Cubic Transmuted Weibull Distribution: Properties and Applications

Abstract

In this paper, a cubic transmuted Weibull ($CTW$) distribution has been proposed by using the general family of transmuted distributions introduced by Rahman et al. (Pak J Stat Oper Res 14:451–469, 2018). We have explored the proposed $CTW$ distribution in details and have studied its statistical properties as well. The parameter estimation and inference procedure for the proposed distribution have been discussed. We have conducted a simulation study to observe the performance of estimation technique. Finally, we have considered two real-life data sets to investigate the practicality of proposed $CTW$ distribution.

Appendix 1: The Hessian Matrix for the \( CTW \) Distribution

The Hessian matrix is given as

$$ H = \left( {\begin{array}{*{20}c} {H_{11} } & {H_{12} } & {H_{13} } & {H_{14} } \\ {H_{21} } & {H_{22} } & {H_{23} } & {H_{24} } \\ {H_{31} } & {H_{32} } & {H_{33} } & {H_{34} } \\ {H_{41} } & {H_{42} } & {H_{43} } & {H_{44} } \\ \end{array} } \right), $$

where the variance–covariance matrix $V$ is obtained by

$$ V = \left( {\begin{array}{*{20}c} {V_{11} } & {V_{12} } & {V_{13} } & {V_{14} } \\ {V_{21} } & {V_{22} } & {V_{23} } & {V_{24} } \\ {V_{31} } & {V_{32} } & {V_{33} } & {V_{34} } \\ {V_{41} } & {V_{42} } & {V_{43} } & {V_{44} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {H_{11} } & {H_{12} } & {H_{13} } & {H_{14} } \\ {H_{21} } & {H_{22} } & {H_{23} } & {H_{24} } \\ {H_{31} } & {H_{32} } & {H_{33} } & {H_{34} } \\ {H_{41} } & {H_{42} } & {H_{43} } & {H_{44} } \\ \end{array} } \right)^{ - 1} , $$

with the elements

$$ \begin{aligned} H_{11} & = - \frac{{\delta^{2} l}}{{\delta \lambda^{2} }} = 3\sum\limits_{i = 1}^{n} {\left( {\frac{{\left( {k - 1} \right)kx_{i}^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 2} }}{{\lambda^{4} }} + \frac{{2kx_{i} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{3} }}} \right)} - \frac{kn}{{\lambda^{2} }} \\ & \quad + \sum\limits_{i = 1}^{n} {\left[ {\frac{{\left\{ { - \frac{{2k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }} - \frac{{2k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }} + 2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}^{2} }}{{\left\{ {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right\}^{2} }}} \right.} \\ & \quad - \frac{1}{{\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} }}\left\{ {\frac{{4k^{2} \beta_{1} x_{i}^{2} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{2k - 2} }}{{\lambda^{4} }}} \right. \\ & \quad + \frac{{2k^{2} \beta_{2} x_{i}^{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{2k - 2} }}{{\lambda^{4} }} + \frac{{2(k - 1)k\beta_{1} x_{i}^{2} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 2} }}{{\lambda^{4} }} \\ & \quad + \frac{{2(k - 1)k\beta_{2} x_{i}^{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 2} }}{{\lambda^{4} }} + \frac{{4k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{3} }} \\ & \quad + \frac{{4k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{3} }} - \frac{{4kx_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} }} \\ & \quad \left. {\left. { - \frac{{4kx_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} }} + 2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}} \right], \\ \end{aligned} $$

$$ \begin{aligned} H_{12} & = - \frac{{\delta^{2} l}}{\delta \lambda \cdot \delta k} = 3\sum\limits_{i = 1}^{n} {\left\{ { - \frac{{x_{i} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} }} - \frac{{kx_{i} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)}}{{\lambda^{2} }}} \right\}} + \frac{n}{\lambda } \\ & \quad + \sum\limits_{i = 1}^{n} {\left[ {\left( {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right)^{ - 2} \left\{ {\left\{ { - \frac{{2k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} }}} \right.} \right.} \right.} \\ & \quad \left. { - \frac{{2k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }} + 2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}\left\{ {2\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)} \right. \\& \quad \left. {\left. {+\,2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)} \right\}} \right\} - \frac{1}{{\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} }} \\ & \quad \times \left\{ { - \frac{{2\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} }} - \frac{{2\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} }}} \right. \\ & \quad - \frac{{2k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)}}{{\lambda^{2} }} - \frac{{2k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)}}{{\lambda^{2} }} \\ & \quad - \frac{{4k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{2k - 1} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)}}{{\lambda^{2} }} - \frac{{2k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{2k - 1} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)}}{{\lambda^{2} }} \\ & \quad \left. {\left. { +\, 2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right) + 2e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)} \right\}} \right], \\ \end{aligned} $$

$$ \begin{aligned} H_{13} & = - \frac{{\delta^{2} l}}{{\delta \lambda \cdot \delta \lambda_{1} }} = - \sum\limits_{i = 1}^{n} {\left[ {\frac{{2kx_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} - 2kx_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} \left( {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right)}}} \right.} \\ & \quad - \left\{ {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right\}^{ - 2} \left\{ {\left\{ {2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right. - e\left. {^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}} \right. \\& \quad \left. {\left. { \times \left\{ { - \frac{{2k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }}} \right.\left. { - \frac{{2k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }} + 2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}} \right\}} \right], \\ \end{aligned} $$

