International Journal of Mathematics and Mathematical Sciences

Volume 2016, Article ID 5898356, 10 pages

http://dx.doi.org/10.1155/2016/5898356

## The Exponentiated Gumbel Type-2 Distribution: Properties and Application

^{1}School of Mathematics, University of Manchester, Manchester M13 9PL, UK^{2}Department of Statistics, Abia State University, PMB 2000, Uturu, Abia State, Nigeria^{3}Department of Mathematics & Statistics, Faculty of Sciences, Federal University, Otuoke, PMB 126, Yenagoa, Bayelsa, Nigeria

Received 17 May 2016; Accepted 10 July 2016

Academic Editor: Niansheng Tang

Copyright © 2016 I. E. Okorie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a generalized version of the standard Gumble type-2 distribution. The new lifetime distribution is called the Exponentiated Gumbel (EG) type-2 distribution. The EG type-2 distribution has three nested submodels, namely, the Gumbel type-2 distribution, the Exponentiated Fréchet (EF) distribution, and the Fréchet distribution. Some statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimates was proposed for estimating the model parameters. The usefulness and flexibility of the Exponentiated Gumbel (EG) type-2 distribution were illustrated with a real lifetime data set. Results based on the log-likelihood and information statistics values showed that the EG type-2 distribution provides a better fit to the data than the other competing distributions. Also, the consistency of the parameters of the new distribution was demonstrated through a simulation study. The EG type-2 distribution is therefore recommended for effective modelling of lifetime data.

#### 1. Introduction

The Gumbel distribution, also known as the type-1 extreme value distribution, has received significant research attention, over the years particularly, in extreme value analysis of extreme events. For a review of the recent developments and applications of the Gumbel distribution, see Pinheiro and Ferrari [1]. There is no question that, before now, the Gumbel type-2 distribution is not popularly used in statistical modelling and the reason may not be far from its lack of fits in data modelling. Generally, standard probability distributions are well known for their lack of fits in modelling complex data sets. On this note, users of this distributions across various fields in general and statistics and mathematics in particular have been fantastically motivated to developing sophisticated probability distributions from the existing ones. Exponentiated distributions have been introduced to solve the problem of lack of fits that is commonly encountered when using the standard probability distributions for modelling complex data sets. Results from this advancement have frequently been proven more reasonable than the one based on the standard distributions. Exponentiating distributions are indeed a powerful technique in statistical modelling that offers an effective way of introducing an additional shape parameter to the standard distribution to achieve robustness and flexibility. This method of generalizing probability distributions is traceable to the work of Gupta et al. [2] who introduced the exponentiated exponential (EE) distribution as a generalized form of the standard exponential distribution by simply raising the cumulative density function (cdf) to a positive constant power. Ever since the introduction of the EE distribution, exponentiated distributions have achieved reasonable feats in modelling data sets from various complex phenomena. A good number of standard probability distributions have their corresponding exponentiated versions. Gupta et al. [2] introduced the Exponentiated Weibull distribution as a generalization of the standard Weibull distribution. Nadarajah and Kotz [3] modified the method by Gupta et al. [2] and introduced the Exponentiated Fréchet distribution as a generalization of the standard Fréchet distribution. Using the same method in Nadarajah and Kotz [3], Nadarajah [4] introduced the Exponentiated Gumbel distribution as a generalization of the standard Gumbel distribution. Mudholkar and Srivastava [5] introduced the Exponentiated Weibull family distribution as a generalization of the Weibull family distribution. Ashour and Eltehiwy [6] developed the exponentiated power Lindley distribution generalizing the power Lindley distribution and so forth. Therefore, this paper is aimed at generalizing the standard Gumbel type-2 distribution to a wider class of distribution so as to improve its performance and encourage its applicability, in modelling varieties of complex data sets.

The cumulative density function of the exponentiated family of distributions according to Nadarajah and Kotz [3] is defined bydifferentiating (1) with respect to gives the corresponding probability density function aswhere and are the vector of parameters and parameter space of the baseline distribution , respectively.

The remaining part of this paper is organized as follows; Section 2 introduces the Gumbel type-2 distribution, its exponentiated version, and special cases (submodels); Section 3 presents some important reliability characteristics of the new distribution and their asymptotic properties; Section 4 presents an explicit derivation of the moments, variance, and moment generating function of the new model; Section 5 presents the th quantile function of the new distribution; Section 6 presents the Rényi’s entropy of the new distribution; Section 7 presents the th order statistics of the new distribution; Section 8 proposes the maximum likelihood estimation method for estimating the parameters of the new distribution; Section 9 presents the application of the new distribution to a real data set and a simulation study; Section 10 is the discussion of results and Section 11 contains the conclusion of the study.

#### 2. Exponentiated Gumbel Type-2 Distribution

*Definition 1. *According to Gumbel [7–9], a random variable is said to follow the Gumbel type-2 distribution if its cumulative density function (cdf) is given bywhile the corresponding probability density function (pdf) is given byUsing (3), we obtain the cdf of the Exponentiated Gumbel (EG) type-2 distribution aswhile the corresponding pdf is given bywhere and are the shape parameters and is the scale parameter.

Figure 1 shows the plots of the pdf (a) and cdf (b) of the EG type-2 distribution for certain parameter values.