Exponentiated Power Lindley Logarithmic: Model, Properties and Applications

Abstract

A new class of lifetime distributions is proposed. Closed form expressions are provided for the density, cumulative distribution, survival and hazard rate functions. Maximum likelihood estimation is discussed and formulas for the elements of the observed information matrix are provided. Simulation studies are conducted. Finally, two real data applications are given showing the flexibility and potentiality of the new distribution

Appendices

Appendix 1: Elements of the Observed Information Matrix

The observed information matrix is

$$\begin{aligned} \mathbf {I} = - \left( \begin{array}{cccc} \frac{\partial ^2 L}{\partial \alpha ^2} &{} \frac{\partial ^2 L}{\partial \alpha \partial \beta } &{} \frac{\partial ^2 L}{\partial \alpha \partial \lambda } &{} \frac{\partial ^2 L}{\partial \alpha \partial \theta }\\ \frac{\partial ^2 L}{\partial \beta \partial \alpha } &{}\quad \frac{\partial ^2 L}{\partial \beta ^2} &{}\quad \frac{\partial ^2 L}{\partial \beta \partial \lambda } &{}\quad \frac{\partial ^2 L}{\partial \beta \partial \theta }\\ \frac{\partial ^2 L}{\partial \lambda \partial \alpha } &{}\quad \frac{\partial ^2 L}{\partial \lambda \partial \beta } &{}\quad \frac{\partial ^2 L}{\partial \lambda ^2} &{}\quad \frac{\partial ^2 L}{\partial \lambda \partial \theta }\\ \frac{\partial ^2 L}{\partial \theta \partial \alpha } &{}\quad \frac{\partial ^2 L}{\partial \theta \partial \beta } &{}\quad \frac{\partial ^2 L}{\partial \theta \partial \lambda } &{}\quad \frac{\partial ^2 L}{\partial \theta ^2} \end{array} \right) . \end{aligned}$$

Let $I_{\vartheta _{1}\vartheta _{2}}=I_{\vartheta _{2}\vartheta _{1}}={\frac{\partial ^{2}}{\partial \vartheta _{1}\partial \vartheta _{2}}I}$. Then, the elements of $\mathbf {I}$ are

$$\begin{aligned} I_{\alpha \alpha }= & {} -{\frac{n}{{\alpha }^{2}}}+\sum _{i=1}^{n}\left\{ {\theta }^{2} \left( \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2} \right. \nonumber \\&\times \left( \ln \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta } }{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) \right) ^{2} \\&\cdot { \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2} }\\&+\, {\theta \left( 1 - \left( 1+{\frac{ \lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }} \right) ^{\alpha } \left( \ln \left( 1 - \left( 1+{\frac{ \lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }} \right) \right) ^{2} }\\&\cdot \left. \left( 1-\theta \ \left( 1 - \left( 1 + {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1}\right\} , \end{aligned}$$

$$\begin{aligned} I_{\alpha \beta }= & {} \sum _{i=1}^{n} \left( - {\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) \\&\times \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+\, \sum _{i=1}^{n} \left\{ {\theta }^{2}\alpha \left( \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2} \ln \left( 1 - \left( 1+ {\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) \right. \\&\cdot \left( - {\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) \right) \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^ {-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+\,\theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \\&\times \left( - {\frac{ \lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ { x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) \\&\cdot \ln \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\times \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}}\right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+\,\theta \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&\times \left( - {\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) \right) \\&\cdot \left. \left( 1-\theta \ \left( 1 - \left( 1 + {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+ {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1} \right\} , \end{aligned}$$

$$\begin{aligned} I_{\alpha \lambda }= & {} \sum _{i=1}^{n} \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{ \beta }}{x_{{i}}}^{\beta } \right) \\&\times \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+\sum _{i=1}^{n} \left\{ \alpha \ { \theta }^{2} \left( \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2} \right. \\&\times \ln \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) \\&\cdot \left( -\left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}}-{ \frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+\,\alpha \ \theta \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda + 1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) { e}^{-\lambda \ {x_{{i}}}^{\beta }} \right. \\&\left. + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}} }^{\beta } \right) \\&\cdot \ln \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) \left( 1-\theta \ \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\times \left( 1 - \left( 1 + {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+\,\theta \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} \right) ^{\alpha } \left( - \left( {\frac{{x_{{i}}}^{ \beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right. \\&\left. +\, \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) \\&\cdot \left. \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta } } \right) ^{-1} \right\} , \end{aligned}$$

