Appendix A
The second and mixed partial derivatives of \(\alpha \),\(\beta \), and \(\lambda \) are obtained from (6) as the following:
$$\begin{aligned} \frac{\partial ^{2}L}{\partial \alpha ^{2}}&=-\frac{m_{1+}m_{2}}{\alpha ^{2} }-\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}\left[ \frac{\partial \omega _{3}(\alpha ;t_{ji}) }{\partial \alpha }\right. \\&\quad \ \left. -(\beta -1)\frac{\partial ^{2}\omega _{1}(\alpha ;t_{ji}) }{\partial \alpha ^{2}}-[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}}\right] , \\ \frac{\partial ^{2}L}{\partial \alpha \partial \beta }&=\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}\left[ \frac{\partial \omega _{1}(\alpha ;t_{ji}) }{\partial \alpha }+[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha \partial \beta }\right] , \\ \frac{\partial ^{2}L}{\partial \alpha \partial \lambda }&=\sum _{i=1}^{m_{2}}r_{2i}\frac{\partial \omega _{2}(\alpha ,\beta ;t_{2i})}{ \partial \alpha }, \\ \frac{\partial ^{2}L}{\partial \beta ^{2}}&=-\frac{m_{1+}m_{2}}{\beta ^{2}} +\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{2}}, \\ \frac{\partial ^{2}L}{\partial \beta \partial \lambda }&=\sum _{i=1}^{m_{2}}r_{2i}\frac{\partial \omega _{2}(\alpha ,\beta ;t_{2i})}{ \partial \beta }, \\ \frac{\partial ^{2}L}{\partial \lambda ^{2}}&=-\frac{m_{2}}{\lambda ^{2}}. \end{aligned}$$
where
$$\begin{aligned} \frac{\partial \omega _{3}(\alpha ;t_{ji})}{\partial \alpha }= & {} \omega _{3}(\alpha ;t_{ji})\ln t_{ji}, \\ \frac{\partial ^{2}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{2}}= & {} D_{1}(\alpha ;t_{ji})\frac{\partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha }, \\ \frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}}= & {} D_{2}(\alpha ,\beta ;t_{ji})\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }, \\ \frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{2}}= & {} D_{3}(\alpha ,\beta ;t_{ji})\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta }, \\ \frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha \partial \beta }= & {} \frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{ \partial \alpha }\left( \frac{1}{\beta }+D_{3}(\alpha ,\beta ;t_{ji})\right) , \\ D_{1}(\alpha ;t_{ji})= & {} \ln t_{ji}-\omega _{3}(\alpha ;t_{ji})[1+\omega _{4}(\alpha ;t_{ji})], \\ D_{2}(\alpha ,\beta ;t_{ji})= & {} D_{1}(t_{ji})+\beta \omega _{3}(\alpha ;t_{ji})\omega _{4}(\alpha ;t_{ji})[1+\omega _{5}(\alpha ,\beta ;t_{ji})], \\ D_{3}(\alpha ,\beta ;t_{ji})= & {} \omega _{1}(\alpha ;t_{ji})[1+\omega _{5}(\alpha ,\beta ;t_{ji})] \end{aligned}$$
and \(\omega _{1}(\alpha ;t_{ji}),\omega _{3}(\alpha ;t_{ji}),\omega _{4}(\alpha ;t_{ji}),\omega _{5}(\alpha ,\beta ;t_{ji}),\frac{\partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha },\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }\) and \(\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta }~\)are as given in (5), (7), and (8).
