Bayesian Inference on the Generalized Gamma Distribution using Conjugate Priors 1

This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techniques to estimate its parameters. Many authors con-sidered estimating the parameters of the generalized gamma distribution in a Bayesian framework using Jeffrey’s priors. Others used different loss functions and the least squares approach. This study uses Bayesian techniques to estimate the three-parameter generalized gamma distribution by using conjugate priors. The random Metropolis algorithm is used to simulate the Bayesian estimates of the three parameters. Then these estimates are compared to the maximum like-lihood estimates using the mean error through simulation. It has been shown in this paper that the obtained estimates using this approach is more accurate than the traditional methods of estimation such as the Maximum likelihood method. The same approach is then used to estimate the parameters of mixtures of the generalized gamma parameters using conjugate priors. Department of Mathematics Faculty of Science, Islamic University of Gaza, Gaza Strip, Palestine * Corresponding author e-mail address: riffim@iugaza.edu.ps K e y w o r d s : Bayesian techniques, generalized gamma distribution, conjugate prior, max-imum likelihood estimator, method-of-moments. Bayesian Inference on the Generalized Gamma Distribution using Conjugate Priors

We consider the three-parameter generalized gamma distribution first introduced by joseph in [11]. The main goal of this paper is to estimate the parameters of this distribution using the conjugate priors and Bayesian techniques. If f (x | θ) is an exponential family, with density f (x | θ) = C(θ)h(x) exp(φ(θ)S(x)), then a conjugate prior distribution for θ exists and the prior distribution P (θ) ∝ C(θ) exp(φ(θ)b) is conjugate to the likelihood of the exponential distribution (see [6]).
A random variable X is said to have a generalized gamma distribution if its probability density function (pdf) has the following form x γ e −ax δ , x > 0, δ > 0, a > 0, γ > −1. (1.1) In the above density a is the scale parameter, γ and δ are the shape parameters. The mean and the variance of this distribution, are respectively given by Many authors considered estimating the parameters of the generalized gamma distribution in a Bayesian framework. For example, Naqash et. al. [16] obtained Bayesian estimators of the unknown parameters of the three parameter generalized gamma distribution, based on several priors using different loss functions. Clifford Cohen and Betty Jones Whitten [7] were concerned with the modifications of both maximum likelihood and moment estimators for parameters of the three-parameter gamma distribution. Stacy and Mihram [23] derived parameter estimation techniques for the generalized gamma distribution. Vani et. al. [26] estimated the three-parameter gamma distribution by using likelihood, spacings and least squares approach. Upadhyay et. al. [25] proposed Bayesian inference in life testing and reliability by using Markov Chain Monte Carlo (MCMC). Pang et. al. [18] used MCMC techniques to carry out a Bayesian estimation procedure using Hirose's simulated data. Pandey and Rao [17] derived Bayesian estimation of scale parameter of generalized gamma distribution using precautionary loss function. Balakrishnan et. al. [4] proposed some simple efficient estimators for the three-parameter gamma distribution. Shukla and Kumar [22] obtained Bayes estimators of the scale parameter of a generalized gamma type model by using several priors. Hirose [9,10] used maximum likelihood parameter estimation and continuation method in the three-parameter gamma distribution. Reshi et. al. [21] derived Bayesian analysis of size-biased generalized gamma distribution. Bai et. al. [3] used methods of moments and maximum likelihood in the three-parameter gamma and lognormal distributions. Ahmad et. al. [2] employed Bayesian method of estimation to estimate the parameters of generalized gamma distribution using Jeffrey's and extension of Jeffrey's priors.
Ramos et. al. [19] proposed an objective Bayesian estimation approach for the parameters of the generalized gamma distribution. Gaurav et. al. [8] compared the Bayesian estimates with the maximum likelihood estimates of the scale parameter in generalized gamma type distribution with known shape parameters under different loss functions. Koutrouvelis and Canavos [12] used the empirical moment generating function for the estimation of the shape, scale, and location parameters of a three-parameter gamma distribution. Tzavelas [24] obtained the maximum likelihood parameter estimation in the three-parameter gamma distribution with the use of Mathematica. Munilla [15] obtained Bayesian conjugate analysis using a generalized inverted Wishart distribution accounts for differential uncertainty among the genetic parameters-an application to the maternal animal model. Ramos [20] obtained Bayesian reference analysis for the Generalized Gamma distribution.

Maximum Likelihood Estimation
The likelihood for a random sample x 1 , x 2 , . . . , x n of size n from the generalized gamma distribution is The corresponding log-likelihood function is Then, we simultaneously solve these equations for γ, δ, and a.