$$\begin{aligned} I_{\alpha \theta }= & {} \sum _{i=1}^{n}\theta \left( \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2} \ln \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{ \beta }} \right) \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{ \beta }} \right) ^{\alpha } \right) ^{-2}\\&+\, \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} \right) ^{\alpha }\ln \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{ \beta }} \right) \\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{ \beta }} \right) ^{\alpha } \right) ^{-1}, \end{aligned}$$

$$\begin{aligned} I_{\beta \beta }= & {} -{\frac{n}{{\beta }^{2}}}+\sum _{i=1}^{n}{\frac{{x_{{i}}}^{\beta } \left( \ln \left( x_{{i}} \right) \right) ^{2}}{1+{x_{{i}}}^{\beta }}}- {\frac{ \left( {x_{{i}}}^{\beta } \right) ^{2} \left( \ln \left( x_{{i}} \right) \right) ^{2}}{ \left( 1+{x_{{i}}}^{\beta } \right) ^{2} }}-\lambda \ \sum _{i=1}^{n} {x_{{i}}}^{\beta } \left( \ln \left( x_{{i} } \right) \right) ^{2}\\&+ \,\left( \alpha -1 \right) \sum _{i=1}^{n}\left\{ - {\frac{\lambda \ {x_{{i}}}^{\beta } \left( \ln \left( x_{{i}} \right) \right) ^{2}{e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}}+2 \ {\frac{{\lambda }^{2} \left( {x_{{i}}}^{\beta } \right) ^{2} \left( \ln \left( x_{{i}} \right) \right) ^{2}{e}^{-\lambda \ {x_{{i}}}^{ \beta }} }{\lambda +1}} \right. \\&- \, \left. \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) \left( \ln \left( x_{{i}} \right) \right) ^{2} \left( {x_{{i}}}^{\beta } \right) ^{2}{e}^{- \lambda \ {x_{{i}}}^{\beta }} {\lambda }^{2} \right. \\&+\, \left. \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \left( \ln \left( x_{{i}} \right) \right) ^{2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1} \right. \\&-\, \left. \left( - {\frac{\lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) ^{2} \right. \\&\times \left. \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2} \right\} \\ \end{aligned}$$

$$\begin{aligned}&+\,\sum _{i=1}^{n} \left\{ {\theta }^{2}{\alpha }^{2} \left( \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2} \right. \\&\cdot \left( - {\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{- \lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_i \right) \right) ^{2}\\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^ {-2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2}\\&+ \, \theta \ { \alpha }^{2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \\&\times \left( -{\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) ^{2}\\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e }^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2}\\&+\,\theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda + 1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&-\, {\frac{\lambda \ {x_{{i}}}^{\beta } \left( \ln \left( x_{{i}} \right) \right) ^{2}{e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} +2 \ {\frac{{\lambda }^{2} \left( {x_{{i}}}^{\beta } \right) ^{2} \left( \ln \left( x_{{i}} \right) \right) ^{2}{e}^{-\lambda \ {x_{{i}}}^{ \beta }} }{\lambda +1}}\\&-\,\left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) \left( \ln \left( x_{{i}} \right) \right) ^{2} \left( {x_{{i}}}^{\beta } \right) ^{2}{e}^{- \lambda \ {x_{{i}}}^{\beta }} {\lambda }^{2}\\&+\, \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \left( \ln \left( x_{{i}} \right) \right) ^{2} \left( 1-\theta \ \left( 1 - \left( 1+{\frac{ \lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\cdot \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} \right) ^{-1}-\theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} \right) ^{\alpha }\\&\cdot \left( - {\frac{\lambda \ {x_{{i}}}^{ \beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{ \lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) \right) ^{2}\\&\cdot \left. \left( 1-\theta \ \left( 1 - \left( 1+ {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+ {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2} \right\} , \end{aligned}$$

