A Generalization of Generalized Gamma Distributions

For the first time, a new generalization of generalized gamma distribution called the modified generalized gamma distribution has been introduced to provide greater flexibility in modeling data from a practical viewpoint. The new distribution generalizes some recently introduced generalizations of the gamma and beta distributions . Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment generating function, mean deviation, mean residual lifetime and expression of the entropies are derived. The distribution is capable of monotonically increasing, decreasing, bathtub-shaped, and upside-down bathtub-shaped hazard rates. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving nonlinear equations only. Two real data sets have been analyzed to show how the proposed models work in practice.


Introduction
In recent years, it is a common practice in the statistical distribution theory to add an extra parameter to an existing family of distribution functions. Such a technique (adding an extra parameter) is adopted to bring in more flexibility to a class of distribution functions. Besides, it can be very useful for data analysis purposes. For instance, Azzalini (1985) created the skew normal distribution by the addition of an extra parameter to the normal distribution so as to add flexibility to the normal distribution. Eugene et al. (2002) put forward the beta generated method that uses the beta distribution with parameters  and  as the generator to enhance the beta generated distributions. Alzaatreh et al.
(2013) proposed a new method for generating families of continuous distributions called T-X family by replacing the beta PDF with a PDF, r(t), of a continuous random variable and applying a function W(F(x)) that satisfies some specific conditions. Of late, Aljarrah et al. (2014) employed quantile functions to generate T-X family of distributions. For assessment of methods for generating distributions see Lee et al. (2013) &Jones (2015).
The gamma distribution is the most popular model for analyzing skewed data. In the last few years, many generalizations of gamma and Weibull distributions are proposed. These generalizations are mainly introduced in order to extend the scope of ordinary gamma and Weibull distributions and to develop a model for failure time to suit any given This paper proposes a new six parameter generalization of GG distribution, called the modified generalized gamma (MGG) distribution. It includes as special sub models such as the generalizations of gamma distribution introduced by Stacy (1962), Hoq et al. (1974), Lee and Gross (1991), Agarwal and Kalla (1996), Agarwal and Al-Saleh (2001), Kalla et al. (2001) and also the generalized beta distribution of the second kind (GB2) (McDonald,(1984)), among others. We are motivated to introduce the MGG distribution because (i) it contains a number of known lifetime sub models such as gamma, Weibull, exponential, Rayleigh, Maxwell, Chi-square, folded normal, beta type-II, Burr XII, Burr III, log-logistic distributions and so on; (ii) it is capable of modeling monotonically increasing, decreasing, bathtub-shaped, and upside-down bathtub-shaped hazard rates; (iii) it can be viewed as a suitable model for fitting the skewed data which may not be properly fitted by other common distributions and can also be used in a variety of problems in different areas such as public health, biomedical studies, and industrial reliability and survival analysis; and (iv) two real data applications show that it compares well with other competing lifetime distributions in modeling lifetime data. This paper is organized in the following way: In Section 2, the MGG is defined and some basic distributional properties of the new model are studied. In Section 3, some wellknown and new lifetime models as members of MGG are derived. Properties of the MGG distribution are studied in Section 4 including, quantile, mode, moments, moment generating function, mean deviation, mean residual life and entropy. The maximum likelihood estimates (MLEs) of the model parameters and the corresponding observed Fisher information matrix are obtained in section 5. The potentiality of the new model is illustrated by means of application to two real data sets in Section 5. Finally, some concluding remarks are addressed in Section 6.

The Modified Generalized Gamma Distribution
The generalized gamma function which is essentially a confluent hypergeometric function has been considered by Kobayashi (1991) iii.
For the case of 0 b  ,1 k  and , h   equation (2) yields the standard form of beta function of the second type as .
Based on equation (2), we can define the following probability density function (pdf) applying the transformation 1 xy    in (5), the pdf of the MGG distribution is given by Plots of the density function (6) for selected parameter values are given in Figure 1.

Special Distributions
The MGG distribution has several distributions as special cases, which makes it distinguishable scientific importance from other distributions. In this section, we investigate the various special models of the MGG distribution.
In this case various special models of the MGG distribution are listed in table 1.
, MGG corresponds to the generalized life testing model given by Hoq et al. (1974) , we get the generalized gamma model defined by Lee and Gross (1991) as (6) becomes the generalized gamma model defined by Agarwal and Kalla (1996) as , where 0, , , , , , 0.
xm       From (11), the following special cases can be derived (i) With 1   and 1 mm  we get the pdf (10).
(ii) If 0   , the density (11) reduces to the general form of Weibull distribution with the following pdf 1 1 ( , , , and h are the shape parameters and  is the scale parameter. The GB2 is most useful for unifying a substantial part of the size distributions literature.It contains a large number of income and loss distributions as special or limiting cases. Full details can be found in Brazauskas (2002), Kleiber and Kotz (2003). Table 2 lists various special models of the MGG distribution when 0, b  1 k  and h  .

