Journal of Probability and Statistics

Volume 2016, Article ID 7208425, 12 pages

http://dx.doi.org/10.1155/2016/7208425

## General Results for the Transmuted Family of Distributions and New Models

^{1}Departamento de Estatística, Universidade Federal do Rio Grande do Norte, 59078-970 Natal, RN, Brazil^{2}Department of Mathematics and Statistics, University of North Carolina Wilmington, Wilmington, NC, USA^{3}Departamento de Estatística, Universidade Federal de Pernambuco, 50740-540 Recife, PE, Brazil

Received 4 October 2015; Accepted 29 December 2015

Academic Editor: Zacharias Psaradakis

Copyright © 2016 Marcelo Bourguignon 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

The transmuted family of distributions has been receiving increased attention over the last few years. For a baseline* G* distribution, we derive a simple representation for the transmuted-*G* family density function as a linear mixture of the* G* and exponentiated-*G* densities. We investigate the asymptotes and shapes and obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, Rényi and Shannon entropies, and order statistics and their moments. We estimate the model parameters of the family by the method of maximum likelihood. We prove empirically the flexibility of the proposed model by means of an application to a real data set.

#### 1. Introduction

Adding parameters to a well-established distribution is a time honored device for obtaining more flexible new families of distributions. Shaw and Buckley [1] pioneered an interesting method of adding a new parameter to an existing distribution that would offer more distributional flexibility. They used the quadratic rank transmutation map (QRTM) in order to generate a flexible family of distributions. The generated family, also called the transmuted extended distribution, includes the parent distribution as a special case and gives more flexibility to model various types of data.

In the last three years, there has been a growing interest in transmuted distributions and several of them have been investigated. A significant amount of work has been attributed towards developing a new transmuted model and subsequently discussing its utilities as enhanced flexibility in modeling various types of real life data, where the parent model does not provide a good fit. Aryal and Tsokos [2] defined the transmuted generalized extreme value distribution and studied some basic mathematical characteristics of the transmuted Gumbel distribution and its applications to climate data. Aryal and Tsokos [3] presented a new generalized Weibull distribution called the transmuted Weibull distribution. Recently, Aryal [4] proposed and studied various structural properties of the transmuted log-logistic distribution. Khan and King [5] introduced the transmuted modified Weibull distribution, which extends the transmuted Weibull distribution [3], and studied its mathematical properties and maximum likelihood estimation of the model parameters. Elbatal [6] proposed the transmuted modified inverse Weibull distribution. Elbatal and Aryal [7] explored the transmuted additive Weibull model, which extends the additive Weibull distribution and some other distributions using the QRTM method [1]. However, several published works did not investigate many properties such as finite mixture of the density function, Rényi and Shannon entropies, extreme values, probability weighted moments (PWMs), and bivariate and multivariate generalization. This paper aims to fill out this gap in the existing literature and contribute with general properties of the transmuted family.

This vast amount of literature merits for a detailed study for the most general transmuted family of distributions, which is our major motivation to carry out this work. In this paper, we derive general mathematical properties for the transmuted family, which hold for any baseline distribution, such as the ordinary, central, and incomplete moments, quantile and generating functions, mean deviations, Rényi and Shannon entropies, extreme values, PWMs, order statistics and their moments, and bivariate and multivariate generalizations. We provide a comprehensive description of these properties with the hope that the transmuted family will attract wider applications in biology, medicine, economics, reliability, and engineering and in other areas of research. We also introduce new distributions based on the transmuted construction.

