + \displaystyle \sum _{I_0} (1-\pi _i^{(k)}) \, \log \left( \frac{1+\eta _i(1-p_i)}{1+\eta _i}\right) \\&+\displaystyle \sum _{I_0}\pi _i^{(k)} \log \left( \frac{\eta _ip_iS(t_i)}{1+\eta _i}\right) .\\ \frac{\partial Q}{\partial \beta _{1l}}= & {} \displaystyle \sum _{I_1}\frac{x_{1il}}{1+\eta _i}-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{1il}\eta _ip_i}{(1+\eta _i)\{1+\eta _i(1-p_i)\}}\\&+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}}{1+\eta _i},\\ \frac{\partial Q}{\partial \beta _{2k}}= & {} \displaystyle \sum _{I_1}x_{2ik}(1-p_i)-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{2ik}\eta _ip_i(1-p_i)}{\{1+\eta _i(1-p_i)\}}\\&+ \displaystyle \sum _{I_0}\pi _i^{(k)}x_{2ik}(1-p_i),\\ \frac{\partial Q}{\partial \gamma _1}= & {} \displaystyle \sum _{I_1}E_i+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{\log (\gamma _2t_i)(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma ^2_1},\\ \frac{\partial Q}{\partial \gamma _2}= & {} \displaystyle \sum _{I_1}H_i-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1\gamma _2},\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \beta _{1l^{\prime }}}= & {} -\displaystyle \sum _{I_1}\frac{x_{1il}x_{1il^{\prime }}\eta _i}{(1+\eta _i)^2}-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{1il}x_{1il^{\prime }}\eta _ip_i\{1-\eta ^2_i(1-p_i)\}}{(1+\eta _i)^2\{1+\eta _i(1-p_i)\}^2}\\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}x_{1il^{\prime }}\eta _i}{(1+\eta _i)^2},\\ \frac{\partial ^2 Q}{\partial \beta _{2k}\partial \beta _{2k^{\prime }}}= & {} -\displaystyle \sum _{I_1}x_{2ik}x_{2ik^{\prime }}p_i(1-p_i)\\ \end{aligned}$$
$$\begin{aligned}&-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{2ik}x_{2ik^{\prime }}\eta _ip_i(1-p_i)^2\{1+\eta _i-\frac{p_i}{(1-p_i)}\}\{1+\eta _i(1-p_i)\}}{\{1+\eta _i(1-p_i)\}^2}\\&- \displaystyle \sum _{I_0}\pi _i^{(k)}x_{2ik}x_{2ik^{\prime }}p_i(1-p_i),\\ \frac{\partial ^2 Q}{\partial \gamma ^2_1}= & {} -\displaystyle \sum _{I_1}\left\{ \frac{2E_i}{\gamma _1}+\frac{1}{\gamma _1^2}+\frac{(\log (\gamma _2t_i))^2(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1^4}\right\} \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{\log (\gamma _2t_i)(\gamma _2t_i)^{\frac{1}{\gamma _1}}\{\log (\gamma _2t_i)+2\gamma _1\}}{\gamma ^4_1},\\ \frac{\partial ^2 Q}{\partial \gamma ^2_2}= & {} -\displaystyle \sum _{I_1}\left\{ \frac{H_i}{\gamma _2} + \frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1^2\gamma _2^2}\right\} +\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}(\gamma _1-1)}{\gamma ^2_1\gamma ^2_2},\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \beta _{2k}}= & {} -\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{1il}x_{2ik}\eta _ip_i(1-p_i)}{\{1+\eta _i(1-p_i)\}^2},\\ \frac{\partial ^2 Q}{\partial \gamma _1\partial \gamma _2}= & {} \displaystyle \sum _{I_1}\left\{ \frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}\log (\gamma _2t_1)}{\gamma ^2_1\gamma _2}-\frac{H_i}{\gamma _1}\right\} + \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma ^2_1\gamma _2}\left( 1+\frac{\log (\gamma _2t_1)}{\gamma _1}\right) ,\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \gamma _s}= & {} 0,\\ \frac{\partial ^2 Q}{\partial \beta _{2k}\partial \gamma _s}= & {} 0\\ \end{aligned}$$
for \(s=1,2\), where
$$\begin{aligned} E_i = \frac{\log (\gamma _2t_i)(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma ^2_1} - \frac{\log (\gamma _2t_i)}{\gamma ^2_1} - \frac{1}{\gamma _1}\quad \text {and} \quad H_i=\frac{1-(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1\gamma _2}. \end{aligned}$$
1.2 Destructive Poisson cure rate model
$$\begin{aligned} Q({\varvec{\theta }},\varvec{\pi }^{(k)})\propto & {} \displaystyle \sum _{I_1}\left\{ \log (\eta _ip_if(t_i))-\eta _ip_iF(t_i)\right\} - \, \displaystyle \sum _{I_0} (1-\pi _i^{(k)})\eta _ip_i\\&+\displaystyle \sum _{I_0}\pi _i^{(k)} \log \left\{ \exp (-\eta _ip_iF(t_i))-\exp (-\eta _ip_i)\right\} .\\ \frac{\partial Q}{\partial \beta _{1l}}= & {} \displaystyle \sum _{I_1}x_{1il}(1-\eta _ip_iF(t_i))-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{1il}\eta _ip_i - \displaystyle \sum _{I_0}\pi _i^{(k)}x_{1il}\eta _ip_iB_i,\\ \frac{\partial Q}{\partial \beta _{2k}}= & {} \displaystyle \sum _{I_1}x_{2ik}(1-p_i)(1-\eta _ip_iF(t_i))-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{2ik}\eta _ip_i(1-p_i)\\&- \displaystyle \sum _{I_0}\pi _i^{(k)}x_{2ik}\eta _ip_i(1-p_i)B_i,\\ \frac{\partial Q}{\partial \gamma _1}= & {} \displaystyle \sum _{I_1}\left( E_i -\eta _ip_iF_1^{\prime }(t_i)\right) -\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{\eta _ip_iF_1^{\prime }(t_i)}{C_i},\\ \frac{\partial Q}{\partial \gamma _2}= & {} \displaystyle \sum _{I_1}\left( H_i-\eta _ip_iF_2^{\prime }(t_i)\right) - \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{\eta _ip_iF_2^{\prime }(t_i)}{C_i},\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \beta _{1l^{\prime }}}= & {} -\displaystyle \sum _{I_1}x_{1il}x_{1il^{\prime }}\eta _ip_iF(t_i)-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{1il}x_{1il^{\prime }}\eta _ip_i\\&+\displaystyle \sum _{I_0}\pi _i^{(k)}x_{1il}x_{1il^{\prime }}\eta _ip_i\left\{ \eta _ip_i\left( B_i^*-B_i^2\right) +B_i\right\} ,\\ \frac{\partial ^2 Q}{\partial \beta _{2k} \partial \beta _{2k^{\prime }}}= & {} -\displaystyle \sum _{I_1}x_{2ik}x_{2ik^{\prime }}p_i(1-p_i)\left\{ \eta _i(1-p_i)F(t_i) + 1-\eta _ip_iF(t_i)\right\} \\&+\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{2ik}x_{2ik^{\prime }}\eta _ip_i(1-p_i)(2p_i-1)\\&+\displaystyle \sum _{I_0}\pi _i^{(k)}x_{1il}x_{2ik}\eta _ip_i(1-p_i)^2\left\{ \eta _ip_i(B_i^*-B_i^2)+\frac{(2p_i-1)B_i}{1-p_i}\right\} ,\\ \frac{\partial ^2 Q}{\partial \gamma ^2_1}= & {} -\displaystyle \sum _{I_1}\left\{ \frac{2E_i}{\gamma _1}+\frac{1}{\gamma _1^2}+\frac{(\log (\gamma _2t_i))^2(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1^4}+\eta _ip_iF_{11}^{\prime \prime }(t_i)\right\} \end{aligned}$$
$$\begin{aligned}&-\displaystyle \sum _{I_0}\pi _i^{(k)}\left\{ \frac{\eta _ip_iF_{11}^{\prime \prime }(t_i)}{C_i}+\frac{\left( \eta _ip_iF_1^{\prime }(t_i)\right) ^2(1-C_i)}{C_i^2}\right\} ,\\ \frac{\partial ^2 Q}{\partial \gamma ^2_2}= & {} -\displaystyle \sum _{I_1}\left\{ \frac{H_i}{\gamma _2} + \frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1^2\gamma _2^2} + \eta _ip_iF_{22}^{\prime \prime }(t_i)\right\} \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\left\{ \frac{\eta _ip_iF_{22}^{\prime \prime }(t_i)}{C_i}+\frac{\eta _i^2p_i^2\left( F_2^{\prime }(t_i)\right) ^2 (1-C_i)}{C_i^2}\right\} ,\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \beta _{2k}}= & {} -\displaystyle \sum _{I_1}x_{1il}x_{2ik}\eta _ip_i(1-p_i)F(t_i) - \displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{1il}x_{2ik}\eta _ip_i(1-p_i)\\&+\displaystyle \sum _{I_0}\pi _i^{(k)}x_{1il}x_{2ik}\eta _ip_i(1-p_i) \left\{ \eta _ip_i\left( B_i^*-B_i^2\right) -B_i\right\} ,\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \gamma _{s}}= & {} -\displaystyle \sum _{I_1}x_{1il}\eta _ip_iF_s^{\prime }(t_i) - \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}\eta _ip_iF_s^{\prime }(t_i)}{C_i}\left\{ 1-\frac{\eta _ip_iS(t_i)(1-C_i)}{C_i}\right\} ,\\ \frac{\partial ^2 Q}{\partial \beta _{2k}\partial \gamma _{s}}= & {} -\displaystyle \sum _{I_1}x_{2ik}\eta _ip_i(1-p_i)F_s^{\prime }(t_i) - \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{2ik}\eta _ip_i(1-p_i)F_s^{\prime }(t_i)}{C_i}\\&\times \left\{ 1-\frac{\eta _ip_iS(t_i)(1-C_i)}{C_i}\right\} ,\\ \frac{\partial ^2 Q}{\partial \gamma _1\partial \gamma _2}= & {} \displaystyle \sum _{I_1}\left\{ \frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}\log (\gamma _2t_i)}{\gamma _1^3\gamma _2}-\frac{H_i}{\gamma _1} - \eta _ip_iF_{12}^{\prime \prime }(t_i)\right\} - \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{\eta _ip_iF_{12}^{\prime \prime }(t_i)}{C_i} \\&+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{\eta ^2_ip^2_iF_1^{\prime }(t_i)F_2^{\prime }(t_i)\left( C_i-1\right) }{C_i^2} \end{aligned}$$
