Appendix
Expressions for Weibull baseline
In this case, we have the following expressions:
\( S_0(t)=e^{-{(\gamma _1 t)}^{1/\gamma _0}},f_0(t)=\frac{(\gamma _1 t)^{1/\gamma _0}}{\gamma _0 t}e^{-{(\gamma _1 t)}^{1/\gamma _0}}; \)
the derivatives of \(S_0\) are given by
$$\begin{aligned}&S_{0; \gamma _1} =\frac{(\gamma _1t_i)^{1/\gamma _0}\mathrm {log}(\gamma _1t_i)}{\gamma _0^2}S_0 ,~~ S_{0; \gamma _0} =-\frac{(\gamma _1t_i)^{1/\gamma _0}}{\gamma _0\gamma _1}S_0 ,~~ \\&S_{0; \gamma _0 \gamma _0} =S_{0; \gamma _0}\frac{[(\gamma _1t_i)^{1/\gamma _0}-1]\mathrm {log}(\gamma _1t_i)-2\gamma _0}{\gamma _0^2}, \\&S_{0; \gamma _1 \gamma _1} =S_{0; \gamma _1}\frac{[(\gamma _1t_i)^{1/\gamma _0}-1]\mathrm {log}(\gamma _1t_i)-\gamma _0}{\gamma _0^2} ,~~~ S_{0; \gamma _1\gamma _0} =S_{0; \gamma _1}\frac{1-\gamma _0-(\gamma _1t_i)^{1/\gamma _0}}{\gamma _0\gamma _1}; \end{aligned}$$
the derivatives of \(\log S_0\) are given by
$$\begin{aligned}&\frac{\partial \mathrm {log}S_0}{\partial \gamma _0}=\frac{(\gamma _1 t_i)^{1/\gamma _0}\mathrm {log}(\gamma _1 t_i)}{\gamma _0^2} ,~~ \frac{\partial \mathrm {log}S_0}{\partial \gamma _1}=-\frac{(\gamma _1 t_i)^{1/\gamma _0}}{\gamma _0\gamma _1} ,~~ \\&\frac{\partial ^2\mathrm {log} S_0}{\partial \gamma _0^2}=- \frac{\partial \mathrm {log}S_0}{\partial \gamma _0}\left( \frac{\mathrm {log}(\gamma _1 t_i)}{\gamma _0^2}+\frac{2}{\gamma _0} \right) , \\&\frac{\partial ^2\mathrm {log} S_0}{\partial \gamma _0\partial \gamma _1}=- \frac{\partial \mathrm {log}S_0}{\partial \gamma _1}\left( \frac{\mathrm {log}(\gamma _1 t_i)}{\gamma _0^2}+\frac{1}{\gamma _0}\right) ,~~ \frac{\partial ^2\mathrm {log} S_0}{\partial \gamma _1^2}=\frac{\partial \mathrm {log}S_0}{\partial \gamma _1}\left( \frac{1}{\gamma _0\gamma _1}-\frac{1}{\gamma _1}\right) ; \end{aligned}$$
the derivatives of \(\log f_0\) are given by
$$\begin{aligned}&\frac{\partial \mathrm {log} f_0}{\partial \gamma _0}=\frac{\partial \mathrm {log} S_0}{\partial \gamma _0}-\frac{1}{\gamma _0}\left( 1 +\frac{\mathrm {log}(\gamma _1t_i)}{\gamma _0}\right) ,~~ \frac{\partial \mathrm {log} f_0}{\partial \gamma _1}=\frac{1}{\gamma _0\gamma _1}+\frac{\partial \mathrm {log} S_0}{\partial \gamma _1} ,\\&\frac{\partial ^2\mathrm {log} f_0}{\partial \gamma _0^2}=\frac{\partial \mathrm {log} S_0^2}{\partial \gamma _0^2}+\frac{1}{\gamma _0^2}+\frac{2 \mathrm {log}(\gamma _1 t_i)}{\gamma _0^3} ,~~ \frac{\partial ^2\mathrm {log} f_0}{\partial \gamma _0\partial \gamma _1}=\frac{\partial \mathrm {log} S_0^2}{\partial \gamma _0 \partial \gamma _1}-\frac{1}{\gamma _0^2\gamma _1} ,~~ \\&\frac{\partial ^2\mathrm {log} f_0}{\partial \gamma _1 ^2}=\frac{\partial \mathrm {log}S_0^2}{\partial \gamma _1^2}-\frac{1}{\gamma _0\gamma _1^2}; \end{aligned}$$
