The Generalized Odd Gamma-G Family of Distributions : Properties and Applications

Recently, new continuous distributions have been proposed to apply in statistical analysis in a way that each one solves a particular part of the classical distribution problems. In this paper, the Generalized Odd Gamma-G distribution is introduced . In particular, G has been considered as the Uniform distribution and some statistical properties such as quantile function, asymptotics, moments, entropy and order statistics have been calculated. We survey the theoretical outcomes with numerical computation by using R software.The fitness capability of this model has been investigated by fitting this model and others based on real data sets. The maximum likelihood estimators are assessed with simulated real data from proposed model. We present the simulation in order to test validity of maximum likelihood estimators .

Based on T-X idea by Alzaatreh et al. (2013), by the following definition, the Generalized The Generalized Odd Gamma-G Family of Distributions Odd Gamma-G distribution (GOGa-G) would be made where α, β > 0 are two additional shape parameters, ξ is the parameter for baseline G and γ(α, x) = x 0 t α−1 e −t dt denote the incomplete gamma function.In this case, the probabilty density function (pdf) of the GOGa-G distribution will be as follows: where g(x; ξ) is the pdf of the G(x; ξ) distribution.From now on, the random variable X with pdf (2) is shown with X ∼ GOGa-G(α, β, ξ).According to (1) and ( 2) hrt of X is as follows: An interpretation of the GOGa-G family (1) can be given as follows: Let T be a random variable describing a stochastic system by the cdf G(x) β (for β > 0).If the random variable X represents the odds ratio, the risk that the system following the lifetime T will be not working at time x is given by G(x) β 1−G(x) β .If we are interested in modeling the randomness of the odds ratio by the Gamma pdf r(t) = 1 Γ(α) t α−1 e −t (for t > 0), the cdf of X is given by which is exactly the cdf (1) of the new family.
Theorem 1 provides some relations of the GOGa family with other distributions.
The basic motivations for using the GOGa family in practice are the following: (i) to make the kurtosis more flexible compared to the baseline model; (ii) to produce a skewness for symmetrical distributions; (iii)to construct heavy-tailed distributions that are not longer-tailed for modeling real data; (iv)to generate distributions with symmetric, leftskewed, right-skewed and reversed-J shaped; (v) to define special models with all types of the hrf; (vi) to provide consistently better fits than other generated models under the same baseline distribution.
In the following, the paper would be like this: In Section 2, a special distribution is introduced by selecting G.In Section 3, the features of the GOGa-model will be assessed using quantile function, asymptotics, functions expansion, quantile power series, moments, entropy and order statistics.In Section 4, MLE calculation method and in Section 5, estimability of the model additional parameters will be discussed using simulation.In Section 6, the proposed model is fitted based on two real data sets and compared to other famous models.

Special models
2.1.The generalized odd gamma-uniform (GOGa-U) Different distributions family can be reached by selecting different Gs in equation (2).Torabi and Hedesh (2012), G has been considered as uniform distribution.In this case, by letting ξ = (a, b) equation (2) will changed as follows: where α, β > 0, a, b ∈ R and a < b.If X be a random variable with density function (4), then it will be displayed by GOGa-U(α, β, a, b).In Figure 1 some density and hazard functions for GOGa-U have been drawn.
One can see in the curves of Figure 1 that the different states of density function including symmetric density function (approximately), mild and high skewed (right and left) and bimodal (in the right bottom curve, one mode is in point zero) have been produced.In Figure 2 one can see some curves of the hazard function of the GOGa-U distribution for some paramentreters.According to Figure 2 you see that the U shape hazard functions are producible by GOGa-U.

The generalized odd gamma-Weibull (GOGa-W)
In GOGa-G, suppose G is as follows Weibull distribution function: In this case, by letting ξ = (λ, k) equation ( 2) will be changed as follows where α, β, λ, k > 0. If X be a random variable with density function (5), then it will be displayed by GOGa-W(α, β, λ, k).In Figure 3 some pdfs for GOGa-W have been drawn.

Quantile function
By considering (1) quantile function (qf) X is obtained as follows: If V ∼ Γ(α, 1) then the solution of nonlinear equation

Expansion for Pdf and Cdf and hrf
Using generalized binomial and taylor expansion one can obtain where .
, denote the pdf of exp-G distribution with power parameter β.
The Generalized Odd Gamma-G Family of Distributions By integrating from equation ( 6) with respect to x, we have where ] and binomial expansion we have: In this case, regarding to (7) cdf extends as follows then where and finally regarding to (8) for cdf we also have

Moments
The rth ordinary moment of X is given by Using (1), we obtain the following: Hereafter, Y k+1 denotes the Exp-G distribution with power parameter (k + 1).Setting r = 1 in (10), We have the mean f X.The last integration can be computed numericaly for most parent distributions.The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.The nth central moment of X, say M n , follows as The cumulants (κ n ) of X follow recursively from where The skewness and kurtosis measures also can be calculates from the ordinary moment using well-known relationships.The moment generating function (mgf) of X, say M X (t) = E e tX , is given by

Incomplete moments
The main application of the first incomplete moment refers to Bonferroniand Lorenz curves.These curves are very useful in economics, reliability, demography, insurance and medicine.
The answers to many important questions in economics require more than just knowing the mean of the distribution, its shape as well.This is obvious both in the study of econometrics and in areas as well.The sth incomplete moments, say ϕ s (t), is given by Using equation ( 8), we obtain The first incomplete of the GOGa-G family, ϕ 1 (t), can be obtained by setting s = 1 in (11).
Another application of the first incomplete moment is related to meanresidual life and mean waiting tie given by m , respectively.