Properties of the MGGD
In this section, we provide some general properties of the MGG distribution including quantile function, mode, moments, mean deviation, mean residual life and mean waiting time, Rényi entropy and order statistics.

Mode and quantile
The p-thquantile function of the MGG distribution is the solution of ( , , , ( ) ) , ( , , ) particularly, the median, denoted by *  , can be obtained from (14) It is noted that, when 0 k  , the mode becomes Equations (14) and (15) are used to obtain the median and the mode for the MGG distribution. Median and mode values are reported in Table 3   From Table 3, it is to be noted that for fixed , , , bk  and  , the median and the mode of MGG distribution are increasing functions of .

Moments, generating function and mean deviation
From (6) it is easy to obtain the r -th moment about zero of MGG distribution as Also, the central moments of MGG distribution can be obtained as follows (2 ) , , , , The moment generating function The mean deviation (MD)can be derived as (16) and (17) are used to obtain the mean, variance, skewness and kurtosis for the MGG distribution. The results are shown in Table (4) using the parameters values proposed in the previous subsection. From Table 4, it is to be noted that for fixed , , , bk  and  , the variance and the skewness are decreasing functions of  , while for fixed , , , bk  and  ,the mean is increasing function of  . Also, it can be seen that the MGG distribution can be positively skewed, negatively skewed, platykurtic or leptokurtic.

Mean Residual Life and Mean Waiting Time
If the random variable X follows the MGG distribution with pdf given in (6), then the mean residual life function, say () t  , is given by is the survival function and substituting (19) in (18), the mean residual function can be written as ( 1/ , , , ( / ) ) ( ) . ( , , , ( / ) ) The mean waiting time of X , say () t  , is defined by

Entropy
The entropy of a random variable X measures the variation of the uncertainty. The Rényi entropy, say for the MGG distribution with PDF given by (6) It is to be noted that when 1   , the Rényi entropy converges to the Shannon entropy.

Order statistics
Suppose that 12 , ,..., n X X X be a random sample of size n, then the PDF of the i-th order statistic : in X , say : () in fx is given by 1 : Substituting the pdf of MGG distribution given by (6) and the corresponding cdf in equation (22) The pdf of the minimum and the maximum order statistics of MGG distribution can be obtained, respectively, from (23) as follows   The joint pdf of the i-th and the l-th order statistics can be written as x C k b y kb

Estimation of Parameters
Here, we consider the estimation of the unknown parameters of the MGG distribution by the method of maximum likelihood. Let 12 , ,....., n x x x be a random sample from the MGG distribution. The total log-likelihood ( ) is given The MLEs of the parameters are the solutions of the nonlinear equations 0  .The observed information matrix is given by whose elements are listed in Appendix.

Applications
Here, we use two real data sets to compare the fits of the MGG distribution with several other com-petitive models namely: is the beta function

Application 1: Carbon Data
The first data set is taken from Nichols and Padgett (2006)

Application 2: Repair Times Data
The second data corresponds to 46 observations reported on active repair times (hours) for an airborne communication transceiver discussed by Alven (1964). The data are: 0. For each model, we estimate the unknown parameters by maximum likelihood method. Tables 5 and 7 lists the MLEs (and the corresponding standard errors in parentheses) of the parameters of all the above models for both the carbon data and repair times data, respectively. We apply formal goodness-of fit tests in order to verify which distribution fits better for both data sets. The statistics we use are:2 ( )   (where()  denotes the loglikelihood function evaluated at the maximum likelihood estimates), Kolmogorov-Smirnov (K-S) and p-values are presented in tables 6 and 8. In general, the smaller the values of these statistics, the better the fit to the data. From these tables we observe that the MGG distribution has the lowest2 ( ),   and K-S and largest p-value among all the other models, and so it could be chosen as the best model. The histogram for both carbon data and repair times data sets and their estimated pdfs for the fitted models are displayed in Figures 3(a) and 4(a)respectively. Also, the plots of the fitted MGG survival and the empirical survival functions for both carbon data and repair times data sets displayed in Figure 3(b) and Figure4(b), respectively. Therefore, the proposed model provides a better fit to these data.

Conclusion
The six-parameter MGG distribution, whose hazard function can be monotonically increasing, decreasing, bathtub and upside down bathtub-shaped depending on the parameter values, is introduced and studied. Some mathematical and statistical properties of the new model are investigated. We estimate the model parameters using maximum likelihood and determine the observed information matrix. The potentiality of the new model is illustrated by means of application to two real data sets. We hope that this model may attract wider applications, since the formulae derived are manageable using modern computer facilities.