Appendix B
In a three-parameter case \(U=U(\alpha ,\beta ,\lambda )\), Lindley’s approximation form reduces to
$$\begin{aligned} E(U(\alpha ,\beta ,\lambda )|\mathbf {t})&=U+[U_{1}a_{1}+U_{2}a_{2}+U_{3}a_{3}+a_{4}+a_{5}] \nonumber \\&\quad \ +\frac{1}{2}[\psi _{1}(U_{1}\sigma _{11}+U_{2}\sigma _{12}+U_{3}\sigma _{13})+\psi _{2}(U_{1}\sigma _{21}+U_{2}\sigma _{22}+U_{3}\sigma _{23}) \nonumber \\&\quad \ +\psi _{3}(U_{1}\sigma _{31}+U_{2}\sigma _{32}+U_{3}\sigma _{33})]. \end{aligned}$$
(10)
where\(\ U_{i}=\frac{\partial U}{\partial \xi _{i}},\sigma _{ij}~\)is the element \((i,\ j)\) in the variance–covariance matrix \((-\ L_{ij})\), \( i,j=1,2,3\), and
$$\begin{aligned} a_{i}= & {} \rho _{1}\sigma _{i1}+\rho _{2}\sigma _{i2}+\rho _{3}\sigma _{i3},~i=1,2,3\\ a_{4}= & {} U_{12}\sigma _{12}+U_{13}\sigma _{13}+U_{23}\sigma _{23},\\ a_{5}= & {} \frac{1}{2}(U_{11}\sigma _{11}+U_{22}\sigma _{22}+U_{33}\sigma _{33}),\\ \psi _{1}= & {} \sigma _{11}L_{111}+2(\sigma _{12}L_{121}+\sigma _{13}L_{131}+\sigma _{23}L_{231})+\sigma _{22}L_{221}+\sigma _{33}L_{331},\\ \psi _{2}= & {} \sigma _{11}L_{112}+2(\sigma _{12}L_{122}+\sigma _{13}L_{132}+\sigma _{23}L_{232})+\sigma _{22}L_{222}+\sigma _{33}L_{332},\\ \psi _{3}= & {} \sigma _{11}L_{113}+2(\sigma _{12}L_{123}+\sigma _{13}L_{133}+\sigma _{23}L_{233})+\sigma _{22}L_{223}+\sigma _{33}L_{333}, \hbox { where }\\ \rho _{i}= & {} \frac{\partial \rho }{\partial \xi _{i}},U_{ij}=\frac{\partial ^{2}U}{\partial \xi _{i}\partial \xi _{j}},L_{ijk}=\frac{\partial ^{3}L}{ \partial \xi _{i}\partial \xi _{j}\partial \xi _{k}}. \end{aligned}$$
To apply Lindley’s approximation form (B.1), we first observe
$$\begin{aligned} \rho (\alpha ,\beta ,\lambda )= & {} \ln \pi (\alpha ,\beta ,\lambda ) \\\propto & {} -\mu \ln \alpha +(\mu -1)\ln \beta -\ln \lambda -\left( \frac{\alpha ^{2}+\beta \nu }{\alpha \nu }\right) . \end{aligned}$$
therefore,
$$\begin{aligned} \rho _{1}=\frac{\partial \rho }{\partial \alpha }=-\frac{\mu }{\alpha }- \frac{1}{\nu }+\frac{\beta }{\alpha ^{2}},\ \ \ \ \ \ \rho _{2}=\frac{ \partial \rho }{\partial \beta }=\frac{\mu -1}{\beta }-\frac{1}{\alpha },\ \ \ \ \ \ \rho _{3}=\frac{\partial \rho }{\partial \lambda }=-\frac{1}{\lambda }. \end{aligned}$$
and
$$\begin{aligned} L_{111}= & {} \frac{\partial ^{3}L}{\partial \alpha ^{3}}=\frac{2(m_{1+}m_{2})}{ \alpha ^{3}}-\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}\left[ \frac{\partial ^{2}\omega _{3}(\alpha ;t_{ji})}{\partial \alpha ^{2}}-(\beta -1)\frac{\partial ^{3}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{3}}\right. \\&\left. -[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{3}}\right] , \\ L_{112}= & {} L_{121}=\frac{\partial ^{3}L}{\partial \alpha ^{2}\partial \beta } =\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}\left[ \frac{\partial ^{2}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{2}}+[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}\partial \beta } \right] , \\ L_{113}= & {} L_{131}=\frac{\partial ^{3}L}{\partial \alpha ^{2}\partial \lambda }=\sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha ^{2}}, \\ L_{123}= & {} L_{132}=L_{231}=\frac{\partial ^{3}L}{\partial \alpha \partial \beta \partial \lambda }=\sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha \partial \beta }, \\ L_{221}= & {} L_{122}=\frac{\partial ^{3}L}{\partial \beta ^{2}\partial \alpha } =\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{2}\partial \alpha }, \\ L_{222}= & {} \frac{\partial ^{3}L}{\partial \beta ^{3}}=\frac{2(m_{1+}m_{2})}{ \beta ^{3}}+\sum _{j=1}^{2}\sum _{i=1}^{m_{j}}[r_{ji}\lambda ^{j-1}-1]\frac{ \partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{3}}, \\ L_{223}= & {} L_{232}=\frac{\partial ^{3}L}{\partial \beta ^{2}\partial \lambda }=\sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \beta ^{2}}, \\ L_{331}= & {} L_{133}=\frac{\partial ^{3}L}{\partial \lambda ^{2}\partial \alpha }=0,\\ L_{332}= & {} L_{233}=\frac{\partial ^{3}L}{\partial \lambda ^{2}\partial \beta }=0, \\ L_{333}= & {} \frac{\partial ^{3}L}{\partial \lambda ^{3}}=\frac{2m_{2}}{ \lambda ^{3}}. \end{aligned}$$
where
