The gamma-normal distribution: Properties and applications

https://doi.org/10.1016/j.csda.2013.07.035Get rights and content

Abstract

In this paper, some properties of gamma-X family are discussed and a member of the family, the gamma-normal distribution, is studied in detail. The limiting behaviors, moments, mean deviations, dispersion, and Shannon entropy for the gamma-normal distribution are provided. Bounds for the non-central moments are obtained. The method of maximum likelihood estimation is proposed for estimating the parameters of the gamma-normal distribution. Two real data sets are used to illustrate the applications of the gamma-normal distribution.

Introduction

There are several methods to generate continuous distributions. Many of these methods are discussed in the book by Johnson et al. (1994, Chapter 12). Since the publication of the book, new methods continue to appear in the literature. Eugene et al. (2002) introduced the beta-generated class of distributions and pointed out that the distributions of order statistics are special cases of beta-generated distributions. Jones (2004) studied some properties of beta-generated distributions. Many beta-generated distributions have been studied (e.g., Famoye et al., 2004, Famoye et al., 2005, Nadarajah and Kotz (2006), Akinsete et al. (2008), Barreto-Souza et al. (2010), and Alshawarbeh et al. (2012)). The method leading to beta-generated distributions was extended by using a generalized beta distribution as the generator (Jones, 2009, Cordeiro and de Castro, 2011). Ferreira and Steel (2006) used inverse probability integral transformation method to generate skewed distributions, which include the skewed normal family introduced by Azzalini, 1985, Azzalini, 2005 as a special class. Recently, Alzaatreh et al. (2013b) developed a new method to generate family of distributions and called it the TX family of distributions. For a review of methods for generating univariate continuous distributions, one may refer to Lee et al. (2013). This article has two purposes. First, we take T as a gamma random variable, X as any continuous random variable and study some general properties of the gamma-X family. Second, we study in detail the gamma-normal distribution, which is a member of the gamma-X family.

Let F(x) be the cumulative distribution function (CDF) of any random variable X and r(t) be the probability density function (PDF) of a random variable T defined on [0,). The CDF of the TX family of distributions defined by Alzaatreh et al. (2013b) is given by G(x)=0log(1F(x))r(t)dt.Alzaatreh et al. (2013b) named the family of distributions defined in (1.1) the ‘Transformed-Transformer’ family (or TX family). When X is a continuous random variable, the probability density function of the TX family is g(x)=f(x)1F(x)r(log(1F(x)))=h(x)r(H(x)). Thus, the family of distributions defined in (1.2) can be viewed as a family of distributions arising from hazard functions. If a random variable T follows the gamma distribution with parameters α and β, r(t)=(βαΓ(α))1tα1et/β,t0. The definition in (1.2) leads to the gamma-X family with the PDF g(x)=1Γ(α)βαf(x)(log(1F(x)))α1(1F(x))1/β1. The CDF of the gamma-X distribution in (1.3) can be written as G(x)=γ{α,log(1F(x))/β}Γ(α), where γ(α,t)=0tuα1eudu is the incomplete gamma function.

The Weibull-X family along with a member, Weibull–Pareto distribution, was studied by Alzaatreh et al. (2013a). By using geometric distribution as the distribution of the random variable X in the TX family, Alzaatreh et al. (2012) derived the family of discrete analogues of continuous random variables. In Section  2, we provide some properties of the gamma-X family. In the remaining sections, the gamma-normal distribution is studied in detail. In Section  3, we study some properties of the gamma-normal distribution including unimodality, quantile function and Shannon’s entropy. Series representation and bounds for the non-central moments of the gamma-normal distribution are studied in Section  3. Section  4 deals with the method of maximum likelihood for estimating the parameters of the gamma-normal distribution. Applications of the distribution to real data sets are provided in Section  5.

Section snippets

The gamma-X family

In this section, some general properties of the gamma-X in (1.3) are discussed. The following are some special cases of the gamma-X family:

  • 1.

    When α=1, the gamma-X family in (1.3) reduces to g(x)=β1f(x)(1F(x))1/β1, which is the distribution of the first order statistic from a random sample of size n(=β1) with PDF f(x).

  • 2.

    Arnold et al. (1998, Chapter 2) defined the CDF of the upper record value Un as GU(u)=P(Unu)=0log(1F(u))[wnew/n!]dw. The corresponding PDF is gU(u)=f(u)[log(1F(u))]n/n! If

Gamma-normal distribution and some of its properties

We now focus on the gamma-X family member with normal random variable X. If X is a normal random variable with PDF ϕ(x) and CDF Φ(x), then (1.3) gives the gamma-normal distribution with parameters α,β,μ and σ as g(x)=1Γ(α)βαϕ(x)[log(1Φ(x))]α1[1Φ(x)](1/β)1,<x<, where α>0,β>0,σ>0 and <μ<.

When α=β=1, the gamma-normal distribution in (3.1) reduces to the normal distribution with parameters μ and σ. Thus, (3.1) is a generalization of the normal distribution. From (1.4), the CDF of the

Parameter estimation for gamma-normal distribution

Let a random sample of size n be taken from the gamma-normal distribution. The log-likelihood function for the gamma-normal distribution in (3.1) is given by logL(α,β,μ,σ)=nlogΓ(α)nαlogβn2log(2π)nlog(σ)12σ2i=1n(xiμ)2+(α1)i=1nlog(log(1Φ(xi)))+(β11)i=1nlog(1Φ(xi)). Using the facts that Φ(x)μ=ϕ(x) and Φ(x)σ=(xμ)ϕ(x)σ, the derivatives of (4.1) with respect to α,β,μ and σ are given by logLα=nlogβnψ(α)+i=1nlog(log(1Φ(xi)))logLβ=nαβ1β2i=1nlog(1Φ(xi))logLμ=n(x̄μ)σ2+

Applications of gamma-normal distribution

In this section, the gamma-normal distribution is applied to two data sets. The first data in Table 3 is from Nichols and Padgett (2006) on the breaking stress of carbon fibers of 50 mm in length. The second data set in Table 5 is from Smith and Naylor (1987) on the strengths of 1.5 cm glass fibres measured at the National Physical Laboratory in England. The maximum likelihood estimates, the log-likelihood value, the AIC (Akaike Information Criterion), the Kolmogorov–Smirnov (K–S) test

Summary

In this article, some properties of the gamma-X family are provided. A special case of the gamma-X family, the gamma-normal distribution is studied. The gamma-normal distribution is a generalization of normal distribution. In general, the gamma-X distribution is a generalization of the X distribution. Various properties of the gamma-normal distribution are investigated, including moments, bounds for non-central moments, hazard function, and entropy. The gamma-normal distribution can be over-,

Acknowledgments

The authors are grateful for the comments and suggestions by the referees and the Associate Editor. Their comments and suggestions have greatly improved the paper.

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