Bayesian Inference
In order to use the Bayesian techniques to estimate the parameters of the generalized gamma distribution, we consider the following cases.

Case 1: Unknown scale parameter a
When the scale parameter a is unknown and both the shape parameters γ and δ are known. By ignoring terms that contain γ, δ in (2.1), the likelihood function is given by: The conjugate prior π (a) is the gamma distribution with hyperparameters r > 0 and k > 0 π (a) = 1 By ignoring terms that contain γ, δ, the posterior distribution π (a | γ, δ, x) is the gamma distribution with hyperparametersŕ = n(γ+1) 3.2 Case 2: Unknown shape parameter γ When the shape parameter γ is unknown and both the scale parameter a and the shape parameter δ are known. By ignoring terms that contain a, δ in (2.1), the likelihood function is given by: The conjugate prior π (γ) with hyperparameters s > 0, b > 0 and t > 0 is (3.5) By ignoring terms that contain a, δ, the posterior distribution π (γ | a, δ, x) with hyperparametersś = s + n,b = b + n andt = tP is

Case 3: Unknown shape parameter δ
When the shape parameter δ is unknown and both the scale parameter a and the shape parameter γ are known. By ignoring terms that contain a, γ in (2.1), the likelihood function is given by: L (x, a, γ | δ) = δ n a n( γ+1 δ ) Γ n γ+1 δ e −a n i=1 x δ i . (3.7) The conjugate prior π (δ) with hyperparameters m > 1, s > 0, b > 0 and c > 0 is By ignoring terms that contain a, γ, the posterior distribution π (δ | a, γ, x) with hy- (3.9) 3.4 Case 4: Unknown both scale parameter a and shape parameter γ When both the scale parameter a and the shape parameter γ are unknown and the shape parameter δ is known. By ignoring terms that contain δ in (2.1), the likelihood function is given by: The joint conjugate prior π (a, γ) with hyperparameters m > 1, (3.11) By ignoring terms that contain δ, the joint posterior distribution π (a, γ | δ,

Case 5: Unknown both scale parameter a and shape parameter δ
When both the scale parameter a and the shape parameter δ are unknown and the shape parameter γ is known. By ignoring terms that contain γ in (2.1), the likelihood function is given by: The joint conjugate prior π (a, δ) with hyperparameters m > 1, (3.14) By ignoring terms that contain γ, the joint posterior distribution π (a, δ | γ, (3.15) 3.6 Case 6: Unknown both shape parameter γ and δ When both the shape parameters γ and δ are unknown and the scale parameter a is known. The likelihood function from (2.1) is given by: The joint conjugate prior π (γ, δ) with hyperparameters m > 1, By ignoring terms that contain a, the joint posterior distribution π

Case 7: Unknown three parameters a, γ and δ
When both the shape parameters γ and δ and the scale parameter a are unknown, the likelihood function is given by: The joint conjugate prior π (a, γ, δ) with hyperparameters m > 1, (3.20) The joint posterior distribution π (a, γ, where P = n i=1 x i .

Finite Mixtures of Generalized Gamma Distribution
The general form of the k-finite generalized gamma mixture is given by . , x n are independent and identically distributed observations according to the k-finite generalized gamma mixture. For each observation x i , i = 1, 2, . . . , n, the indicator parameter z i is introduced as follows: Since z 1 , z 2 , . . . , z n are independent, we write the joint indicator density as the following form: . . , (x n , z n ) denote the complete data. Since for fixed i, only one of z ij equal to 1, the joint pdf of the observed observation x i and the unobserved indicator parameter z i can be written as Since (x 1 , z 1 ) , (x 2 , z 2 ) , . . . , (x n , z n ) are independent, the complete data likelihood of this mixture is given by: i , for all j = 1, 2, . . . , k.

z i Posterior
When the indicator parameter z i is unknown, for all observation x i , i = 1, 2, . . . , n and the scale parameter a, the shape parameters γ, δ and the weight parameter λ are known.

λ Posterior
When the weight parameter λ is unknown and the scale parameter a and the shape parameters γ, δ are known. By ignoring terms that contain a, γ, δ in (4.6), the complete data likelihood function is given by: 9) where N j is the number of the observations, which belong to the j th component, for j = 1, 2, . . . , k and we compute N j as the following The conjugate prior π (λ) is a Dirichlet distribution with hyperparameters µ = (µ 1 , µ 2 , . . . , µ k ) where 0 < λ j < 1, µ j > 0, for all j = 1, 2, . . . , k and k j=1 λ j = 1.

