A New Extension of Weibull Distribution with Application to Lifetime Data

Abstract

The Weibull distribution has been generalized by many authors in recent years. Here, we introduce a new generalization, called alpha-power transformed Weibull distribution that provides better fits than the Weibull distribution and some of its known generalizations. The distribution contains alpha-power transformed exponential and alpha-power transformed Rayleigh distributions as special cases. Various properties of the proposed distribution, including explicit expressions for the quantiles, mode, moments, conditional moments, mean residual lifetime, stochastic ordering, Bonferroni and Lorenz curve, stress–strength reliability and order statistics are derived. The distribution is capable of modeling monotonically increasing, decreasing, constant, bathtub, upside-down bathtub and increasing–decreasing–increasing hazard rates. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non-linear equations only. Two data sets have been analyzed to show how the proposed models work in practice. Further, a bivariate extension based on Marshall–Olkin and copula concept of the proposed model are developed but the properties of the distribution not considered in detail in this paper that can be addressed in future research.

Appendix

The elements of the observed information matrix $J(\varvec{\theta })$ for the three parameters $(\alpha ,\beta ,\lambda )$ are given by

$$\begin{aligned} J(\varvec{\theta })=\left( \begin{array}{ccc} J_{\alpha \alpha } &{} J_{\alpha \beta } &{} J_{\alpha \lambda } \\ &{} J_{\beta \beta } &{} J_{\beta \lambda } \\ &{} &{} J_{\lambda \lambda } \end{array} \right) \end{aligned}$$

whose elements are

$$\begin{aligned} J_{\alpha \alpha }&=\dfrac{\partial ^{2}\log \ell }{\partial \alpha ^{2}}= -\frac{n (\log \alpha +1)}{(\alpha \log \alpha )^{2}}+ \frac{n}{\alpha -1}- \frac{\sum _{i=1}^{n}(1-e^{-\beta x_{i}^{\lambda }})}{\alpha ^2}\\ J_{\beta \beta }&=\dfrac{\partial ^{2}\log \ell }{\partial \beta ^{2}}=-\frac{n}{\beta ^2}- \log (\alpha )\sum _{i=1}^{n} x_{i}^{2\lambda } e^{-\beta x_{i}^{\lambda }},\\ J_{\lambda \lambda }&=-\dfrac{\partial ^{2}\log \ell }{\partial \lambda ^{2}}= -\frac{n}{\lambda ^2} -\beta \sum _{i=1}^{n} x_{i}^{\lambda }(\log x_{i})^{2}+ \beta \log (\alpha )\sum _{i=1}^{n} [x_{i}^{\lambda }(\log x_{i})^{2}e^{-\beta x_{i}^{\lambda }}\\&\qquad - \beta x_{i}^{2\lambda }(\log x_{i})^{2}e^{-\beta x_{i}^{\lambda }}],\\ J_{\alpha \beta }&=\dfrac{\partial ^{2}\log \ell }{\partial \alpha \partial \beta }=\frac{1}{\alpha }\sum _{i=1}^{n} x_{i}^{\lambda }e^{-\beta x_{i}^{\lambda }},\\ J_{\alpha \lambda }&=\dfrac{\partial ^{2}\log \ell }{\partial \alpha \partial \lambda }=\frac{\beta }{\alpha }\sum _{i=1}^{n} x_{i}^{\lambda }(\log x_{i})e^{-\beta x_{i}^{\lambda }} ,\\ J_{\beta \lambda }&=\dfrac{\partial ^{2}\log \ell }{\partial \beta \partial \lambda }= -\sum _{i=1}^{n} x_{i}^{\lambda }(\log x_{i})+\log (\alpha )\sum _{i=1}^{n} x_{i}^{\lambda }(\log x_{i})e^{-\beta x_{i}^{\lambda }}[1-\beta x_{i}^{\lambda }e^{-\beta x_{i}^{\lambda }}]. \end{aligned}$$