the derivatives of S are given by
$$\begin{aligned}&\frac{\partial S}{ \partial \gamma _0}=\frac{\partial S_0}{ \partial \gamma _0} G_0 , ~~~~ \frac{\partial S}{ \partial \gamma _1} =\frac{\partial S_0}{ \partial \gamma _1} G_0 , ~~~~ \frac{\partial S}{\partial \alpha _l}= \frac{x_{il}F_0S}{G} , \\&\frac{\partial ^2 S}{\partial \gamma _0^2 } =\frac{\partial ^2 S_0}{\partial \gamma _0^2 }G_0-2\frac{\partial S}{ \partial \gamma _0}\frac{\partial S_0}{ \partial \gamma _0}G_1, \\&\frac{\partial ^2 S}{\partial \gamma _0 \partial \gamma _1} =\frac{\partial ^2 S_0}{\partial \gamma _0 \partial \gamma _1}G_0-2\frac{\partial S}{ \partial \gamma _0}\frac{\partial S_0}{ \partial \gamma _1}G_1 , ~~~~ \frac{\partial ^2 S}{\partial \gamma _1^2}=\frac{\partial ^2 S_0}{\partial \gamma _1 ^2}G_0-2\frac{\partial S}{ \partial \gamma _1}\frac{\partial S_0}{ \partial \gamma _1}G_1, \\&\frac{\partial ^2 S}{\partial \gamma _0 \partial \alpha _l}=\frac{\partial S }{\partial \gamma _0}x_{il}G_2,~~~ \frac{\partial ^2 S}{\partial \gamma _1 \partial \alpha _l}= \frac{\partial S}{ \partial \gamma _1} x_{il} G_2 ,~~~ \frac{\partial ^2 S}{\partial \alpha _l\partial \alpha _{l'}}=\frac{\partial S}{\partial \alpha _l}x_{il'}G_2; \end{aligned}$$
the derivatives of \(\log S\) are given by
$$\begin{aligned}&\frac{\partial \mathrm {log}S}{ \partial \gamma _0}=\frac{\partial \mathrm {log}S_0}{ \partial \gamma _0} \frac{1}{G } , ~~~~ \frac{\partial \mathrm {log} S}{ \partial \gamma _1} =\frac{\partial \mathrm {log}S_0}{ \partial \gamma _1} \frac{1}{G} , \\&\frac{\partial ^2 \mathrm {log}S}{\partial \gamma _0 ^2} =\left( \frac{\partial ^2 \mathrm {log}S_0}{\partial \gamma _0^2} -\frac{\partial \mathrm {log}S_0}{\partial \gamma _0} \frac{\partial S_0}{\partial \gamma _0} G_1\right) \frac{1}{G} ,\\&\frac{\partial ^2 \mathrm {log}S}{\partial \gamma _0 \partial \gamma _1} =\left( \frac{\partial ^2 \mathrm {log}S_0}{\partial \gamma _0\partial \gamma _1} -\frac{\partial \mathrm {log}S_0}{\partial \gamma _0} \frac{\partial S_0}{\partial \gamma _1} G_1\right) \frac{1}{G} , \\&\frac{\partial ^2 \mathrm {log}S}{\partial \gamma _1^2 }= \left( \frac{\partial ^2 \mathrm {log}S_0}{\partial \gamma _1^2} -\frac{\partial \mathrm {log}S_0}{\partial \gamma _1} \frac{\partial S_0}{\partial \gamma _1} G_1\right) \frac{1}{G} ,\\&\frac{\partial ^2 \mathrm {log}S}{\partial \gamma _0 \partial \alpha _{il}}=-\frac{\partial S}{\partial \gamma _0} x_{il} ,~~~ \frac{\partial ^2 \mathrm {log}S}{\partial \gamma _1 \partial \alpha _{il}}=- \frac{\partial S}{ \partial \gamma _1} x_{il} , \\&\frac{\partial \mathrm {log}S}{\partial \alpha _{il}}= \frac{x_{il}F_0}{G} ,~~~ \frac{\partial ^2 \mathrm {log}S}{\partial \alpha _{il}\partial \alpha _{il'}}=- \frac{\partial S}{\partial \alpha _{il}} x_{il'}, \end{aligned}$$
