A New Class of Gamma distribution

This paper presents a new class of probability distributions generated from the gamma distribution. For the new class proposed, we present several statistical properties, such as the risk function, the density expansions, Moment-generating function, characteristic function, the moments of order m, central moments of order m, the log likelihood and its partial derivatives and also entropy, kurtosis, symmetry and variance. These same properties are determined for a particular distribution within this new class that is used to illustrate the capability of the proposed new class through an application to a real data set. The database presented in Choulakian and Stephens (2001) was used. Six models are compared and for the selection of these models were used the Akaike Information Criterion (AIC), the Akaike Information Criterion corrected (AICc), Bayesian Information Criterion (BIC), Hannan Quinn Information Criterion (HQIC) and tests of Cramer-Von Mises and Anderson-Darling to assess the models fit. Finally, we present the conclusions from the analysis and comparison of the results obtained and the directions for future work.


Introduction
The Gamma distribution is used in a variety of applications including queue, financial and weather models. It can naturally be considered as the distribution of the waiting time between events distributed according to a Poisson process. It is a biparamétrica distribution, whose density is given by: distribution function its generalization or exponentialization ( ) is obtained by ( ) = ( ), with > 0. Gupta et al. (1998) proposed and studied some properties exponentiated Gamma distribution. Cordeiro et al. (2011) extended the exponentiated Gamma distribution defining a new distribution called Exponentiated Generalized Gamma Distribution with four parameters, which is capable of modeling bathtub shaped failure rate phenomena. Zografos and Balakrishnan (2009) defined a family of probability distributions based on the integration of a Gamma distribution as follows: where ( ) is an arbitrary distribution function. When = + 1and = 1this distribution coincides with the distribution of the n-th highest value record (ALZAATREH et al. 2014).
A new family of probability distributions which is also based on the integration of the Gamma distribution has been proposed by Silva (2013). This author defined this new family as follows: where ( ) is an arbitrary distribution function. When = + 1 and = 1 this distribution coincides with the distribution of the nth smallest value record (ALZAATREH et al. 2014).
Following the line of work of Zografos and Balakrishnan (2009) and Silva (2013), our goal in this work is to propose a new family of distributions based on Gamma distribution. The family of distributions proposed here is the following: where 1 ( ) is an arbitrary distribution function and 1 ( ) and has the same support as the distribution 1 ( ). We shall call this new class Then, to illustrate the applicability of the proposed new family, we consider the particular case of the distribution obtained when considering that 1 ( ) is the distribution function of an exponential random variable. The statistical properties of this new distribution are also derived and to illustrate its potentiality, an application to a set of real data is performed. For this, we used the database presented in the work Choulakian and Stephens (2001) to see if the models are well adjusted to this data. As comparative criteria of fitness of the models, it was considered: the Akaike (AIC) (AKAIKE, 1972), the Akaike Fixed (AIC) (BURNHAM AND ANDERSON, 2002), the Bayesian information criterion (BIC) (SCHWARTZ, 1978), the Hannan-Quinn information criterion (HQIC) (HANNAN AND QUINN, 1979), and the Cramer-von Mises (DARLING, 1957) and Anderson-Darling (ANDERSON and DARLING, 1952) tests. Both hypothesis tests, Anderson-Darling and Cramér-von Mises, are discussed in detail by Chen and Balakrishnan (1995) and belong to the class of quadratic statistics based on the empirical distribution function, because they work with the squared differences between the empirical distribution and the hypothetical. This paper is organized as follows. In Section 2, we describe the statistical properties of the proposed Gamma class. In section 3, we studied a special case of this new class in the case in which the distribution function is an exponential random variable.
We study the properties of this new distribution and perform an application to real data illustrating its potential. Finally, we present our conclusions in Section 4.

Model functional class range (1-G 1) / G 1
The 1− 1 1 Gamma class is defined by the cumulative distribution function: which is equivalent to If the distribution 1 ( )has density 1 ( ) the class will have a probability density function given by Such functions can be rewritten as a sum of exponentiated distributions, as follows. As we have to

Furthermore, as
(1 − 1 ( )) + −1 = ∑ ( it follows that , we can rewrite the distribution function as Therefore, If the distribution 1 ( ) is discrete, 1 ( )is also discrete and we have that ( = ) = ( ) − ( −1 ). Therefore, In addition, we can obtain the risk function of the new Gamma class as follows: Using the density and distribution function expansions, we can get the statistical properties of the new class, as discussed below.