for \(s=1, 2\), where \(E_i\) and \(H_i\) are as defined before with
$$\begin{aligned} B_i= & {} \frac{S_{\text {pop}}(t_i)F(t_i)-p_{0i}}{S_{\text {pop}}(t_i)-p_{0i}}, \quad B_i^*=\frac{S_{\text {pop}}(t_i)F^2(t_i)-p_{0i}}{S_{\text {pop}}(t_i)-p_{0i}},\\ C_i= & {} 1-\exp (-\eta _ip_iS(t_i)),\\ F_1^{\prime }(t_i)= & {} \frac{\partial F(t_i)}{\partial \gamma _1}=-\frac{S(t_i)(\gamma _2t_i)^{\frac{1}{\gamma _1}}\log (\gamma _2t_i)}{\gamma _1^2}, \qquad F_2^{\prime }(t_i)=\frac{\partial F(t_i)}{\partial \gamma _2}=\frac{S(t_i)(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1\gamma _2},\\ F_{11}^{\prime \prime }(t_i)= & {} \frac{\partial ^2 F(t_i)}{\partial \gamma _1^2}=F_1^{\prime }(t_i)\left( E_i-\frac{1}{\gamma _1}\right) , \quad F_{22}^{\prime \prime }(t_i)=\frac{\partial ^2 F(t_i)}{\partial \gamma _2^2}=F_2^{\prime }(t_i)\left( H_i-\frac{1}{\gamma _2}\right) ,\\ F_{12}^{\prime \prime }(t_i)= & {} \frac{\partial ^2 F(t_i)}{\partial \gamma _1\partial \gamma _2}=F_2^{\prime }(t_i)E_i. \end{aligned}$$
1.3 Destructive COM-Poisson cure rate model
$$\begin{aligned} \mathbf Q ({\varvec{\theta }},\varvec{\pi }^{(k)})\propto & {} \displaystyle \sum _{I_1}\left[ \log (p_if(t_i)) + \log z_{f10}-\log \{z(1-p_iF(t_i))\}\right] \\&+\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\left( \log z_p-\log z\right) +\displaystyle \sum _{I_0}\pi _i^{(k)}\log \left( \frac{z_f-z_p}{z}\right) .\\ \frac{\partial Q}{\partial \beta _{1l}}= & {} \displaystyle \sum _{I_1}x_{1il}\left\{ \frac{z_{f20}}{z_{f10}} - \frac{z_1}{z}\right\} +\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{1il}\left\{ \frac{z_{p10}}{z_p} - \frac{z_1}{z}\right\} \\&+\displaystyle \sum _{I_0}\pi _i^{(k)} x_{1il}\left\{ \frac{z_{f10} - z_{p10} }{z_f-z_p}-\frac{z_1}{z}\right\} ,\\ \frac{\partial Q}{\partial \beta _{2k}}= & {} \displaystyle \sum _{I_1}x_{2ik}(1-p_i)\left\{ \frac{1}{1-p_iF(t_i)}-p_iF(t_i)\left( \frac{z_{f21}}{z_{f10}}\right) \right\} \\&-\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{2ik}p_i(1-p_i)z_{p11}}{z_p}-\displaystyle \sum _{I_0}\pi _i^{(k)}x_{2ik}p_i(1-p_i) \end{aligned}$$
$$\begin{aligned}&\times \left\{ \frac{ F(t_i)z_{f11} - z_{p11}}{z_f-z_p}\right\} ,\\ \frac{\partial Q}{\partial \gamma _1}= & {} \displaystyle \sum _{I_1}\left\{ E_i - p_iF_1^{\prime }(t_i)\left( \frac{z_{f21}}{z_{f10}} - \frac{1}{1-p_iF(t_i)}\right) \right\} - \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{p_iF_1^{\prime }(t_i)z_{f11}}{z_f-z_p},\\ \frac{\partial Q}{\partial \gamma _2}= & {} \displaystyle \sum _{I_1}\left\{ H_i-p_iF_2^{\prime }(t_i)\left( \frac{z_{f21}}{z_{f10}}-\frac{1}{1-p_iF(t_i)}\right) \right\} -\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{p_iF_2^{\prime }(t_i)z_{f11}}{z_f-z_p},\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \beta _{1l^{\prime }}}= & {} \displaystyle \sum _{I_1}x_{1il}x_{1il^{\prime }}\left\{ \frac{z_{f30}}{z_{f10}}-\left( \frac{z_{f20}}{z_{f10}} \right) ^2+\left( \frac{z_1}{z}\right) ^2-\frac{z_2}{z}\right\} \\&+\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{1il}x_{1il^{\prime }}\left\{ \frac{z_{p20}}{z_p} - \left( \frac{z_{p10}}{z_p}\right) ^2+\left( \frac{z_1}{z}\right) ^2-\frac{z_2}{z}\right\} \end{aligned}$$
$$\begin{aligned}&+\displaystyle \sum _{I_0}\pi _i^{(k)} x_{1il}x_{1il^{\prime }}\left\{ \frac{z_{f20} - z_{p20}}{z_f-z_p}-\left( \frac{z_{f10} - z_{p10}}{z_f-z_p}\right) ^2+\left( \frac{z_1}{z}\right) ^2 - \frac{z_2}{z}\right\} ,\\ \frac{\partial ^2 Q}{\partial \beta _{2k}\partial \beta _{2k^{\prime }}}= & {} \displaystyle \sum _{I_1}\frac{x_{2ik}x_{2ik^{\prime }}(1-p_i)^2}{(1-p_iF(t_i))^2}\left\{ p_iF(t_i) - \frac{p_i(1-p_iF(t_i))}{1-p_i}\right\} \\&+\displaystyle \sum _{I_1}\frac{x_{2ik}x_{2ik^{\prime }}p^2_iF(t_i)(1-p_i)z_{f21}}{z_{f10}}\\&+\displaystyle \sum _{I_1}\frac{x_{2ik}x_{2ik^{\prime }}p_iF(t_i)(1-p_i)^2}{z_{f10}}\left\{ p_iF(t_i)\left( z_{f22}-\frac{z_{f21}^2}{z_{f10}}\right) -z_{f21}{}\right\} \\&+\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{2ik}x_{2ik^{\prime }}(1-p_i)^2p_i(p_iz_{p12}-z_{p11})}{z_p}\\&+\displaystyle \sum _{I_0}(1-\pi _i^{(k)})\frac{x_{2ik}x_{2ik^{\prime }}p_i^2(1-p_i)z_{p11}}{z_p}\left\{ 1-(1-p_i)\frac{z_{p11}}{z_p}\right\} \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{2ik}x_{2ik^{\prime }}p_i(1-p_i)^2}{z_f-z_p}\\&\times \left\{ F(t_i)z_{f11}-z_{p11}-p_i (F^2(t_i)z_{f12} - z_{p12})\right\} \\&+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{2ik}x_{2ik^{\prime }}p^2_i(1-p_i)(F(t_i)z_{f11}-z_{p11})}{z_f-z_p}\\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{2ik}x_{2ik^{\prime }}p^2_i(1-p_i)^2(F(t_i)z_{f11}-z_{p11})^2}{(z_f-z_p)^2}, \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \gamma _1^2}= & {} -\displaystyle \sum _{I_1}\left\{ \frac{2E_i}{\gamma _1}+\frac{(\log (\gamma _2t_i))^2(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma ^4_1}+\frac{1}{\gamma ^2_1}+p_iF_{11}^{\prime \prime }(t_i)\frac{z_{f21}}{z_{f10}}\right\} \\&+\displaystyle \sum _{I_1}\left[ p^2_i(F_1^{\prime }(t_i))^2\left\{ \frac{z_{f22}}{z_{f10}} - \left( \frac{z_{f21}}{z_{f10}}\right) ^2\right\} + \frac{p_i}{1-p_iF(t_i)}\right. \\&\left. \times \left\{ F_{11}^{\prime \prime }+ \frac{p_i\left( F_1^{\prime }(t_i)\right) ^2}{1-p_iF(t_i)}\right\} \right] \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{p_i}{z_f-z_p}\left\{ F_{11}^{\prime \prime }(t_i)z_{f11}-p_i\left( F_1^{\prime }(t_i)\right) ^2\left( z_{f12}-\frac{z_{f11}^2}{z_f-z_p}\right) \right\} , \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 Q}{\partial \gamma _2^2}= & {} -\displaystyle \sum _{I_1}\left\{ \frac{H_i}{\gamma _2}+\frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}}{\gamma _1^2\gamma _2^2}+p_iF_{22}^{\prime \prime }(t_i)\frac{z_{f21}}{z_{f10}}\right\} \\&+\displaystyle \sum _{I_1}\left[ p^2_i(F_2^{\prime }(t_i))^2\left\{ \frac{z_{f22}}{z_{f10}}-\left( \frac{z_{f21}}{z_{f10}}\right) ^2\right\} +\frac{p_i}{1-p_iF(t_i)}\right. \\&\times \left. \left\{ F_{22}^{\prime \prime }+\frac{p_i\left( F_2^{\prime }(t_i)\right) ^2}{1-p_iF(t_i)}\right\} \right] \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{p_i}{z_f-z_p}\left\{ F_{22}^{\prime \prime }(t_i)z_{f11}-p_i\left( F_2^{\prime }(t_i)\right) ^2\left( z_{f12}-\frac{z_{f11}^2}{z_f-z_p}\right) \right\} ,\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \beta _{2k}}= & {} \displaystyle \sum _{I_1}\frac{x_{1il}x_{2ik}p_iF(t_i)(1-p_i)}{z_{f10}}\left( \frac{z_{f21} z_{f20}}{z_{f10}}-z_{f31}\right) \\&+\displaystyle \sum _{I_0}(1-\pi _i^{(k)})x_{1il}x_{2ik}p_i(1-p_i)\left\{ \frac{z_{p11}z_{p10}}{z_p^2} - \frac{z_{p21}}{z_p}\right\} \\&+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}x_{2ik}p_i(1-p_i)}{\left( z_f-z_p\right) ^2}\left[ z_{f10}\left\{ F(t_i)z_{f11}-z_{p11}\right\} \right. \\&\left. -z_{p10}\left\{ F(t_i)z_{f11}-z_{p11}\right\} \right] \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}x_{2ik}p_i(1-p_i) (F(t_i)z_{f21}-z_{p21})}{z_f-z_p},\\ \frac{\partial ^2 Q}{\partial \beta _{1l}\partial \gamma _s}= & {} \displaystyle \sum _{I_1}x_{1il}p_iF_s^{\prime }(t_i)\left\{ (1-p_iF(t_i))\left( \frac{z_{f21}}{z_{f10}}\right) ^2 - \frac{z_{f31}}{z_{f10}}\right\} \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}p_i(1-p_i)F_s^{\prime }(t_i)z_{f11} z_{p11}}{\left( z_f-z_p\right) ^2}\\&+ \displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}p_i(1-p_iF(t_i))F_s^{\prime }(t_i)z_{f11}^2}{\left( z_f-z_p\right) ^2}\\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{1il}p_i(1-p_i)F_s^{\prime }(t_i)z_{f21}}{z_f-z_p},\\ \frac{\partial ^2 Q}{\partial \beta _{2k}\partial \gamma _s}= & {} -\displaystyle \sum _{I_1}x_{2ik}p_iF_s^{\prime }(t_i)(1-p_i)\left\{ \frac{z_{f21}}{z_{f10}} - p_iF(t_i)\frac{z_{f22}}{z_{f10}}\right\} \\&-\displaystyle \sum _{I_1}x_{2ik}p_iF_s^{\prime }(t_i)(1-p_i)\left\{ p_iF(t_i)\left( \frac{z_{f21}}{z_{f10}}\right) ^2 - \frac{1}{\left( 1-p_iF(t_i)\right) ^2}\right\} \\&-\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{2ik}p^2_iF_s^{\prime }(t_i)(1-p_i)}{(z_f-z_p)^2}\left\{ F(t_i)z_{f11}^2-z_{f11}z_{p11}\right\} \end{aligned}$$
