Beta Exponentiated Gamma Distribution: Some Properties and Estimation

The exponentiated gamma (EG) distribution is one of the important families of distributions in lifetime tests. In this paper, a new generalized version of this distribution which is called the beta exponentiated gamma (BEG) distribution has been introduced. The new distribution is more flexible and has some interesting properties. A comprehensive mathematical treatment of the BEG distribution has been provided. We derived the rth moment and moment generating function for this distribution. Moreover, we discussed the maximum likelihood estimation of this distribution under a simulation study.


Introduction
Gamma distributions are some of the most popular models for hydrological processes. One of the important families of distributions in lifetime tests is the exponentiated gamma (EG) distribution. The exponentiated gamma (EG) distribution has been introduced by Gupta et al. (1998), which has cumulative distribution function (CDF) and a probability density function (pdf) of the form, respectively; ( , , ) 1 (1 ) , 0, 0 and 0 where  and  are scale and shape parameters respectively. The corresponding probability density function (pdf) is given by 1 2 ( , , ) Shawky and Bakoban (2008) discussed the exponentiated gamma distribution as an important model of life time models and derived Bayesian and non-Bayesian estimators of the shape parameter, reliability and failure rate functions in the case of complete and type-II censored samples. In addition, the order statistics from exponentiated gamma distribution and associated inference was discussed by Shawky where 0 a  and 0 b  are two additional parameters whose role is to introduce skewness and to vary tail weight. And is the beta function. The application of the 1 ()  yields X with CDF (2). The class of generalized distributions (3) has been receiving considerable attention over the last years, in particular, after the recent studies by Eugene et al. (2002) and Jones (2004). Eugene et al. (2002) introduced what is known as the beta normal () BN distribution by taking () Gx in (3) to be the CDF of the normal distribution and derived some of its first moments. Many authors considered various forms of G and studied their properties: Nadarajah and Kotz (2004) introduced the beta Gumbel () BGu distribution by taking () Gx to be the CDF of the Gumbel distribution and provided closed form expressions for the moments, the asymptotic distribution of the extreme order statistics and discussed the maximum likelihood estimation procedure. Nadarajah and Gupta (2004) , . The properties of () Fx for any beta G distribution defined from a parent () Gx in (3) could, in principle, follow from the properties of the hyper geometric function which are well established in the literature; see, for example, Section 9.1 of Gradshteyn and Ryzhik (2000).
The probability density function (pdf) corresponding to (3) can be put in the form we noted that () fx will be most tractable when the CDF () Gx and pdf ( ) ( ) d dx g x G x  have simple analytic expressions. Except for some special choices for () Gxin Equation (3), as is the case when () Gxis given by Equation (1), it seems that the pdf () fx will be difficult to deal with in generality. Now we introduce the four parameter beta exponentiated Gamma () BEG distribution by taking () Gx in (3) to be the CDF (1). The CDF of the BEG distribution is given by 1 ( and   1 1 2 The probability density function in Equation (6) does not involve any complicated function. If X is a random variable with pdf (6), we write ( , , , ) BGE distribution generalizes some well-known distributions in the literature. If 1 ab , we get Exponentiated gamma distribution, also when the shape parameter 1 ab     in both (1) and (2) give the CDF and pdf of gamma distribution with shape parameter 2   and scale parameter For more details about this distribution, see: Shawky and Bakoban (2008).
The rest of the paper is organized as follows. In section 2, we demonstrate that the () BEG density function can be expressed as a linear combination of the exponentiated gamma. This result is important to provide mathematical properties of the BEG model directly from those properties of the exponentiated gamma distribution. In section 3, we 20  discussed some important statistical properties of the () BEG distribution such as quantile, and the ordinary moments and measures of skewness and kurtosis. The distribution of the order statistics is expressed in section 4. The maximum likelihood estimates of the four parameter index to the distribution are presented in section 5. The section 6 contains the simulation study. Finally the discussion and conclusion regarding the study have been presented in the section 7.