$$\begin{aligned} I_{\beta \lambda }= & {} -\sum _{i=1}^{n}{x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) + \left( \alpha -1 \right) \\&\times \sum _{i=1}^{n} \left\{ \left( -{\frac{{x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda + 1}}+{\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{ \left( \lambda +1 \right) ^{2}}}+{ \frac{{e}^{-\lambda \ {x_{{i}}}^{\beta }}\ln \left( x_{{i}} \right) \left( {x_{{i}}}^{\beta } \right) ^{2}\lambda }{\lambda +1}} \right. \right. \\&+\, \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}}-{\frac{ \lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \\&-\, \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \left( {x_{{i}}}^{\beta } \right) ^{2} \lambda \ \ln \left( x_{{i}} \right) \\&+\,\left. \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }}{x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&-\, \left( - {\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) \\&\cdot \left. \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) \right. \\&\times \left. \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2} \right\} \end{aligned}$$

$$\begin{aligned} \quad \quad&+\sum _{i=1}^{n} \left\{ {\theta }^{2}{\alpha }^{2} \left( \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2}\right. \\&\times \left( - {\frac{ \lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) {e}^{-\lambda \ { x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) \\&\cdot \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}}-{ \frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) \\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2}\\&\cdot \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2}+\theta \ {\alpha }^{2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&\cdot \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}}- {\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) { e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}} }^{\beta } \right) \\&\cdot \left( - {\frac{\lambda \ {x_{ {i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{ \beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}} }^{\beta } \ln \left( x_{{i}} \right) \right) \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2} \end{aligned}$$

$$\begin{aligned} \quad \quad&+ \theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&\cdot \left( - {\frac{{x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} +{\frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{-\lambda \ { x_{{i}}}^{\beta }}}{ \left( \lambda +1 \right) ^{2}}} +{\frac{{e}^{- \lambda \ {x_{{i}}}^{\beta }}\ln \left( x_{{i}} \right) \left( {x_{{i} }}^{\beta } \right) ^{2}\lambda }{\lambda +1}} \right. \\&+\,\left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ { x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) \\&-\, \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \left( { x_{{i}}}^{\beta } \right) ^{2} \lambda \ \ln \left( x_{{i}} \right) \\&+\,\left. \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{ x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) \right) \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \nonumber \\&\times \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta } } \right) ^{-1}\\&-\theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta } } \right) ^{\alpha } \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda + 1}}-{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2 }}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right. \\&\left. +\, \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }}{x_{{i}}}^{\beta } \right) \\&\cdot \left( - { \frac{\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) {e}^{- \lambda \ {x_{{i}}}^{\beta }}}{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \lambda \ {x_{{i}}}^{\beta } \ln \left( x_{{i}} \right) \right) \\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{ \lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\cdot \left. \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} \right) ^{-2} \right\} , \end{aligned}$$

$$\begin{aligned} I_{\beta \theta }= & {} \sum _{i=1}^{n} \left\{ \theta \ \alpha \left( 1 - \left( 1+{\frac{ \lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }} \right) ^{2 \alpha } \right. \\&\times \left. \left( - {\frac{\ln \left( x_{{i}} \right) {x_{{i}}}^{\beta }{e}^{-\lambda \ {x_{{i}}}^{ \beta }}\lambda }{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta }\ln \left( x_{{i}} \right) \right) \right. \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1 +{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta } } \right) ^{-1}\\&+\, \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&\times \left( - {\frac{\ln \left( x_{{i}} \right) {x_{{i}}}^{\beta } {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda }{\lambda +1}} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\lambda \ {x_{{i}}}^{\beta } \ln \left( x_{ {i}} \right) \right) \\&\cdot \left. \left( 1-\theta \ \left( 1 - \left( 1 + {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1} \right\} , \end{aligned}$$

$$\begin{aligned} I_{\lambda \lambda }= & {} -2\ {\frac{n}{{\lambda }^{2}}}+{\frac{n}{ \left( \lambda +1 \right) ^{2}}} + \left( \alpha -1 \right) \sum _{i=1}^{n} \left( - \left( - 2\ { \frac{{x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} + 2\ {\frac{ \lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{3}}} \right) { e}^{-\lambda \ {x_{{i}}}^{\beta }} \right. \\&\left. + 2\ \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } - \left( 1+{\frac{\lambda \ {x_{{i}}}^{ \beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \left( {x_{{i}}}^{\beta } \right) ^{2} \right) \\&\times \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&-\, \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) { e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}} }^{\beta }\ \right) ^{2}\\&\times \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2}\\&+\,\sum _{i=1}^{n} \left\{ {\theta }^{2}{\alpha }^{2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{2 \alpha } \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} - {\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\right. \right. \\&\left. + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta }\ \right) ^{2} \\ \end{aligned}$$

