“Bayesian Statistical Inference for a Mixture of Two Independent Generalized Exponential Distributions”

Abstract

Inference problems concerning any parameter or set of parameters 𝜃 can be easily dealt with using Bayesian analysis. The idea is that since the posterior distribution supposedly contains all the available information about 𝜃 (both sample and prior information) any inferences concerning 𝜃 should consists solely of features of this distribution. The simplest inferential use of the posterior distribution is to report a point estimate for 𝜃 with an associate measure of accuracy. In all but very stylized problems, the integrals required for Bayesian computation require analytic or numerical approximation. These include asymptotic approximations, numerical integrations and Monte Carlo importance sampling. The method that we shall use in this thesis is the Monte Carlo methods, which estimate features of the posterior or predictive distribution of interest by using samples drawn from that distribution, or suitably reweighted samples drawn from some other appropriately chosen distribution. Often, particularly in high dimensional problems, this may be only feasible approach. In section (1) of this chapter, we define the generalized exponential distribution (GED) and some of its properties with a review of literature. Some basic concepts of finite mixture, maximum likelihood, reliability, order statistics, complete and censored data sets, Bayesian prediction and others are given in section (2). Finally, our aim of this thesis and a description to the problem of study is presented in section (3). 1. Generalized exponential distribution (GED) In this thesis we consider a population with density given by a mixture of two components each a generalized exponential distribution. Inferences about the parameters of this mixture are discussed under different types of 2 sampling. Two sample Bayesian prediction is also discussed. First we introduce the generalized exponential distribution (GED) and some of its properties, then a review of literature will be presented. The three – parameter gamma and the three – parameter Weibull are the most popular distributions for analyzing lifetime data. The three parameters, in both distributions, represent location, scale and shape. Because of these parameters both distributions have quite a bit of flexibility for analyzing skewed data. Unfortunately both distributions have drawbacks. Recently, Gupta and Kudu (1997) considered a special case (exponentiated exponential or generalized exponential distribution) of the exponentiated Weibull model assuming the location parameter to be zero, and compared its performance with the two parameter Weibull family, mainly through data analysis and computer simulations. Generalized exponential distribution has a right skewed unimodal density function and monotone hazard function similar to the density functions and hazard functions of the gamma and Weibull distributions. It was observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. (Gupta & Kundu, 2007) mentioned that, certain cumulative distribution functions were used during the first half of the nineteenth century by Gompertz (1825) and Vernhulst (1838, 1845 and 1847) to compare known human mortality tables and represent mortality growth. The following is an example: 𝐺(𝑡)=(1−𝜌𝑒−𝑡𝜆)𝛼 𝑓𝑜𝑟 𝑡>1𝜆ln𝜌 (1.1) Where 𝜌,𝛼 𝑎𝑛𝑑 𝜆 are all positive real numbers. In the twentieth century, Ahuja and Nash (1967) also considered this model and made some further generalization. The generalized exponential distribution or the exponentiated exponential distribution is defined as a particular case of the Gompertz-Verhulst distribution function (1.1), when 𝜌=1. 3 1.1. Definitions The definitions of the distribution function (CDF), the probability density function (PDF) and the survival function of the generalized exponential distribution (GED) are given. [For more details, see (Gupta & Kundu, 2003)] Definition (1.1.1): Distribution function The CDF of a two parameters GED is given by: 𝐹𝐺𝐸(𝑥;𝛼,𝜆)=(1−𝑒−𝜆𝑥)𝛼 ;𝛼,𝜆>0 ,𝑥>0 (1.2) Definition (1.1.2): Density function A continuous random variable 𝑋 is said to have GED if the PDF of 𝑋 is given by: 𝑓𝐺𝐸(𝑥;𝛼,𝜆)=𝛼𝜆(1−𝑒−𝜆𝑥)𝛼−1 𝑒−𝜆𝑥 ;𝛼,𝜆>0 , 𝑥>0 (1.3) Definition (1.1.3): Survival function Let 𝑋 be a continuous random variable with GED, then the survival function of 𝑋 is given by: 𝑅𝐺𝐸(𝑥;𝛼,𝜆)=1−(1−𝑒−𝜆𝑥)𝛼 ;𝛼,𝜆>0 , 𝑥>0 (1.4) Where 𝛼 𝑎𝑛𝑑 𝜆 play the role of the shape and reciprocal scale parameters, respectively. 1.2. Properties Naturally the shape of the density function does not depend on λ (see Figure 1.1). For different values of α, the density functions of the generalized exponential distribution can take different shapes. For α≤1, it is strictly decreasing function. When α=1, it coincides with the one-parameter exponential distribution. And for α>1, it is a unimodal, skewed, right tailed similar to the weibull or gamma density function. It is observed 4 that even for very large shape parameter, it is not symmetric. (see Figure 1.2) 𝑓(𝑥) 𝑥 Figure 1.1 Density functions of the GE distribution for different values of λ when 𝛼 is constant 𝑓(𝑥) 𝑥 Figure 1.2: Density functions of the GE distribution for different values of 𝛼 when λ is constant 1 2 3 4 0.2 0.4 0.6 0.8 1 2 3 4 0.2 0.4 0.6 0.8 λ = 0.2 λ =1 λ =2 𝛼=0.5 𝛼=3.5 𝛼=1.5 𝛼=1 𝛼=4.5 𝛼=0.8 𝛼 = 0.5 𝜆 = 1 5 The generalized exponential distribution has some nice physical interpretations. Consider a parallel system, consisting of n components, i.e the system works only when at least one of the n components works. If the lifetime distributions of the components are independent identically distributed (i.i.d) exponential random variables, then the lifetime distribution of the system becomes 𝐹(𝑥;𝑛,𝜆)=(1−𝑒−𝜆𝑥)𝑛 ,𝑥>0 (1.5) For 𝜆>0. Clearly, (1.5) represents the generalized exponential distribution function with 𝛼=𝑛. Therefore contrary to the weibull distribution function, which represents a series system, the generalized exponential distribution function represents a parallel system. 