The rest of the paper is organized as follows. In Section 2, we discuss the general theory behind the transmuted distribution and present useful representations for the density and cumulative functions. In Section 3, we investigate its asymptotes and shapes. In Section 4, we provide an algorithm for generating samples from the transmuted family based on its quantile function (qf). In Section 5, we derive expressions for the moments and generating function. In Section 6, we obtain mean deviations and provide some examples. In Section 7, we present two special transmuted models. In Section 8, we discuss the limiting behavior of the extreme statistics. In Section 9, we derive the PWMs. In Section 10, we obtain the order statistics. We derive expressions for the Shannon and Rényi entropies and Kullback-Leibler divergence measure in Section 11. We introduce in Section 12 the bivariate and multivariate extensions of the univariate transmuted family. In Section 13, we use the maximum likelihood method to estimate the model parameters. In Section 14, we fit some special models of the transmuted family to a real data set to prove empirically its usefulness. In Section 15, we offer some concluding remarks.

#### 2. Distribution and Density Functions

Let and be the cumulative distribution functions (cdfs) of two models with a common sample space. The general rank transmutation as given in Shaw and Buckley [1] is defined as and . Note that the qf is defined by for . Functions and both map the unit interval into itself and, under suitable assumptions, are mutual inverses and satisfy and (for ). The QRTM is defined by , from which it follows that . Differentiating gives , where and are the probability density functions (pdfs) corresponding to the cdfs and , respectively. For more details about the QRTM approach, see Shaw and Buckley [1].

A random variable has the* transmuted-G* () family if the pdf and cdf are defined through the QRTM method by (for )where is the parent cdf and is the parent pdf. Both functions depend on the parameter vector . For , it reduces to the parent model. Hereafter, the random variable following (1) with parameter and baseline vector of parameters is denoted by . The computations for fitting family (1) to real data in practical problems can be easily performed using the* AdequacyModel* script in the R software.

For an arbitrary baseline cdf , a random variable is said to have the Exp- distribution with power parameter , say , if its pdf and cdf are given by respectively. Note that . The properties of exponentiated distributions have been studied by many authors in recent years. See, for example, Mudholkar and Srivastava [8] for exponentiated Weibull, Gupta et al. [9] for exponentiated Pareto, Gupta and Kundu [10] for exponentiated exponential, Nadarajah [11] for exponentiated Gumbel, Kakde and Shirke [12] for exponentiated lognormal, and Nadarajah and Gupta [13] for exponentiated gamma distributions.

Theorem 1. *The density function of can be expressed as the linear mixture where .*

Corollary 2. *If , then .*

Theorem 1 is important to obtain some measures of from those of exponentiated distributions. This result plays an important role in the paper, since we can obtain, for example, the moments, generating function, and mean deviations of . Established explicit expressions for these measures can be simpler than using numerical integration.

The hazard rate function (hrf) of is given bywhere is the baseline hrf. The multiplying quantity is a kind of correction factor for the baseline hrf.

Equation (5) can deal with general situations for modeling survival data with various hrf shapes. From this equation, we note that is decreasing in for and it is increasing in for . Additionally, we have for and , respectively.

Equation (5) can be expressed as where , , and and are the hrfs of the and Exp- distributions, respectively.

#### 3. Asymptotes and Shapes

Proposition 3. *The asymptotics of (1) and (2) as are*(i)*,*(ii)*,*(iii)*, where .*

*Proposition 4. The asymptotics of (1) and (2) as are(i),(ii),(iii).*

*The shapes of the density and hazard functions of can be described analytically. The critical points of the pdf are the roots of the equation*

*There may be more than one root to (8). Let . We haveIf is a root of (9), then it corresponds to a local maximum if for all and for all . It corresponds to a local minimum if for all and for all . It refers to a point of inflection if either for all or for all .*

*The critical points of the hrf of are obtained from*

*Again, there may be two roots to (10). Let . We haveIf is a root of (11), then it corresponds to a local maximum if for all and for all . It corresponds to a local minimum if for all and for all . It refers to a point of inflection if either for all or for all .*

*4. Quantile Function and Simulation*

*4. Quantile Function and Simulation*

*The qf of the family is given bywhere is the inverse of the baseline cdf. The family is easily simulated by Algorithm 1.*