ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.17(2014) No.1,pp.34-40 Transformation method for Estimating P (X<Y ) in the Case of Three Parameter Rayleigh Distribution Ali Abolhasani 1 , Hossein Jabbari Khamnei 2 ∗ 1 Department of Mathematics, Faculty of Sciences, Azarbaijan Shahid Madani University, Tabriz, Iran 2 Department of Statistics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran (Received 8 December 2012, accepted 5 July 2013) Abstract: In this paper, we will use transformation method to find MLE of P (X<Y ), where X and Y are independent random variables from three parameter generalized rayleigh distribution. The results of simula- tion shows that the mean of errors, by using transformation method, is often around zero. Keywords: stress-strength model; transformation method; three parameter generalized rayleigh distribution; incomplete Beta function Mathematics Subject Classification: 62N05 1 Introduction The problem of estimating the probability that a random variable is less than another independent random variable arises in reliability studies. When X and Y represent the random variables of stress and strength of a component, respectively, the reliability of the system is equal to R = P (X<Y ). In the last four decades, estimation of R = P (X<Y ) is very common between statisticians. Awad et al (1981), determined the maximum likelihood estimator of R when X and Y have bivariate exponential distributions. Sathe and Shah (1981) and Tong (1977), considered the problem of estimating R when X and Y are independent exponential random variables. Constantine and Karson (1986), considered the estimation of R when X and Y are independent gamma variables. Kundu and Gupta (2005) considered this problem when X and Y have generalized exponential distributions. Tong (1974) discusses on estimation of R in exponential case. Basu (1981), has obtained the MLE for R when the distribution of variables are gamma and exponential. Johnson (1988) had a good review on estimation of R for exponential families. Dinh et al (1991) discusses on the UMVUE of R in bivariate normal case. Cramer and Kamps (1997), considered the UMVUE of R when the distribution of X and Y are exponential. Ahmad et al (1977) have studied about empirical Bayes estimation of R for Burr-type X distribution. Surles and Padgett (1998) have made inference for R when the distribution of variables are scaled Burr-type X. Comparison of different estimations of R for a scaled Burr-type X, have been made by Raqab and Kundu (2005). The main aim of this paper is finding the MLE of R = P (X<Y ) when X and Y are independent random variables from a three parameter generalized rayleigh, using transformation method. The rest of paper is organized as follow: In section 2, we will explain transformation method and state its role in stress- strength problems. In section 3, we will present three parameter generalized distribution and introduce a monotone function to transform three parameter generalized distribution into gamma distribution. Also, we will use transformation method two times for obtaining estimation of P (X<Y ). In section 4, simulation studies are presented. 2 Transformation method As Kotz et al (2003) states in chapter 2, one can use transformation method to find point and interval estimation for R. Transformation method have been overlooked by statisticians, but rarely applied to stress-strength models before. In this * Corresponding author. E-mail address:h jabbari@tabrizu.ac.ir Copyright c ⃝World Academic Press, World Academic Union IJNS.2014.02.15/789