where \( G=1+S_0(y_ie^{\pmb {x}_i'\gamma _1}-1), \)
\(G_0=\frac{f}{f_0};~~G_1=\frac{y_ie^{\pmb {\beta }'\pmb {z}_i}-1}{G}, ~~ G_2= 2\frac{F_0}{G}-1, ,~~ G_3=(\gamma _1 t_i)^{1/\gamma _0} S_0G_1+1;\)
and finally, the derivatives of \(\log f_0\) are given by
$$\begin{aligned} \frac{\partial \mathrm {log}f}{\partial \gamma _0}&= \frac{\partial \mathrm {log} f_0}{\partial \gamma _0} -2 \frac{\partial S_0}{\partial \gamma _0} G_1, ~~ \frac{\partial \mathrm {log} f}{\partial \gamma _1} = \frac{\partial \mathrm {log} f_0}{\partial \gamma _1} -2 \frac{\partial S_0}{\partial \gamma _1} G_1, \\ \frac{\partial ^2\mathrm {log} f}{\partial \gamma _0^2}&= \frac{\partial ^2\mathrm {log}f_0}{\partial \gamma _0^2}-2 \frac{\partial ^2 S_0}{\partial \gamma _0^2} G_1+2 \left( \frac{\partial S_0}{\partial \gamma _0} G_1\right) ^2, \\ \frac{\partial ^2 \mathrm {log} f}{\partial \gamma _0 \partial \gamma _1}&= \frac{\partial ^2 \mathrm {log}f_0}{\partial \gamma _0 \partial \gamma _1}-2 \frac{\partial ^2 S_0}{\partial \gamma _0\partial \gamma _1} G_1+2 \frac{\partial S_0}{\partial \gamma _0}\frac{\partial S_0}{\partial \gamma _1}( G_1)^2, \\ \frac{\partial ^2 \mathrm {log} f}{\partial \gamma _0 \partial \alpha _l}&=-2 \frac{\partial S_0}{\partial \gamma _0} \frac{ x_{il}y_ie^{\pmb {\alpha }'\pmb {x}_i} }{G^2}, \\ \frac{\partial ^2 \mathrm {log} f}{\partial \gamma _1^2}&= \frac{\partial ^2 \mathrm {log}f_0}{\partial \gamma _1^2}-2 \frac{\partial ^2 S_0}{\partial \gamma _1^2} G_1+2 \left( \frac{\partial S_0}{\partial \gamma _1} G_1\right) ^2, \\ \frac{\partial ^2 \mathrm {log} f}{\partial \gamma _1 \partial \alpha _l}&=-2 \frac{\partial S_0}{\partial \gamma _1} \frac{ x_{il}y_ie^{\pmb {\alpha }'\pmb {x}_i}}{G^2} ,\\ \frac{\partial \mathrm {log}f}{\partial \alpha _l}&= x_{il}\left( 2\frac{ 1-S_0}{G}-1\right) , ~~~ \frac{\partial ^2 \mathrm {log}f}{\partial \alpha _l \partial \alpha _{l'}}=- \frac{2x_{il}x_{il'} S_0F_0y_ie^{\pmb {\alpha }'\pmb {x}_i}}{G^2 }. \end{aligned}$$
Expressions for Log-Logistic Baseline
In this case, we have the following expressions:
$$\begin{aligned} S_0=\frac{\gamma _0^{\gamma _1}}{t_i^{\gamma _1}+\gamma _0^{\gamma _1}} ,&~~ S=\frac{\gamma _0^{\gamma _1}y_ie^{\pmb {\alpha }'\pmb {x}}}{y_i \gamma _0^{\gamma _1}e^{\pmb {\alpha }'\pmb {x}}+t_i^{\gamma _1}}, ~~ \nonumber \\ f_0= \frac{\gamma _0^{\gamma _1}\gamma _1t_i^{\gamma _1-1}}{(t_i^{\gamma _1}+\gamma _0^{\gamma _1})^2} ,&~~ f= \frac{\gamma _0^{\gamma _1}\gamma _1t_i^{\gamma _1-1}e^{\pmb {\alpha }'\pmb {x}_i}}{(t_i^{\gamma _1}+\gamma _0^{\gamma _1}e^{\pmb {\alpha }'\pmb {x}_i})^2}. \end{aligned}$$
