The Gamma-Rayleigh Distribution and Applications to Survival Data

Abstract

Studies on probability distribution functions and their properties are needful as they are very important in modeling random phenomena. However, research has shown that some real life data can be modeled more adequately by distributions obtained as combination of two random variables with known probability distributions. This paper introduces the Gamma-Rayleigh distribution (GRD) as a new member of the Gamma-X family of generalized distributions. The Transformed-Transformer method is used to combine the Gamma and Rayleigh distributions. Various properties of the resulting twoparameter Gamma-Rayleigh distribution, including moments, moment generating function, survival function and hazard function are derived. Results of simulation study reveals that the distribution is unimodal, skewed and normal-type for some values of the shape parameter. The distribution is also found to relate with the Gamma, Rayleigh and Generalized-Gamma distributions. The method of maximum likelihood has been used to estimate the shape and scale parameters of the distribution. To illustrate its adequacy in modelling real life data the distribution is fitted to two survival data sets. The results show that the distribution produced fits that are competitive and compared better, in some cases, to the Gamma, Rayleigh, Weibull and Lognormal distributions.

Keywords: Gamma-X family, Gamma-Rayleigh distribution, Maximum Likelihood estimators, Survival data.