International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 2013 1 ISSN 2250-3153 www.ijsrp.org Bayesian Analysis of Rayleigh Distribution A. Ahmed, S.P Ahmad and J.A. Reshi Department of Statistics, University of Kashmir, Srinagar, India Email: reshijavaid19@gmail.com Abstract: The Rayleigh distribution is often used in physics related fields to model processes such as sound and light radiation, wave heights, and wind speed, as well as in communication theory to describe hourly median and instantaneous peak power of received radio signals. It has been used to model the frequency of different wind speeds over a year at wind turbine sites and daily average wind speed. In the present paper, we consider the estimation of the parameter of Rayleigh distribution. Bayes estimator is obtained by using Jeffrey’s and extension of Jeffrey’s prior under squared error loss function and Al-Bayyati’s loss function. Maximum likelihood estimation is also discussed. These methods are compared by using mean square error through simulation study with varying sample sizes. Index Terms: Rayleigh distribution, Jeffrey’s prior and extension of Jeffrey’s prior, loss functions. I. INTRODUCTION he Rayleigh distribution (RD) is considered to be a very useful life distribution. Rayleigh distribution is an important distribution in statistics and operations research. It is applied in several areas such as health, agriculture, biology, and other sciences. One major application of this model is used in analyzing wind speed data. This distribution is a special case of the two parameter Weibull distribution with the shape parameter equal to 2.This model was first introduced by Rayleigh (1980), Siddiqui (1962) discussed the origin and properties of the Rayleigh distribution. Inference for model Rayleigh model has been considered by Sinha and Howlader (1993), Mishra et al. (1996) and Abd Elfattah et al. (2006). The probability density function of Rayleigh distribution is given as: ) 1 . 1 ( 0 , 0 2 exp ) ; ( 2 2 2                 y for y y y f Recently Bayesian estimation approach has received great attention by most researchers. Bayesian analysis is an important approach to statistics, which formally seeks use of prior information and Bayes Theorem provides the formal basis for using this information. In this approach, parameters are treated as random variables and data is treated fixed. Ghafoor et al. (2001)] and Rahul et al. (2009) have discussed the application of Bayesian methods. An important pre-requisite in Bayesian estimation is the appropriate choice of prior(s) for the parameters. However, Bayesian analysts have pointed out that there is no clear cut way from which one can conclude that one prior is better than the other. Very often, priors are chosen according to ones subjective knowledge and beliefs. However, if one has adequate information about the parameter(s) one should use informative prior(s); otherwise it is preferable to use non informative prior(s). In this paper we consider the extended Jeffrey’s prior proposed by Al-Kutubi (2002) as:       R c I g c    1 , 1 Where              2 2 ; log    y f nE I is the Fisher’s information matrix. For the model (1.1), the prior is given by  1 2 c n k g          T