(49)
The derivatives of \(S(t_i,\pmb {\gamma })\) are given by,
$$\begin{aligned} \frac{\partial S_0(t_i;\pmb {\gamma })}{\partial \gamma _0}&=F_0S_0\frac{\gamma _1}{\gamma _0} ,~~~ \frac{\partial S_0(t_i;\pmb {\gamma })}{\partial \gamma _1}= F_0 S_0\mathrm {log}\frac{\gamma _0}{t_i} ,\\ \frac{\partial ^2 S_0(t_i;\pmb {\gamma })}{\partial \gamma _0 ^2}&=-\frac{\partial S_0(t_i;\pmb {\gamma })}{\partial \gamma _0}\frac{1+\gamma _1S_0(t_i;\pmb {\gamma })}{\gamma _0}+F_0S_0\frac{\gamma _1}{\gamma _0}F_0\frac{\gamma _1}{\gamma _0} ,\\ \frac{\partial ^2 S_0(t_i;\pmb {\gamma })}{\partial \gamma _0 \partial \gamma _1}&=\frac{\partial S_0(t_i;\pmb {\gamma })}{\partial \gamma _0}\left( \frac{1}{\gamma _1}+S_0(t_i;\pmb {\gamma })\mathrm {log}\frac{t_i}{\gamma _0}\right) +F_0S_0\frac{\gamma _1}{\gamma _0} S_0\mathrm {log}\frac{\gamma _0}{t_i} ,\\ \frac{\partial ^2 S_0(t_i;\pmb {\gamma })}{\partial \gamma _1 ^2}&=\frac{\partial S_0(t_i;\pmb {\gamma })}{\partial \gamma _1 }S_0(t_i;\pmb {\gamma })\mathrm {log}\frac{t_i}{\gamma _0}+F_0 \mathrm {log}\frac{\gamma _0}{t_i}F_0 S_0\mathrm {log}\frac{\gamma _0}{t_i}; \end{aligned}$$
the derivatives of \(\mathrm {log} S(t_i,\pmb {\gamma })\) are given by
$$\begin{aligned} \frac{\partial \mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _0}&=\frac{t_i^{\gamma _1}(\gamma _1/\gamma _0)}{\gamma _0^{\gamma _1}+t_i^{\gamma _1}} ,~~~ \frac{\partial \mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _1}= \frac{t_i^{\gamma _1}\mathrm {log}(\gamma _0/t_i)}{\gamma _0^{\gamma _1}+t_i^{\gamma _1}} ,\\ \frac{\partial ^2\mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _0^2}&=-\frac{\partial \mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _0}\frac{1+\gamma _1S_0(t_i;\pmb {\gamma })}{\gamma _0} ,~~~ \\ \frac{\partial \mathrm {log}S_0^2(t_i;\pmb {\gamma })}{\partial \gamma _0 \partial \gamma _1}&=\frac{\partial \mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _0}\left( \frac{1}{\gamma _1}+S_0(t_i;\pmb {\gamma })\mathrm {log}\frac{t_i}{\gamma _0}\right) ,\\ \frac{\partial ^2 \mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _1 ^2}&=\frac{\partial \mathrm {log}S_0(t_i;\pmb {\gamma })}{\partial \gamma _1 }S_0(t_i;\pmb {\gamma })\mathrm {log}\frac{t_i}{\gamma _0}; \end{aligned}$$
the derivatives of \(\mathrm {log} f(t_i,\pmb {\gamma })\) are given by