$$ \begin{aligned} H_{14} & = - \frac{{\delta^{2} l}}{{\delta \lambda \cdot \delta \lambda_{2} }} = - \sum\limits_{i = 1}^{n} {\left[ {\frac{{2kx_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} - 4kx_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k - 1} }}{{\lambda^{2} \left( {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right)}}} \right.} \\ & \quad - \left( {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right)^{ - 2} \left\{ {\left\{ {4e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right. - e\left. {^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3} \right\}} \right. \\ & \quad \times \left. {\left. {\left\{ { - \frac{{2k\beta_{1} x_{i} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }}} \right. - \left. {\frac{{2k\beta_{2} x_{i} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} }}{{\lambda^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{1 - k} }} + 2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}} \right\}} \right], \\ \end{aligned} $$

$$ \begin{aligned} H_{{22}} & = - \frac{{\delta ^{2} l}}{{\delta k^{2} }} = 3\sum\limits_{{i = 1}}^{n} {\left( {\frac{{x_{i} }}{\lambda }} \right)^{k} } \log ^{2} \left( {\frac{{x_{i} }}{\lambda }} \right) + \frac{n}{{k^{2} }} \\ & \quad + \sum\limits_{{i = 1}}^{n} {\left[ {\frac{1}{{\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3\lambda _{2} }}\left\{ {2\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log ^{2} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)} \right.} \right.} \\ & \quad + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {\frac{{x_{i} }}{\lambda }} \right)^{k} \log ^{2} \left( {\frac{{x_{i} }}{\lambda }} \right) + 4\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {\frac{{x_{i} }}{\lambda }} \right)^{{2k}} \log ^{2} \left( {\frac{{x_{i} }}{\lambda }} \right) \\ & \quad \left. { + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {\frac{{x_{i} }}{\lambda }} \right)^{{2k}} \log ^{2} \left( {\frac{{x_{i} }}{\lambda }} \right)} \right\} - \left\{ {\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3\lambda _{2} } \right\}^{{ - 2}} \\ & \quad \times \left. {\left\{ {2\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {\frac{{x_{i} }}{\lambda }} \right)^{k} \log \left( {\frac{{x_{i} }}{\lambda }} \right) + 2\beta _{2} e\left. {^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {\frac{{x_{i} }}{\lambda }} \right)^{k} \log \left( {\frac{{x_{i} }}{\lambda }} \right)} \right\}} \right\}^{2} } \right], \\ \end{aligned} $$

$$ \begin{aligned} H_{{23}} & = - \frac{{\delta ^{2} l}}{{\delta k \cdot \delta \lambda _{1} }} = - \sum\limits_{{i = 1}}^{n} {\left[ {\frac{{2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right) - 2e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)}}{{\left( { - \lambda _{1} - \lambda _{2} + 1} \right)e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\left( {\lambda _{1} + 2\lambda _{2} } \right)e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3\lambda _{2} }}} \right.} \\ & \quad - \left. {\frac{{\left\{ {\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)} \right\}\left\{ {2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} } \right\}\left\{ {2\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} } \right\}}}{{\left\{ {\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3\lambda _{2} } \right\}^{2} }}} \right], \\ \end{aligned}$$

$$ \begin{aligned} H_{{24}} = - \frac{{\delta ^{2} l}}{{\delta k \cdot \delta \lambda _{2} }} & = - \sum\limits_{{i = 1}}^{n} {\left[ {\frac{{4e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right) - 2e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)}}{{\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3\lambda _{2} }}} \right.} \\ & \quad - \left. {\frac{{\left\{ {\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} \log \left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)} \right\}\left\{ {4e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3} \right\}\left\{ {2\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} } \right\}}}{{\left\{ {\beta _{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} + 2\beta _{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{k} }} - 3\lambda _{2} } \right\}^{2} }}} \right], \\ \end{aligned}$$

$$ H_{33} = - \frac{{\delta^{2} l}}{{\delta \lambda_{1}^{2} }} = \sum\limits_{i = 1}^{n} {\left\{ {2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}^{2} \left\{ {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right\}^{ - 2} } , $$

$$ H_{34} = - \frac{{\delta^{2} l}}{{\delta \lambda_{1} \cdot \delta \lambda_{2} }} = \sum\limits_{i = 1}^{n} {\frac{{\left\{ {2e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} } \right\}\left\{ {4e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3} \right\}}}{{\left\{ {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right\}^{2} }}} , $$

$$ H_{44} = - \frac{{\delta^{2} l}}{{\delta \lambda_{2}^{2} }} = \sum\limits_{i = 1}^{n} {\left\{ {4e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3} \right\}^{2} \left\{ {\beta_{1} e^{{2\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} + 2\beta_{2} e^{{\left( {{{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } \lambda }} \right. \kern-0pt} \lambda }} \right)^{k} }} - 3\lambda_{2} } \right\}^{ - 2} } , $$

where $\beta_{1} = \left( {1 - \lambda_{1} - \lambda_{2} } \right)$ and $\beta_{2} = \left( {\lambda_{1} + 2\lambda_{2} } \right)$.