Entropy
Entropy is an index for measuring variation or uncertainty of a random variable.The measure of entropy, Rrnyi (1961), is defined as follows for γ > 0 and γ = 1.The Shannon entropy measure is also defined by E {− log [f (x)]} that is a special state of the Rényi entropy when γ ↑ 1.
where i+j) dx.In Figure 5 one can see some curves of the entropy function of the GOGa-U distribution for some parameters.

Order statistics
Order statistics make their appearance in many areas of statistical theory and practice.Suppose X 1 , . . ., X n is a random sample from any GOGa-G distribution.Let X i:n denote the ith order statistic.The pdf of X i:n can be expressed as where c = 1 B(i,n−i+1) .We use the result 0.314 of Gradshteyn and Ryzhik (2000) for a power series raised to a positive integer n (for n ≥ 1) where the coefficients c n,i (for i = 1, 2, . ..) are determined from the recurrence equation (with By using equations ( 9), ( 12), ( 13), We can demonstrate that the density function of the ith order statistic of any GOGa-G distribution can be expressed as follows: where h r+k+1 (x) denotes the exp-G density function with parameter r + k + 1, b r is given by equation ( 9) and the quantities f j+i−1,k can be determined given that f j+i−1,0 = b j+i−1 0 and recursively for k ≥ 1 We can obtain the ordinary and incomplete moments, generating function and mean deviations of the GOGa-G order statistics from equation ( 14) and some properties of the exp-G model.

The maximum likelihood estimator
The MLE is one of the most common point estimators.This estimator is very applicable in confidence intervals and hypothesis testing.By MLE, various statistics is built for assessing the goodness-of-fit in a model, such as: the maximum log-likelihood ( ˆ max ), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Anderson-Darling (A * ) and Cramér-von Mises (W * ), described by Chen and Balakrishnan (1995).The lower values of these statistics indicate that the model have better fitting.We use these statistics in section 5.
The Generalized Odd Gamma-G Family of Distributions To calculating the MLE, let x 1 , x, ..., x n are observations from pdf (2).In this case, by letting θ = (α, β, ξ) we have By numerically solving the following equations, the maximum likelihood estimators can be obtained.

Simulation study
In this section, the Maximum likelihood estimators for additional parameters α and β in pdf (4) for three different states, has been assessed by simulating: (α, β) = (0.6, 1.6), (α, β) = (2, 2) and (α, β) = (13, 0.1).In each three case, the uniform distribution parameters in ( 4) are (a, b) = (0, 10).The density functions for one of the three states, has been indicated in Figure 6.One can see three different states of GOGa-U density functions, means skewed to the left, right and the symmetric .
To verify the validity of the maximum likelihood estimator, Mean Square Error of the Estimate (MSE), Coverage Probability (CP) and Coverage Lenght (CL) have been used.For example, as described in Section 3.1, for (α, β) = (0.6, 1.6), N = 10000 times have been simulated samples of n = 30, 40, ..., 500 of GOGa-U(0.6,1.6, 0, 10).To estimate the numerical value of the maximum likelihood, the optim function (in the stat package) and L-BFGS-B method in R software has been used.If θ = (α, β), for any simulation by n volume and i = 1, 2, ..., N , the maximum likelihood estimates are obtained as θ i = ( α i , β i ).The standard deviation of estimations, which is obtained through the information matrix is shown by In this case, the MLE, Bias, MSE, CP and CL are calculated by the following formula Bias In Figures 7 represent the Biases, MSEs, CPs and CLs plots for (α, β) = (0.6, 1.6) .As expected, the biases and MSE of estimated parameters converges to zero while n growing.
The CPs plots should converge to 0.95 and CLs plots should be descending they are correct in Figures 7. Plots of parameters vector (α, β) = (2, 2) and (α, β) = (13, 0.1) have the same position that one can see in Appendix 7.1.

The myelogenous leukemia data for AG negative
This sub-section is related to study of AG data which presented by Feigl and Zelen (1965) that include 16 observations.Observed survival times (weeks) for AG negative were identified by the presence of Auer rods and significant granulative of the leukemic cells in the bone marrow at diagnosis.For the AG negative patients these factors were absent.The data set is: 56,65,17,17,16,22,3,4,2,3,8,4,3,30,4,43.

Conclusions
In many applied areas there is a clear need for extended forms of the well-known distributions.Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis.We propose Generalized Odd Gamma-G (GOGa-G) family of distributions.Many well-known models emerge as special cases of the GOGa-G family by using special parameter values.Some mathematical properties of the new class including explicit expansions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, entropies and order statistics are provided.The model parameters are estimated by the maximum likelihood estimation method.We prove empirically by means of an application to a real data set that special cases of the proposed family can give better fits than other models generated from well-known families.

Acknowledgement
The support of Research Committee of Persian Gulf University is greatly acknowledged.

Figure 5 :
Figure 5: Curves of the GOGa-U entropy function for some parameter values.

Figure 6 :
Figure 6: Three density functions for simulation study.

Figure 8 :
Figure 8: Histogram and estimated pdfs for the AG negative data.

Table 1 :
Alzaatreh (2015)eria for the AG negative data.The second data set which contains 202 observation can be seen inWeisberg (2005)that have been used inAlzaatreh (2015)(article not yet published).These data are the sum of skin folds in 202 athletes collected at the Australian Institute of Sports and are as follows: