An extension of the gamma distribution Rodrigo R. Pescim Departamento de Ciˆ encias Exatas Universidade de S˜ao Paulo, 13418-900, Piracicaba, SP, Brazil e-mail:rrpescim@gmail.com Saralees Nadarajah School of Mathematics, University of Manchester, Manchester M13 9PL, UK e-mail:mbbsssn2@manchester.ac.uk Abstract The gamma distribution has been widely used in many research areas such as engineering and survival analysis. We present an extension of this distribution, called the Kummer beta gamma distribution, having greater flexibility to model scenarios involving skewed data. We derive analytical expressions for some mathematical quantities. The estimation of parameters is approached by the maximum likelihood method and Bayesian analysis. The likelihood ratio and formal goodness-of-fit tests are used to compare the presented distribution with some of its sub-models and non-nested models. A real data set is used to illustrate the importance of the distribution. Keywords: Bayesian approach; Gamma distribution; Kummer beta generalized distribution; Maxi- mum likelihood method; Moment; Order statistic. 1 Introduction Ng and Kotz (1995) proposed a three-parameter distribution on (0, 1), the so-called Kummer beta (KB) distribution that provides greater flexibility to extremes. The KB cumulative distribution function (cdf) is defined by F KB (x)= K ∫ x 0 t a−1 (1 − t) b−1 e −ct dt, (1) where a> 0, b> 0 and −∞ <c< ∞. Here, K −1 = Γ(a)Γ(b) Γ(a + b) 1 F 1 (a; a + b; −c) and 1 F 1 (a; a + b; −c)= Γ(a + b) Γ(a)Γ(b) ∫ 1 0 t a−1 (1 − t) b−1 e −ct dt = ∞ ∑ k=0 (a) k (−c) k (a + b) k k! is the confluent hypergeometric function (Abramowitz and Stegun, 1968), where Γ(·) denotes the gamma function and (d) k = d(d + 1) ··· (d + k − 1) denotes the ascending factorial. The probability density function (pdf) corresponding to (1) is f KB (x)= Kx a−1 (1 − x) b−1 e −cx , 1