$$\begin{aligned}&\frac{\partial \mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _0}=\frac{\gamma _1}{\gamma _0}\frac{t_i^{\gamma _1}-\gamma _0^{\gamma _1}}{\gamma _0^{\gamma _1}+t_i^{\gamma _1}} ,~~ \frac{\partial \mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _1}=\frac{\gamma _0^{\gamma _1}-t_i^{\gamma _1}}{\gamma _0^{\gamma _1}+t_i^{\gamma _1}}\mathrm {log}\frac{t_i}{\gamma _0}+\frac{1}{\gamma _1} ,\\&\frac{\partial ^2\mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _0^2}=-\frac{\partial \mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _0}\frac{1}{\gamma _0}-\frac{2\gamma _0^{\gamma _1}t_i^{\gamma _1}}{(\gamma _0^{\gamma _1}+t_i^{\gamma _1})^2}\frac{\gamma _1^2}{\gamma _0^2} ,\\&\frac{\partial ^2 \mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _0\partial \gamma _1}=\frac{\partial \mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _0}\frac{1}{\gamma _1}+\frac{2\gamma _0^{\gamma _1}t_i^{\gamma _1}}{(\gamma _0^{\gamma _1}+t_i^{\gamma _1})^2}\frac{\gamma _1}{\gamma _0}\mathrm {log}\frac{t_i}{\gamma _0} ,\\&\frac{\partial ^2 \mathrm {log} f_0(t_i,\pmb {\gamma })}{\partial \gamma _1^2}=-\frac{2x_{il}\gamma _0^{\gamma _1}t_i^{\gamma _1}}{(\gamma _0^{\gamma _1}+t_i^{\gamma _1})^2}\left( \mathrm {log}\frac{t_i}{\gamma _0}\right) ^2-\frac{1}{\gamma _1^2}; \end{aligned}$$
the derivatives of \( S(t_i|y_i)\) are given by
$$\begin{aligned} \frac{\partial S(t_i)}{\partial \gamma _0}&=F(t_i)S(t_i)\frac{\gamma _1}{\gamma _0} ,~~~ \frac{\partial S(t_i)}{\partial \gamma _1}=-S(t_i) F(t_i)\mathrm {log}\frac{t_i}{\gamma _0} ,~~~ \frac{\partial S(t_i)}{\partial \alpha _{l}}=x_{il}F(t_i)S(t_i) ,\\ \frac{\partial ^2 S(t_i)}{\partial \gamma _0^2}&=\frac{\partial S(t_i)}{\partial \gamma _0} \frac{\gamma _1(F(t_i)-S(t_i))-1}{\gamma _0} ,~~~\\ \frac{\partial S^2(t_i)}{\partial \gamma _0 \partial \gamma _1}&=\frac{\partial S(t_i)}{\partial \gamma _0}\left( \frac{1}{\gamma _1}-(F(t_i)-S(t_i))\mathrm {log}\frac{t_i}{\gamma _0}\right) ,\\ \frac{\partial ^2 S(t_i)}{\partial \gamma _0 \partial \alpha _{h}}&=\frac{\partial S(t_i)}{\partial \gamma _0}(F(t_i)-S(t_i))x_{il} ,~~~~~~~ \frac{\partial ^2 S(t_i)}{\partial \gamma _1^2}=\frac{\partial S(t_i)}{\partial \gamma _1 }(S(t_i)-F(t_i))\mathrm {log}\frac{t_i}{\gamma _0} , \\ \frac{\partial ^2 S(t_i)}{\partial \gamma _1 \partial \alpha _{h}}&=\frac{\partial S(t_i)}{\partial \gamma _1}(F(t_i)-S(t_i))x_{il} ,~~~~~~~ \frac{\partial ^2 S(t_i)}{\partial \alpha _{l} \partial \alpha _{l'}}= \frac{\partial S(t_i)}{\partial \alpha _{l} }(F(t_i)-S(t_i))x_{il'}; \end{aligned}$$
the derivatives of \(\mathrm {log} S(t_i|y_i)\) are given by
