Journal of Probability and Statistics

Volume 2018, Article ID 2931326, 12 pages

https://doi.org/10.1155/2018/2931326

## Extended Odd Fréchet-G Family of Distributions

Department of Statistics, Faculty of Mathematical Sciences, University for Development Studies, Tamale, Ghana

Correspondence should be addressed to Suleman Nasiru; moc.liamg@tatsnamelus

Received 8 August 2018; Accepted 10 November 2018; Published 2 December 2018

Academic Editor: Luis A. Gil-Alana

Copyright © 2018 Suleman Nasiru. 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

The need to develop generalizations of existing statistical distributions to make them more flexible in modeling real data sets is vital in parametric statistical modeling and inference. Thus, this study develops a new class of distributions called the extended odd Fréchet family of distributions for modifying existing standard distributions. Two special models named the extended odd Fréchet Nadarajah-Haghighi and extended odd Fréchet Weibull distributions are proposed using the developed family. The densities and the hazard rate functions of the two special distributions exhibit different kinds of monotonic and nonmonotonic shapes. The maximum likelihood method is used to develop estimators for the parameters of the new class of distributions. The application of the special distributions is illustrated by means of a real data set. The results revealed that the special distributions developed from the new family can provide reasonable parametric fit to the given data set compared to other existing distributions.

#### 1. Introduction

The fundamental reason for parametric statistical modeling is to identify the most appropriate model that adequately describes a data set obtained from experiment, observational studies, surveys, and so on. Most of these modeling techniques are based on finding the most suitable probability distribution that explains the underlying structure of the given data set. However, there is no single probability distribution that is suitable for different data sets. Thus, this has triggered the need to extend the existing classical distributions or develop new ones. Barrage of methods for defining new families of distributions have been proposed in literature for extending or generalizing the existing classical distributions in recent time. Some of these methods include Weibull-G [1], odd generalized exponential family [2], odd Lindley-G family [3], Topp-Leone odd log-logistic-G family [4], odd Burr-G family [5], odd Fréchet-G family [6], odd gamma-G family [7], transformed-transformer method [8], exponentiated transformed-transformer method [9], exponentiated generalized transformed-transformer method [10], alpha power transformed family [11], alpha logarithmic transformed family [12], Kumaraswamy-G family [13], beta-G family [14], Kumaraswamy transmuted-G family [15], transmuted geometric-G family [16], and beta extended Weibull family [17]. These methods are developed with the motivation of defining new models with different kinds of failure rates (monotonic and nonmonotonic), constructing heavy-tailed distributions for modeling different kinds of data sets, developing distributions with symmetric, right skewed, left skewed, reversed J shape, and consistently providing a reasonable parametric fit to given data sets.

Recently, [6] developed the odd Fréchet family of distributions and defined its cumulative distribution function (CDF) aswhere is the baseline CDF and is a vector of associated parameters. Using the transformed-transformer method proposed by [8], an extension of the odd Fréchet family of distributions called the extended odd Fréchet-G (EOF-G) family of distributions is developed by integrating the Fréchet probability density function (PDF). Hence, the CDF of the EOF-G family is defined aswhere and are extra shape parameters. The corresponding PDF of the new family is obtained by differentiating equation (2) and is given byThe associated hazard rate function of the EOF-G family is defined asHereafter, a random variable following the EOF-G distribution is denoted by and for the purpose of simplicity, can be written as . The CDF of the EOF-G family of distributions is tractable which makes it easy to generate random numbers provided that the CDF of the baseline distribution is also tractable. The quantile of the EOF-G family is given bywhere is the baseline quantile function. When , the EOF-G family of distributions reduces to the odd Fréchet family of distributions. Adopting the interpretation of the CDF of the odd Weibull family as given in [18], the physical interpretation of the CDF of the EOF-G family is given as follows: Suppose is a lifetime random variable with continuous CDF, . The odds ratio that an individual (component) having the lifetime will die (fail) at time is . Given that the variability of these odds of death is denoted by the random variable and that it follows the Fréchet distribution, thenwhich is given in (2). The rest of the paper is organized as follows: In Section 2, special distributions of the EOF-G family are discussed. In Section 3, the mixture representation of the PDF and CDF of the EOF-G family is given. The statistical properties of the new family are derived in Section 4. In Section 5, the estimators for the parameters of the family are developed using the technique of maximum likelihood estimation. Monte Carlo simulations are performed in Section 6 to assess the performance of the estimators. In Section 7, the application of the special distributions is demonstrated using real data set. Finally, the concluding remarks of the study are given in Section 8.

#### 2. Special Distributions of the EOF-G Family

In this section, two special distributions of the EOF-G family are discussed.

##### 2.1. EOF-Nadarajah-Haghighi (EOFNH) Distribution

Suppose the baseline CDF is that of the Nadarajah-Haghighi distribution; that is, with corresponding PDF and positive parameters . The PDF of the EOFNH distribution is given bywhere are shape parameters, is a scale parameter, and . Figure 1 shows the plots of the PDF of the EOFNH distribution for some selected parameter values. The density function exhibits different kinds of shapes.