Expansion for the Density Function
In this section, we presented two formulae for the CDF of the BEG distribution depending if the parameter Using the expansion (7) in (5), the CDF of the BEG distribution when 0 b  is real non-integer follows 1 ( Equation (8) reveals the property that the CDF of the BEG distribution can be expressed as an infinite weighted sum of CDFs of EG distributions, (7) in (5) , we get Again, the same property of Equation (8) holds but now the sum is finite. Expression (8) and (9) are the main results of this section. Also the pdf (6) can be expressed in mixture from in terms of CDFs of the exponentiated gamma distributions, if b is real noninteger, we have

Statistical Properties
This section is devoted to studying statistical properties of the () BEG distribution, specifically the quantile function, moments and moment generating function

Quantile Function
The quantile q x of the () BEG distribution can be easily given as 1 (1 ) is given by the following relation

Moments
In this subsection we discussed the th r moment for () BEG distribution. Moments are necessary and important in any statistical analysis, especially in applications. It can be used to study the most important features and characteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis).

Theorem (3.1)
, ( , , , ) ab   then the th r moment of X is given by the following

Proof:
Let X be a random variable with density function (10). The th r ordinary moment of the () BEG distribution is given by Thus the th r moment is given by   which completes the proof .
Based on the first four moments of the () BEG distribution, the measures of skewness () A  and kurtosis () k  of the () BEG distribution can obtained as .

Moment Generating function
In this subsection we derived the moment generating function of () BEG distribution.

Theorem (3.2):
If X has () BEG distribution, then the moment generating function () X Mt has the following form

Proof.
We start with the well known definition of the moment generating function given by (19) substituting from (14) and (15)

Distribution of the order statistics
In this section, we derive closed form expressions for the pdfs of the th r order statistic of the () BEG distribution, also, the measures of skewness and kurtosis of the distribution of the th r order statistic in a sample of size n for different choices of ; nr are presented in this section. Let 12 , ,..., n X X X be a simple random sample from () BEG distribution with pdf and CDF given by (6) and (9) (20) where ( , ) Fx and ( , ) fx  are the CDF and pdf of the () BEG distribution given by (6), (9), respectively, and (.,.) B is the beta function, since 0 ( , ) 1 Fx    , for 0 x  , by using the binomial series expansion of   substituting from (6) and (9)  can be calculated.

Estimation and Inference
In this section, we determined the maximum likelihood estimates (MLEs) of the parameters of the () BEG distribution from complete samples only. Let 12 , ,..., n X X X be a random sample of size n from (  The log-likelihood can be maximized either directly or by solving the nonlinear likelihood equations obtained by differentiating (22). The components of the score vector are given by where (.)  is the digamma function. We can find the estimates of the unknown parameters by maximum likelihood method by setting these above non-linear equations (23) -(26) to zero and solve them simultaneously. As the closed for expressions for the estimators cannot be derived we have obtained the estimated values numerically.

Simulation Study
The simulation study has been conducted in order to have the numerical estimates for the parameters of the beta exponentiated gamma distribution.    We have used the real life data set regarding the breaking strengths of 64 single carbon fibers of length 10, presented Lawless (2003) for the analysis in the following table.

Discussion and Conclusion
The aim of this paper is to introduce a new distribution named beta exponentiated gamma distribution and to discuss its properties. The maximum likelihood estimates for the parameters of the distribution has been obtained. Tables 1-2 The table 3 suggests that the distribution is positively skewed and leptokurtic under different parametric values. In addition the newly introduced distribution is much more flexible than its available counterparts, so it will be very useful for modeling failure time data. The flexibility of the beta exponentiated gamma model, however, occurs at the cost of its increased complexity, but a comprehensive programming in different computer software can tackle with these complexities. In future this work can be extended by considering the analysis of the distribution under censored samples.