$$\begin{aligned}&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1} } \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2}\\&+\theta \ {\alpha }^{2} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&\cdot \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}}-{\frac{ \lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) { e}^{-\lambda \ {x_{{i}}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}} }^{\beta } \right) ^{2} \\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\cdot \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2}+\theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda + 1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \\&\times \left( - \left( - 2\ {\frac{{x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} + 2\ {\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda + 1 \right) ^{3}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right. \\ \end{aligned}$$

$$\begin{aligned}&+\,2\ \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}}-{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ { x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \left( {x_{{i}}}^{\beta } \right) ^{2}\\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\cdot \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1} - \theta \ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha }\\&\cdot \left( - \left( { \frac{{x_{{i}}}^{\beta }}{\lambda +1}}-{\frac{\lambda \ {x_{{i}}}^{ \beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) ^{2} \\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1}\\&\cdot \left. \left. \left( 1 - \left( 1 + {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-2} \right) \right\} , \end{aligned}$$

$$\begin{aligned} I_{\theta \theta }= & {} -{\frac{n}{{\theta }^{2}}}+{\frac{n}{ \left( 1-\theta \right) ^{2} \ln \left( 1-\theta \right) }}+{\frac{n}{ \left( 1-\theta \right) ^{ 2} \left( \ln \left( 1-\theta \right) \right) ^{2}}}\\&+\sum _{i=1}^{n} 1 \left( \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{2} \\&\times \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ { x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta } } \right) ^{\alpha } \right) ^{-2} \end{aligned}$$

and

$$\begin{aligned} I_{\lambda \theta }= & {} \sum _{i=1}^{n}\alpha \ \theta \left( 1 - \left( 1+{\frac{ \lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i} }}^{\beta }} \right) ^{ 2 \alpha } \left( - \left( {\frac{{x_{{i}}}^{\beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}\right. \\&\left. + \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{ \frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-2} \left( 1 - \left( 1 + {\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{-1}\\&+ \alpha \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i }}}^{\beta }} \right) ^{\alpha } \left( -\left( {\frac{{x_{{i}}}^{ \beta }}{\lambda +1}} -{\frac{\lambda \ {x_{{i}}}^{\beta }}{ \left( \lambda +1 \right) ^{2}}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right. \\&\left. +\, \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }}{x_{{i}}}^{\beta } \right) \\&\cdot \left( 1-\theta \ \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta }} \right) ^{\alpha } \right) ^{-1} \left( 1 - \left( 1+{\frac{\lambda \ {x_{{i}}}^{\beta }}{\lambda +1}} \right) {e}^{-\lambda \ {x_{{i}}}^{\beta } } \right) ^{-1}. \end{aligned}$$

Appendix 2: Code of MLE

ML.epll$<-$function(start, x){

# start  :  Initial values for the parameters to be optimized over

n $=$ length(x)

loglik.epll $<-$ function(mu){

alpha $=$ mu[1];beta $=$ mu[2];lambda $=$ mu[3];theta $=$ mu[4];

-(n*log(alpha)+n*log(beta)+n*log(theta)+2*n*log(lambda)

-n*log(lambda+1)-n*log(theta/(1-theta))+(beta-1)

*(sum(log(x)))+sum $({\texttt {log}}({\texttt {1}}+{\texttt {x}}^{\texttt {beta}}))-$

$lambda*(sum(x^{beta}))+ (alpha-1)*(sum(log(1-(1+lambda*x^{beta}/(lambda+1))*exp(-lambda*x^{beta}))))$ $+sum(log(1/(1-theta*(1-(1+lambda*x^{beta}/(lambda+1))*exp(-lambda*x^beta))^{alpha})$ $+theta*(1-(1+lambda*x^{beta}/(lambda+1))*exp(-lambda*x^{beta}))^alpha/(1-theta*(1-(1+lambda*x^{beta}/(lambda+1))*exp(-lambda*x^beta))^alpha)^2)))\}$

$Mle = nlminb(start,loglik.epll,lower =rep(0,4), upper =$

$c(Inf,Inf,Inf,1) )\$ par$

$names(Mle)= paste(c(``alpha{\hbox {''}},``beta{\hbox {''}},``lambda{\hbox {''}},``theta{\hbox {''}}))$

Mle

}