1.3. Review of literature The two-parameter generalized exponential distribution is a particular member of the three-parameter exponentiated Weibull distribution, introduced by Mudholkar and Srivastava (1993). Moreover, the exponentiated Weibull distribution is a special case of the general class of exponentiated distributions proposed by Gupta et al. (1998) as (𝑡)= [G(t)]𝛼 , where G(t) is the base line distribution function. It is observed by Gupta and Kundu (2001a) that the two-parameter generalized exponential distribution can be used quite effectively to analyze positive lifetime data, particularly, in place of the two-parameter gamma or two-parameter Weibull distributions. Moreover, when the shape parameter 𝛼=1, it coincides with the one-parameter exponential distribution. (Gupta & Kundu, 1999) provided the graphs of the generalized exponential density functions for different values of α. The plots of the hazard functions for different values of α can be obtained as in (Gupta & Kundu, 1999). The hazard function of the generalized exponential distribution behaves exactly the same way as the hazard functions of the gamma distribution, which is quite different from the hazard function of the Weibull distribution, see (Gupta & Kundu, 1999) for details. Several properties of the reversed 6 hazard function of the generalized exponential distribution are obtained in Nanda and Gupta (2001). For the generalized exponential distribution, the reversed hazard function 𝑟(𝑥;𝛼,𝜆) is in a convenient form and it can easily be used to compute Fisher information matrix, see Gupta and Kundu (2006). (Raqab & Ahsanallah, 2001)considered different order statistics of the generalized exponential distribution. The moment generating functions of the different order statistics and the product moments are obtained by (Raqab & Ahsanallah, 2001).They tabulated different product moments and used them to compute the best linear unbiased estimators of the location and scale parameters of the generalized exponential distribution. (Raqab, 2002)considered the three-parameter (including the location) generalized exponential distribution and obtained the best linear unbiased estimators of the location and scale parameters using the moments of the records statistics. Since the moment generating function of the generalized exponential distribution is not in a very convenient form, the distribution of the sum of n.i.i.d generalized exponential random variables cannot be obtained very easily. It is observed that if X follows 𝐺𝐸 (𝛼,1), then 𝑒−𝑋 has a Beta distribution. Since the product of independent Beta random variables has been well studied in thel iterature, it is used effectively to compute the distribution of sum of the n.i.i.d generalized exponential random variables. It is observed (Gupta & Kundu, 1999)that the distribution of the sum of n.i.i.d generalized exponential random variables can be written as the infinite mixture of generalized exponential distributions. The exact mixing coefficients and the parameters of the corresponding generalized exponential distributions are obtained by (Gupta & Kundu, 1999). The MLE of λ can be obtained by maximizing the profile log-likelihood function with respect to λ. It is observed by (Gupta & Kundu, 2002)that the profile likelihood function of λ is a unimodal function and its maximum can be easily obtained by using a very simple iterative procedure. Since for the generalized exponential distribution, the distribution function has a very convenient form, the least squares and the weighted least-squares methods can be used quite effectively to compute 7 the estimators of the unknown parameters, see Gupta and Kundu (2001b) for details. Graphical techniques have been used in (Gupta & Kundu, 2002)for constructing confidence intervals/regions for the unknown parameters when both parameters are unknown. The exact expressions of the approximate Bayes estimates, both under squared errors and LINUX loss functions are obtained by (Gupta & Kundu, 2007). Recently, Kundu and Gupta (2005) developed the inference procedures on 𝑅 = 𝑃(𝑌<𝑋) [in the statistical literature R is known as the stress-strength parameter] both under classical and Bayesian framework, when X and Y are independent generalized exponential distributions. In a series of papers Gupta and Kundu (2003a, b, 2004; Kundu et al., 2005) studied the closeness of the generalized exponential distribution with Weibull, gamma and log-normal distributions. Recently, Gupta and Kundu (2006) compared in detail the Fisher information of the Weibull and generalized exponential distributions for both complete and censored samples. 2. Basic concepts This section is devoted