Expansion for the moments of order for the (1-G1)/G1 Gamma Class
The following is the development of the expansion calculations for the moments of order for the In particular, we have the following expansion of the mean for the

Expansion for the moment generating function for the (1-G1)/G1 Gamma
Class.
The following is the development of the expansion calculations for the moment generating function for the 1− 1 1 Gamma Class. As Using the fact that Therefore, Similarly, one can establish the following expansion for the characteristic function for the (1-G1)/G1 Gamma Class.

Class.
We'll look at the development of the expansion calculations for central moments of order to the 1− 1 1 Gamma Class. As ■ In particular, we need to expand the range of variance for the 1− 1 1 Gamma Class is given by:

Expansion to the general rate for the (1-G1)/G1 Gamma Class
We'll look at the development of the expansion calculations for the general coefficient for the In particular, as = (3)we will have the expansion for the asymmetry coefficient for the Gamma Class is given by: Similarly, as = (4),we have the expansion for the kurtosis coefficient for Gamma Class is given by:

1.6 Derivative of the log-likelihood function with respect to the parameters for the (1-G1)/G1 Gamma Class
Once met some regularity conditions, the maximum likelihood estimators can be obtained by equating the derivative of the log-likelihood function with respect to each parameter to zero. We'll look at the calculations of the derivative of the log-likelihood function with respect to the parameters for the

Entropy Rényi using the (1-G1)/G1 Gamma Class
Entropy is a measure of uncertainty in the sense that the higher the entropy value the lowest the information and the greater the uncertainty, or the greater the randomness or disorder. The following is the expansion entropy calculations for the 1− 1 1 Gamma Class, using the Rényi entropy, which is given by Substituting the expressions of density and cumulative distribution function, we Using the following expansion

Construction of a distribution from the (1-G1)/G1 Gamma Class
In this section, we will examine a particular distribution of the 1− 1 1 Gamma Class proposed here. We will consider the particular case in which 1 ( ) = 1 − − , > 0, that is called the (1-Exp)/Exp Gamma Distribution.

Risk function using the (1-Exp) / Exp Gamma distribution
We can also obtain the risk function using the Gamma distribution generated from some values assigned to parameters.

Other properties of the Gamma distribution (1-Exp) / Exp
Using the and expansion, we can obtain the statistical properties of Expansion to the general coefficient In particular, the expansion for the coefficient of asymmetry for the 1− Gamma distribution, = (3), is given by: While the kurtosis expansion coefficient of the 1− Gamma distribution, = (4), is given by: Derivatives from the log-likelihood function with respect to the parameters

Application
In this section, we will show an application to real data for the proposed Gamma distribution. The data used in this research are from the excesses of flood peaks (in m 3 / s) Wheaton River near Carcross in the Yukon Territory, Canada. 72 exceedances of the years 1958 to 1984 were recorded, rounded to one decimal place. These data were analyzed by Choulakian and Stephens (2001), and are presented in Table 3.1.1. It is worth mentioning that this data set has also been analyzed by means of the distributions of Pareto, Weibull three parameters, the generalized Pareto and beta -Pareto (AKINSETE et al, 2008).
In Table 3

Conclusion
As concluding remarks, we note that the class of (1-G1)/G1 gamma probability distributions developed in this work is a novel way of generalizing the gamma distribution and can be applied in different areas depending on the choice of the distribution G1. As future work, we intend to carry out more detailed comparisons between the novel distribution family proposed in this paper and the family of distributions investigated in Zografos and Balakrishnan (2009) and Silva (2013), which are also based on the integration of the gamma distribution.
In this work, we study in detail only a distribution of the Gamma distribution. We derive the properties of this distribution and applied to a set of real data obtaining better fit than that obtained in a previous study by Akinsete et al. (2008). We intend to conduct the study of new distributions within this class as future work.
We note that after adding several parameters to a model it can better be adjusted to a particular phenomenon due to its greater flexibility. On the other hand, one should not forget that there may be a problem for the estimation of the parameters since it can occur both computational and identifiability problems in parameter estimation. Thus, the ideal is to choose a model that reflects well the phenomenon / experiment with the minimum number of parameters. In the case of the proposed class, only two additional parameters are added to the set of parameters of the G distribution.