$$\begin{aligned}&+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{x_{2ik}p_iF_s^{\prime }(t_i)(1-p_i)}{z_f-z_p}\left\{ p_iF(t_i)z_{f12}-z_{f11}\right\} ,\\ \frac{\partial ^2 Q}{\partial \gamma _1\partial \gamma _2}= & {} \displaystyle \sum _{I_1}\left\{ \frac{p_iF_{12}^{\prime \prime }(t_i)}{(1-p_iF(t_i))}-\frac{H_i}{\gamma _1}\right\} \\&+\displaystyle \sum _{I_1}\left\{ \frac{(\gamma _2t_i)^{\frac{1}{\gamma _1}}\log (\gamma _2t_i)}{\gamma ^3_1\gamma _2} + \frac{p^2_iF_1^{\prime }(t_i)F_2^{\prime }(t_i)}{(1-p_iF(t_i))^2}\right\} \\&+\displaystyle \sum _{I_1}p^2_iF_1^{\prime }(t_i)F_2^{\prime }(t_i)\frac{z_{f22}}{z_{f10}}-\displaystyle \sum _{I_1}p_iF_{12}^{\prime \prime }(t_i)\frac{z_{f21}}{z_{f10}}\\&-\displaystyle \sum _{I_1}p^2_iF_1^{\prime }(t_i)F_2^{\prime }(t_i)\left( \frac{z_{f21}}{z_{f10}}\right) ^2\\&+\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{p^2_iF_1^{\prime }(t_i)F_2^{\prime }(t_i)}{z_f-z_p}\left\{ z_{f12} - \frac{z_{f11}^2}{z_f-z_p}\right\} -\displaystyle \sum _{I_0}\pi _i^{(k)}\frac{p_iF_{12}^{\prime \prime }(t_i)z_{f11}}{z_f-z_p} \end{aligned}$$
for \(s=1, 2\), where \(E_i, \;H_i,\;F_{1}^{\prime }(t_i), \;F_{2}^{\prime }(t_i), \;F_{11}^{\prime \prime }(t_i), \;F_{12}^{\prime \prime }(t_i) \; \text {and} \; F_{22}^{\prime \prime }(t_i)\) are all as defined before with
$$\begin{aligned} z= & {} Z(\eta _i ,\phi ),\quad z_p=Z[\eta _i(1-p_i),\phi ],\quad z_F=Z[\eta _i(1-p_iF(t_i)),\phi ],\\ z_{2}= & {} \sum ^{\infty }_{j=1}\frac{j^2\eta _i^j}{(j!)^{\phi }},\quad z_{1}=\sum ^{\infty }_{j=1}\frac{j\eta _i^j}{(j!)^{\phi }},\\ z_{p10}= & {} \sum ^{\infty }_{j=1}\frac{j[\eta _i(1-p_i)]^j}{(j!)^{\phi }},\\ z_{p11}= & {} \sum ^{\infty }_{j=1}\frac{j\eta _i^j(1-p_i)^{j-1}}{(j!)^{\phi }},\quad z_{p20}=\sum ^{\infty }_{j=1}\frac{j^2[\eta _i(1-p_i)]^j}{(j!)^{\phi }},\\ z_{p21}= & {} \sum ^{\infty }_{j=1}\frac{j^2\eta _i^j(1-p_i)^{j-1}}{(j!)^{\phi }},\\ z_{p12}= & {} \sum ^{\infty }_{j=2}\frac{j(j-1)\eta _i^j(1-p_i)^{j-2}}{(j!)^{\phi }},\quad z_{f10}=\sum ^{\infty }_{j=1}\frac{j[\eta _i(1-p_iF(t_i))]^j}{(j!)^{\phi }},\\ z_{f11}= & {} \sum ^{\infty }_{j=1}\frac{j\eta _i^j(1-p_iF(t_i))^{j-1}}{(j!)^{\phi }},\\ z_{f20}= & {} \sum ^{\infty }_{j=1}\frac{j^2[\eta _i(1-p_iF(t_i))]^j}{(j!)^{\phi }},\quad z_{f21}=\sum ^{\infty }_{j=1}\frac{j^2\eta _i^j(1-p_iF(t_i))^{j-1}}{(j!)^{\phi }},\\ z_{f12}= & {} \sum ^{\infty }_{j=2}\frac{j(j-1)\eta _i^j(1-p_iF(t_i))^{j-2}}{(j!)^{\phi }},\\ z_{f22}= & {} \sum ^{\infty }_{j=2}\frac{j^2(j-1)\eta _i^j(1-p_iF(t_i))^{j-2}}{(j!)^{\phi }},\quad z_{f31}=\sum ^{\infty }_{j=1}\frac{j^3\eta _i^j(1-p_iF(t_i))^{j-1}}{(j!)^{\phi }},\\ z_{f30}= & {} \sum ^{\infty }_{j=1}\frac{j^3[\eta _i(1-p_iF(t_i))]^j}{(j!)^{\phi }}. \end{aligned}$$
