The gamma-normal distribution: Properties and applications
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 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 as a gamma random variable, as any continuous random variable and study some general properties of the gamma- family. Second, we study in detail the gamma-normal distribution, which is a member of the gamma- family.
Let be the cumulative distribution function (CDF) of any random variable and be the probability density function (PDF) of a random variable defined on . The CDF of the family of distributions defined by Alzaatreh et al. (2013b) is given by Alzaatreh et al. (2013b) named the family of distributions defined in (1.1) the ‘Transformed-Transformer’ family (or family). When is a continuous random variable, the probability density function of the family is 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 follows the gamma distribution with parameters and , . The definition in (1.2) leads to the gamma- family with the PDF The CDF of the gamma- distribution in (1.3) can be written as where is the incomplete gamma function.
The Weibull- 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 in the 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- 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- family
In this section, some general properties of the gamma- in (1.3) are discussed. The following are some special cases of the gamma- family:
- 1.
When , the gamma- family in (1.3) reduces to , which is the distribution of the first order statistic from a random sample of size with PDF .
- 2.
Arnold et al. (1998, Chapter 2) defined the CDF of the upper record value as The corresponding PDF is If
Gamma-normal distribution and some of its properties
We now focus on the gamma- family member with normal random variable . If is a normal random variable with PDF and CDF , then (1.3) gives the gamma-normal distribution with parameters and as where and .
When , the gamma-normal distribution in (3.1) reduces to the normal distribution with parameters and . 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 be taken from the gamma-normal distribution. The log-likelihood function for the gamma-normal distribution in (3.1) is given by Using the facts that and , the derivatives of (4.1) with respect to and are given by
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- family are provided. A special case of the gamma- family, the gamma-normal distribution is studied. The gamma-normal distribution is a generalization of normal distribution. In general, the gamma- distribution is a generalization of the 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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