$$\begin{aligned} \frac{\partial ^{2}\omega _{3}(\alpha ;t_{ji})}{\partial \alpha ^{2}}= & {} \frac{\partial \omega _{3}(\alpha ;t_{ji})}{\partial \alpha }\ln t_{ji} \\ \frac{\partial ^{3}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{3}}= & {} \left[ \frac{\partial ^{2}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{2}}-\left( \frac{ \partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha }\right) ^{2}\right] D_{1}(\alpha ;t_{ji}) \\&-\frac{\partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha }\frac{ \partial \omega _{3}(\alpha ;t_{ji})}{\partial \alpha }, \\ \frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{3}}= & {} \left[ \frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}}-\left( \frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha } \right) ^{2}\right] D_{2}(\alpha ,\beta ;t_{ji}) \\&-\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }\frac{ \partial \omega _{3}(\alpha ;t_{ji})}{\partial \alpha }+(\beta -1)D_{1}(\alpha ;t_{ji}) \\&\times \frac{\partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha }\frac{ \partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }, \\ \frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}\partial \beta }= & {} \left[ D_{2}(\alpha ,\beta ;t_{ji})-\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }\right] \frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha \partial \beta } \\&+\frac{\partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha }\frac{ \partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }, \\ \frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{2}\partial \alpha }= & {} D_{3}(\alpha ,\beta ;t_{ji})\left[ \frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta \partial \alpha } +\omega _{5}(\alpha ,\beta ;t_{ji})\right. \\&\left. \times \left[ \omega _{1}(\alpha ;t_{ji})\frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha }-\frac{\partial \omega _{1}(\alpha ;t_{ji})}{\partial \alpha }\right] \right] , \\ \frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{3}}= & {} D_{3}(\alpha ,\beta ;t_{ji})\left[ \omega _{1}(\alpha ;t_{ji})\omega _{5}(\alpha ,\beta ;t_{ji})\right. \\&\left. +D_{3}(\alpha ;t_{ji})\right] \frac{\partial \omega _{2}(\alpha ,\beta ;t_{ji})}{ \partial \beta } \end{aligned}$$
Furthermore,
$$\begin{aligned} \psi _{1}= & {} \left[ \frac{2(m_{1+}m_{2})}{\alpha ^{3}}-\sum _{j=1}^{2} \sum _{i=1}^{m_{j}}\left[ \frac{\partial ^{2}\omega _{3}(\alpha ;t_{ji})}{\partial \alpha ^{2}}-(\beta -1)\frac{\partial ^{3}\omega _{1}(\alpha ;t_{ji})}{ \partial \alpha ^{3}}\right. \right. \\&\left. \left. -[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{3}}\right] \right] \sigma _{11}+2\left[ \sum _{j=1}^{2}\sum _{i=1}^{m_{j}}\left[ \frac{\partial ^{2}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{2}}\right. \right. \\&\left. \left. +[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}\partial \beta }\right] \right] \sigma _{12}+2\left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha ^{2}}\right] \sigma _{13} \\&+2\left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha \partial \beta }\right] \sigma _{23}+\left[ \sum _{j=1}^{2}\sum _{i=1}^{m_{j}}[r_{ji}\lambda ^{j-1}-1]\frac{ \partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{2}\partial \alpha }\right] \sigma _{22},\\ \psi _{2}= & {} \left[ \sum _{j=1}^{2}\sum _{i=1}^{m_{j}}\left[ \frac{\partial ^{2}\omega _{1}(\alpha ;t_{ji})}{\partial \alpha ^{2}}+[r_{ji}\lambda ^{j-1}-1]\frac{ \partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \alpha ^{2}\partial \beta }\right] \right] \sigma _{11} \\&+2\left[ \sum _{j=1}^{2}\sum _{i=1}^{m_{j}}[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{2}\partial \alpha } \right] \sigma _{12} \\&+2\left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha \partial \beta }\right] \sigma _{13}+2\left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \beta ^{2}}\right] \sigma _{23} \\&+\left[ \frac{2(m_{1+}m_{2})}{\beta ^{3}}+\sum _{j=1}^{2} \sum _{i=1}^{m_{j}}[r_{ji}\lambda ^{j-1}-1]\frac{\partial ^{3}\omega _{2}(\alpha ,\beta ;t_{ji})}{\partial \beta ^{3}}\right] \sigma _{22}, \\ \psi _{3}= & {} \left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha ^{2}}\right] \sigma _{11}+2\left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \alpha \partial \beta }\right] \sigma _{12} \\&+\left[ \sum _{i=1}^{m_{2}}r_{2i}\frac{\partial ^{2}\omega _{2}(\alpha ,\beta ;t_{2i})}{\partial \beta ^{2}}\right] \sigma _{22}+\frac{2m_{2}}{\lambda ^{3}} \sigma _{33}. \end{aligned}$$
