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The Gamma-Rayleigh Distribution and Applications to Survival Data Available online at http://www.ajol.info/index.php/njbas/index ISSN 0794-5698 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142 DOI: http://dx.doi.org/10.4314/njbas.v25i2.14 The Gamma-Rayleigh Distribution and Applications to Survival Data *1E. E. E. Akarawak, 2I. A. Adeleke, 3R. O. Okafor 1,3Department of Mathematics, University of Lagos, Akoka (Nigeria) 2Department of Actuarial Science and Insurance, University of Lagos, Akoka (Nigeria) Full Length Research Article Research Full Length [*Corresponding Author: Email: [email protected]; : +2348137678288] 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 two- parameter 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. INTRODUCTION Gupta and Kundu, 2001; Pal et al., 2006) have Studies on probability distribution functions and shown that distributions of combined random their properties are needful as they are very variables are more flexible, perform better and important in modeling random phenomena. have wider applicability. However, research has shown that some real life data that cannot be modeled adequately by Motivated by the recent developments in existing standard distributions are sometimes generating new distributions and the need for found to follow distributions of some continuous extension and generalizations to combinations of two or more random variables more complex situations, this research paper with known probability distributions. introduces and studies the Gamma-Rayleigh Distribution (GRD). The Gamma and Rayleigh Developments on generating new distributions to distributions are well-known survival model naturally occurring phenomena have led to distributions; however, there might be some a number of new distributions being defined and survival data situations in which the two studied. Some of the earlier works include those distributions may not fit so well. Combining of Marsaglia (1965), Press (1969), Basu and them might therefore yield better results. Since Lochner (1972), and Lee et al. (1979). The trend estimation of parameters of a newly generated has been on the increase in recent years due to distribution is fundamental to application, the increasing need for adequate distributions to method of maximum likelihood is used in this model some data arising in practice (Gupta and work to estimate parameters of GRD. The Kundu, 1999; Eugene et al., 2004; Famoye et al., distribution is then applied to two survival data 2005; Akinsete et al., 2008; Alzaatreh et al., sets. MATLAB R2011b has been used for 2013b, Adeleke et al., 2013; Akarawak et al., implementations. 2013 and Akarawak et al., 2015). Furthermore, several studies (Mudholkar and Srivastava, 1993; 130 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142 Methodology given as: This article is organized as follows: derivation g (v) f (v)[ln{1 F(v)}] 1[1 F(v)] 1 and presentation of aspects of the Gamma- V () Rayleigh Distribution; parameter estimation of (2.6) the Gamma-Rayleigh Distribution and study of where f(v) and F(v) are the pdf and cdf of any the simulation and applications of the random variable V. distribution. 2.2 Derivation of the Gamma-Rayleigh THE GAMMA-RAYLEIGH DISTRIBUTION Distribution (GRD) (GRD) 2.2.1 Derivation of the pdf and cdf of GRD Method: The Transformed-Transformer The pdf and cdf of the Gamma-Rayleigh Technique of generating New Family of distribution is derived in this section as a class Continuous Distributions of Gamma-V family of generalized distributions. The Transformed-Transformer method was Theorem 2.1: recently introduced by Alzaatreh et al. (2013a) Let the pdf of a gamma distribution be for generating families of continuous distributions. Let F(v) be the cumulative f (x) (x) 1 ex and V Rayleigh distribution function (cdf) of any random () variable V and p( y ) the probability density distributed random variable with pdf v2 v2 v 2 2 function (pdf) of a random variable Y defined f (v) e 2 , and cdf F(v) 1 e 2 . on 0,. The cdf of a generalized family of 2 distributions is given as Then the pdf of the Gamma-Rayleigh log(1F (v)) distribution is given by: G(v) p(y)dy , (2.1) 0 2 g (v) v2 1 exp(v2 ); v 0;, 0 The family of distributions defined by (2.1) is V () called the ‘Transformed-Transformer’ family (or , (2.7) the Y-V family), where the random variable Y is Where, () is the gamma function. being transformed by another random variable V (the transformer) into a new random variable Proof: with the support . According to Alzaatreh The pdf of the Gamma-V family of distribution is et al. (2013a), the corresponding pdf of the given by: generalized distribution in (2.1) is given by f (v) g(v) p(log[1 F(v)]) (2.2) 1 F(v) , (2.8) h(v)p(log[1 F(v)]) (2.3) Let V follow the Rayleigh distribution with pdf h(v)p[H(v)], (2.4) Where h(v) is the hazard function and H(v) is , and cdf , the cumulative hazard function of V with the 2 cdfF(v), which makes g(v) as arising from a v 2 2 weighted hazard function. One of the Y-V such that 1 F(v) e . Then the pdf in (2.8) families defined by Alzaatreh et al. (2013a) is becomes: 2 2 1 2 1 the Gamma-V family, obtained by letting Y v v v v 2 2 2 2 2 2 follow the Gamma distribution with parameters gV (v) e lne e () 2 and and having the pdf, 1 y , f (y) y e ; y> 0; α , λ >0. (2.5) 2 2 v 2 1 v ( 1) () 2 v 2 ve2 e 2 , According to Alzaatreh (2013a), the pdf of the 2 2 () 2 Gamma-V generalized family of distribution is 131 Akarawak et al: The Gamma-Rayleigh Distribution and Applications to Survival Data……………… 2 2 2 v2 v v v 2 1 u 2 1 2 2 2 where, (,v ) u e du is the 2 2 2 0 2 1 2 v e e e , (2 ) () lower incomplete gamma function and 2 v 1 u 2 2 1 2 2 () u e du is the complete gamma 0 2 v e , 2 () function. 2 v 2 2 1 2 2 2 v e , (2.9) Proof: 2 () We prove the result using the fact that v Replacing 2 by , (2.9) becomes: G(v) g(x)dx , 2 0 2 2 2 v 2 g (v) v2 1ev . G(v) x 2 1ex dx , (2.14) V () () 0 Any random variable V that has the probability Changing variable as in (2.11), the result density function given in (2.7) is said to have follows. the Gamma-Rayleigh distribution with shape parameter and scale parameter and 2.2.2 Relationship with Other written as V ~ GRD( , ). Distributions The Gamma-Rayleigh distribution has two Theorem 2.2 parameters and . The function given in (2.7) is a valid probability density function. Corollary 2.1: When = 1 the pdf of Gamma- Rayleigh distribution reduces to the pdf of the Proof: 2 1 Rayleigh distribution with parameter . Required to prove that g(v)dv 1. 2 0 Preposition 2.1: The Gamma-Rayleigh 2 2 g(v)dv v2 1ev dv , (2.10) distribution is a special case of the generalized 0 () 0 gamma distribution when p = 2, d = 2 and a= 1 1/ 2 2 u 2 . Let, u v v , so that du Proof of Preposition 2.1: , (2.11) dv 1 1 The generalized gamma distribution (Stacy, 2 2 2 u 1962) has a pdf of the form: Then (2.10) reduces to: p 2 1 p d 1 x 1/ 2 x exp 2 u u du a d a g(v)dv e 1 1 0 0 () 2 2 f (x) ; x 0; a, p,d 0 2 u d , p 1 () u 1eu du 1. (2.12) . (2.15) () 0 () When p = 2, d = 2 and a= , (2.15) Theorem 2.3: reduces to the Gamma-Rayleigh pdf given in The cumulative distribution function (cdf) of the (2.7). Gamma-Rayleigh distribution is given by: Corollary 2.2: If the random variable X follows (,v2 ) the Gamma-Rayleigh distribution with G(v) ; (2.13) () parameters and , then the Random variable Y = X2 follows the gamma distribution with parameters and . 132 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142 Proof of Corollary 2.2: y 1ey 2 , (2.16) Let X ~ GRD ( , ) and Y X such that () 1 2 dx 1 where (2.16) is the pdf of the gamma x y .
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