A New Extension of Extended Exponential Distribution with Applications

Abstract

We introduce a new lifetime distribution, called the alpha-power transformed extended exponential distribution which generalizes the extended exponential distribution proposed by Nadarajah and Haghighi (Statistics 45:543–558, 2011) to provide greater flexibility in modeling data from a practical point of view. The new model includes the exponential; extended exponential, and $\alpha$ power transformed exponential (Mahdavi and Kundu in Commun Stat Theory Methods, 2017) distributions as a special case. This distribution exhibits five hazard rate shapes such as constant, increasing, decreasing, bathtub and upside-down bathtub. Various properties of the proposed distribution, including explicit expressions for the quantiles, moments, conditional moments, stochastic ordering, Bonferroni and Lorenz curve, stress–strength reliability and order statistics are derived. The maximum likelihood estimators of the three unknown parameters of alpha-power transformed extended exponential distribution and the associated confidence intervals are obtained. A simulation study is carried out to examine the performances of the maximum likelihood estimates in terms of their bias and mean squared error using simulated samples. Finally, the potentiality of the distribution is analyzed by means of two real data sets. For the two real data sets, this distribution is found to be superior in its ability to sufficiently model the data as compared to the Weibull distribution, Generalized exponential distribution, Marshall–Olkin extended exponentiated exponential distribution and exponentiated Nadarajah–Haghighi distributions.

Appendix

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

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

whose elements are

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