A Bayesian Approach for Estimating Parameter of Rayleigh Distribution

This paper is concerned with estimating the parameter of Rayleigh distribution (special case of two parameters Weibull distribution) by adopting Bayesian approach under squared error (SE), LINEX, MLINEX loss function. The performances of the obtained estimators for different types of loss functions are then compared. Better result is found in Bayesian approach under MLINEX loss function. Bayes risk of the estimators are also computed and presented in graphs.


Introduction
Rayleigh distribution is named after the British Nobel prize winner physicist Lord Rayleigh . This distribution has got remarkable attention in the field of reliability theory and survival analysis, probability theory and operations research. Apart from this, in communication theory to model multiple paths of dense scattered signals reaching a receiver and in the physical sciences to model wind speed, wave heights [8], sound/light radiation, radio signals and wind power, the application of this distribution is noticeable. In addition to, in engineering, to measure the lifetime of an object, where the lifetime depends on the object's age. For example, resistors, transformers, and capacitors in aircraft radar sets.
Rayleigh distribution is also used in mixture models. Most of the researchers worked on the classical and Bayesian analysis of two and three components mixture models. Saleem and Aslam [9] discussed the use of the informative and the non-informative priors for Bayesian analysis of the two component mixture of Rayleigh distributions. Aslam et al. [2] studied Bayesian analysis of a three component mixture of Rayleigh distributions with unknown mixing proportions. Boudjerd et al. [4] applied Bayesian estimation of the Rayleigh distribution under different loss function. Sindhu et al. [10] employed Bayesian inference of mixture of two Rayleigh distributions. After analyzing its significance, this paper has been intended to get the best estimate of the parameter of this distribution considering varied loss functions using simulated and real data.
A continuous random variable X is said to have a Rayleigh [11] distribution if its probability density function is given by ; otherwise. Where, 2  is the only one scale parameter of this distribution. Replacing ; otherwise Now Monte Carlo techniques for generating sample from Rayleigh distribution is applied with the help of inverse transform method.

Sample Generation from Rayleigh Distribution
Let X be a Rayleigh variate having the p.d.f ; otherwise The cumulative distribution function of this distribution is (3)

Bayes estimator under squared error (SE) loss function
Now let the loss function be squared [5] error defined as Hence under SE loss function Bayes estimator of θ is Squared error loss function is applicable when the loss is symmetric in nature.

Bayes estimator under LINEX loss function
Let us consider the following LINEX [3] loss function which is applied in real estate assessment Where, represents the estimation error and c is the shape parameter of the loss function. For LINEX loss function, the Bayes estimator of  is given by  is the Bayes estimator under LINEX loss function.

Bayes estimator under MLINEX loss function
When the loss is asymmetric, then the MLINEX loss function is applied. MLINEX loss function is the modification of LINEX loss function. Now, let us consider the MLINEX loss [6] function defined as which is the Bayes estimator under MLINEX loss function.

Bayes risk of the estimator
 be a random sample of size n drawn from a density function L E R and the Bayes risk [7] of estimator   with respect to the loss function

Empirical Study
The estimated values of the parameter and Bayes risk of the estimators are computed by Monte-Carlo simulation method from Rayleigh distribution. A short algorithm of the above simulation is given below: i) Generate a random sample of size n using the following formula and specific value of  .
ii) Obtain Bayes estimator under different loss function. iii) Repeat the above steps 1000 times and denote the Bayes estimates of 2  as The numerical results and their graphs are as follows.
and different values of n, the Bayes estimator under MLINEX loss function provides better estimate than the other loss functions.      The variation in the performance of the estimators for various sample sizes and varied parameters are observed from Tables 1-8. From Figs. 1-8, the Bayes risk of the estimators of different loss function, the MLINEX loss function is minimum. Therefore, it can be concluded that, Bayes estimator under MLINEX loss function is better than all other estimators in the study.

Real Study
For fitting the distribution, weather data have been used in this paper. Wind speed (kph) data have been chosen in the period of 2014-2017 of Dhaka Airport.    Table 9 shows that for all cases Bayes risk of Bayesian approach under MLINEX loss function is the smallest than other approaches. If we want to predict about the wind speed on specific region then Rayleigh distribution has been used and for fitting the distribution if we use Bayesian approach under MLINEX loss function then it will give better result. Because MLINEX loss function shows the smallest Bayes risk.

Conclusion
In this study, we have considered the Bayesian estimation approach to estimate the parameter of Rayleigh distribution. In Bayesian approach, squared error (SE), linear exponential (LINEX) and modified linear exponential (MLINEX) loss functions have been used. We conducted a comprehensive simulation and real data to judge the relative performance of the Bayes estimator under different loss functions at different sample sizes and varied parameters of prior distribution. From simulated results and real study, smallest risk has been observed by Bayesian approach under MLINEX loss function than other loss functions about all cases. Also Figs. 1-8, downward shape has been displayed by MLINEX loss function than SE and LINEX loss functions. That means, Bayesian approach under MLINEX loss function gives better results than other loss functions. Therefore, Bayesian approach under MLINEX loss function can be suggested to estimate the parameter of Reyleigh distribution.