ISSN: 2319-8753 International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 2, Issue 11, November 2013 Copyright to IJIRSET www.ijirset.com 6244 Bayesian Analysis of Gamma Model with Laplace Approximation Romana Shehla 1 , Athar Ali Khan 2 Research Scholar, Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, Uttar Pradesh, India 1 Professor, Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, Uttar Pradesh, India 2 Abstract: Among all the important failure time distributions, Gamma distribution is the one which is flexible, widely understood and offers a good fit to continuous, skewed reliability data. But the non availability of simple and closed forms of reliability and hazard functions of gamma model had restricted its use. This paper provides a solution to these problems in R by employing Laplace approximation for asymptotic evaluation of integrals. However, the whole demonstration is in Bayesian framework. The asymptotic results have been cross-verified with the simulation results. Real-world problems have been used for illustrative purposes. Keywords: Bayesian, Gamma distribution, Laplace approximation, LaplacesDemon, Marginal posterior density, R. I. INTRODUCTION The parametric inference procedures for reliability data often involves distributions like weibull and exponential that have been widely accepted as important failure-time distributions. There are several other statistical models that can be successfully utilised for some particular types of data. The two-parameter gamma distribution is one of these probability distributions and is well suited for continuous, skewed responses. The gamma distribution fits a wide variety of failure- time data quite efficiently. Its relationship with the exponential distribution also has enhanced its importance as the sum of iid exponential random variables also follows gamma distribution. The pdf of gamma distribution is of the form                    t exp t ) ( 1 ) , | t ( f 1 (1) where 0   determines the shape and 0   , the scale of the distribution. The reliability function of the gamma model is given as  ) t , ( t R     (2) where, .) , (.  is the incomplete gamma function and the corresponding hazard function is defined as                   t exp t ) t , ( 1 ) t ( h 1 (3) Although there is a vast literature available on estimation of the gamma parameters within the classical approach, we have worked here on the Bayesian inference of the gamma parameter(s). To evaluate characteristics of posterior such as densities, means and variances, is a very tedious task. When gamma model is used as a failure-time distribution, Bayesian computations become far more difficult as it involves incomplete gamma functions. Also, in the absence of conjugate prior- likelihood pair, evaluation of posterior quantities cannot be performed in closed forms and thus requires some intensive