Joint Posterior of a, γ, δ
When the weight parameter λ is known and the shape parameters γ, δ and the scale parameter a are unknown. By ignoring terms that contain λ in (4.6), the complete data likelihood function is given by: The joint conjugate prior π (a, γ, δ, λ) with hyperparameters m j > 1, s j > 0, b j > 0, c j > 0 and t j > 0 is By ignoring terms that contain λ, the joint posterior distribution π (a, γ, δ, λ | x, z) with where P j = n i=1 x z ij i , for j = 1, 2, . . . , k.

Simulation Results
In this section, a simulation study using Monte Carlo methods is presented to compare the efficiency of MLE method with Bayesian method of estimation using by computing the mean of the sum of the modulus of the bias (MBias), and the root-mean square error (RMSE), where the smaller RMSE and MBias indicates a better overall quality of the estimates. To take care of small, medium and large data sets, we investigate the performance of the proposed prior distributions through a simulation study with samples of size 100, 150, 200, 250 and 300 generated from the generalized gamma distribution with parameters (a = 6, γ + 1 = 3, δ = 5). To find the MLE estimators, the Newton-Raphson method was adopted. The parameters (a, γ + 1, δ) are estimated with random walk Metropolis method (RWM) of estimation using the joint prior in (3.20) with hyperparameters (c = 2, m = 4.5, t = 6, s = 4 and b = 3 , where the simulation study was carried out 10, 000 times for (a, γ + 1, δ). The working of the random walk Metropolis method is greatly affected by the standard deviations of the distribution that gives the proposals. A large standard deviation allows for large values for the proposal, and then the ratio is often small, which results in small acceptance probabilities and frequent rejection of the proposal. Table 1 present the estimates (Est.) and the RMSE and MBias values by MLE and RWM method. The smaller RMSE and MBias for each sample size is highlighted in bold. Looking at these tables we observe that: In general when sample size increases, the relative MBias and the RMSE inecrease for random walk Metropolis method, and decrease for MLE method. Moreover, we obtained that the random walk Metropolis method is uniformly better than MLE.
On the other hand, We consider the GG distribution to analyze a number of real lifetime data sets. The parameters a, γ + 1 and δ are estimated with random walk Metropolis method (RWM) of estimation for all fifteen real lifetime data sets 1 − 4. For each case, 10, 000 samples were simulated using the joint prior in (3.20) with hyperparameters (c = 3, m = 5, t = 4, s = 5 and b = 2) to be used to get the posterior summaries. Table 3 display the means, medians, standard deviations and CI 95% of the RWM estimators. Data set 1: This data set related to the cycles to failure for a batch of 60 electrical appliances in a life test introduced by Lawless [13]. The data is as follows :  0.014, 0.034, 0.059, 0.061, 0.069, 0.080, 0.123, 0.142, 0.165, 0.210, 0.381, 0.464, 0.  Data set 3: This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee and Wang [14]. The data is as follows: Data set 4: This data set that represents the remission times (in months) of a random sample of 128 bladder cancer patients (see [14]  Also, we choose samples of size 50, 100, 150, 200 and 250 to generate the data set of two-component mixture generalized gamma distribution with parameters (a 1 = 3, a 2 = 4, γ 1 + 1 = 2, γ 2 + 1 = 6, δ 1 = 5, δ 2 = 2, K = 2, λ = 0.5). The parameters λ, a 1 , a 2 , γ 1 + 1, γ 2 + 1, δ 1 and δ 2 are estimated with MLE and random walk Metropolis method (RWM). Then we compare the performance of MLE and Bayes estimators. For each simulated sample, 1000 iterations were performed using random walk Metropolis method. Bayes estimator is obtained by using the joint prior (4.23) and (4.11) with hyperparameters (c 1 = 3, m 1 = 5, t 1 = 4, s 1 = 5, b 1 = 2, µ 1 = 1, c 2 = 2, m 2 = 4.5, t 2 = n methodâ 1γ1 + 1δ 1â2γ2 + 1δ 2λ  Table 4: MBias and RMSE of the MLE estimates and the RWM estimators for twocomponent mixture Generalized Gamma distribution 6, s 2 = 4, b 2 = 3, µ 2 = 1) and the proposal is generated from six-variate normal distribution that has independent components with means of zero and known standard deviations. To avoid the local maximum, we restart the EM algorithm 20 times and choose the result with the highest log-likelihood. Table 4 present the estimates (Est.) and the RMSE and MBias values by MLE and RWM method. The smaller RMSE and MBias for each sample size is highlighted in bold. It is observed that Bayes estimator fairs better than MLE in all cases.