$$\begin{aligned} \frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _0}&=F(t_i)\frac{\gamma _1}{\gamma _0} ,~~~ \frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _1}= F(t_i)\mathrm {log}\frac{\gamma _0}{t_i} ,~~~ \frac{\partial \mathrm {log}S(t_i)}{\partial \alpha _{l}}=x_{il}F(t_i) ,\\ \frac{\partial ^2 \mathrm {log}S(t_i)}{\partial \gamma _0^2}&=-\frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _0}\frac{1+\gamma _1S(t_i)}{\gamma _0} ,~~~ \frac{\partial ^2 \mathrm {log}S(t_i)}{\partial \gamma _0 \partial \gamma _1}=\frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _0}\left( \frac{1}{\gamma _1}+S(t_i)\mathrm {log}\frac{t_i}{\gamma _0}\right) ,\\ \frac{\partial ^2 \mathrm {log}S(t_i)}{\partial \gamma _0 \partial \alpha _{h}}&=-\frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _0}S(t_i)x_{il} ,~~~~~~~ \frac{\partial ^2 \mathrm {log}S(t_i)}{\partial \gamma _1^2}=\frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _1 }S(t_i)\mathrm {log}\frac{t_i}{\gamma _0} , \\ \frac{\partial ^2 \mathrm {log}S(t_i)}{\partial \gamma _1 \partial \alpha _{l}}&=-\frac{\partial \mathrm {log}S(t_i)}{\partial \gamma _1}S(t_i)x_{il} ,~~~~~~~ \frac{\partial ^2 \mathrm {log}S(t_i)}{\partial \alpha _{l} \partial \alpha _{l'}}=-x_{il'} \frac{\partial S(t_i)}{\partial \alpha _{l} }; \end{aligned}$$
and finally, the derivatives of \(\mathrm {log} f(t_i|y_i)\) are given by
$$\begin{aligned}&\frac{\partial \mathrm {log} f(t_i)}{\partial \gamma _0}=\frac{\gamma _1}{\gamma _0}V_i ,~~ \frac{\partial \mathrm {log} f(t_i)}{\partial \gamma _1}=\frac{1}{\gamma _1}-V_i\mathrm {log}\frac{t_i}{\gamma _0} ,~~~ \frac{\partial \mathrm {log} f(t_i)}{\partial \alpha _{l}}=x_{il}V_i ,\\&\frac{\partial ^2 \mathrm {log} f(t_i)}{\partial \gamma _0^2}=-\frac{\partial \mathrm {log} f(t_i)}{\partial \gamma _0}\frac{1}{\gamma _0}-W_i\frac{\gamma _1^2}{\gamma _0^2} ,~~~ \frac{\partial ^2 \mathrm {log} f(t_i)}{\partial \gamma _0\partial \gamma _1}=\frac{\partial \mathrm {log} f(t_i)}{\partial \gamma _0}\frac{1}{\gamma _1}+W_i\frac{\gamma _1}{\gamma _0}\mathrm {log}\frac{t_i}{\gamma _0} ,\\&\frac{\partial ^2 \mathrm {log} f(t_i)}{\partial \gamma _0\partial \alpha _{l}}=-W_i\frac{\gamma _1}{\gamma _0}x_{il} ,~~~ \frac{\partial ^2 \mathrm {log} f(t_i)}{\partial \gamma _1^2}=-W_i\left( \mathrm {log}\frac{t_i}{\gamma _0}\right) ^2-\frac{1}{\gamma _1^2} ,\\&\frac{\partial ^2 \mathrm {log} f(t_i)}{\partial \gamma _1\partial \alpha _{l}}=W_i\mathrm {log}\frac{t_i}{\gamma _0}x_{il} ,~~~ \frac{\partial \mathrm {log} f^2(t_i)}{\partial \alpha _{l}\partial \alpha _{l'}}=-x_{il}x_{il'}W_i, \end{aligned}$$
where \(V_i=F(t_i)-S(t_i)\),and \(W_i=\frac{2\gamma _0^{\gamma _1}y_ie^{\pmb {\alpha }'\pmb {x}}t_i^{\gamma _1}}{(\gamma _0^{\gamma _1}y_ie^{\pmb {\alpha }'\pmb {x}}+t_i^{\gamma _1})^2}\).
