MINIMAX ESTIMATION OF THE PARAMETER OF THE RAYLEIGH DISTRIBUTION UNDER QUADRATIC LOSS FUNCTION

This paper is concerned with the problem of finding the minimax estimator of the parameter θ of the Rayleigh distribution for quadratic loss function by applying the theorem of Lehmann (1950).


INTRODUCTION
In an expository paper, Siddique (1962) discussed the origin and properties of the Rayleigh distribution. Polovko (1968) and Dyer and Whisenand (1973) noted the importance of this distribution in electro vacuum devices and communication engineering. Dey and Das (2007) obtained Bayesian predictive intervals of the parameter of Rayleigh distribution. The probability density function of the Rayleigh distribution is given by: where θ is the parameter of the distribution. Podder et al. (2004) studied the minimax estimator of the parameter of the Pareto distribution under Quadratic and MLINEX loss functions. In this paper, we shall estimate the parameter of Rayleigh distribution by using the technique of minimax approach under quadratic loss function, which is essentially a Bayesian approach. The most important element in the minimax approach is the specification of a distribution function on the parameter space, which is called prior distribution. In addition to the prior distribution, the minimax estimator for a particular model depends strongly on the loss function assumed. The basic difference between the philosophy of the minimax and classical estimation is that in minimax estimation the parameter of the distribution is assumed to be a random variable, where as classical estimation regards it as a fixed point.
In this paper, we derive the minimax estimator of the parameter θ of the Rayleigh distribution under quadratic loss function. The derivation depends primarily on Lehmann's Theorem (Lehmann, 1950) and can be stated as follows.
Theorem: Let τ = {Fθ; θεΘ} be a family of distribution functions and D a class of estimators of θ. Suppose that d* ε D is a Bayes' estimator against a prior distribution ξ*(θ) on the parameter space Θ and the risk function R(d*, θ) = constant on Θ; then d* is a minimax estimator of θ.
The major result of this paper is contained in this theorem and its discussion is given below.

MAJOR RESULTS
Theorem 2.1: Let X = (X 1 ,X 2 , . . . , X n ) be 'n' independently and identically distributed random variables drawn from the density (1.1). Then θˆM QL = where θ is the parameter to be estimated and d is the estimate of θ .
Note that a loss function of the form L (θ,d) = (θ -d) 2 is not a minimax estimator for the above distribution.
First we have to prove Theorem 2.1. We use Lehmann's Theorem, which was stated before. Here, we consider the quadratic loss function (QLF) of the form which is a non-negative symmetric and continuous loss function of θ and d . In order to prove the theorem, it will be sufficient to show that is a minimax estimator of θ for the loss function (2.1). For this, first we have to find the Bayes' estimator d of θ. Then, if we can show that the risk function of d is constant, the Theorem 2.1 will be followed. Let us assume that θ has non-informative prior density defined as When c =3, we get the ALI prior for the Rayleigh pdf (1.1) because of Hartigan (1964).
As pointed out in (1.1), the likelihood function of the distribution of f(x| θ) is given by Now, for the QLF (2.1), the Bayes' estimator of θ is given by The risk function of the estimator d is The estimated values of the parameter and MSE of the estimators are compared by the Monte-Carlo Simulation Method, using the Rayleigh distribution.

CONCLUSION
It can be seen from Tables 1, 3, and 5, along with Figures 1, 3 and 5, that the minimax estimator under squared error loss function and the classical maximum likelihood estimator have approximately the same MSEs when the value of 'c' is positive and sample sizes n>30. Also, it can be seen from Tables 2, 4, and 6, along with Figures 2, 4, and 6, that for small as well as for large sample sizes, the classical maximum likelihood estimator appears to be better than that of minimax estimator under quadratic loss function when the value of 'c' is negative. It is also to be noted that Hartigan's prior gives better results than Jeffrey's prior when n ≥ 25.

ACKNOWLEDGEMENT
The author is thankful to the Editor-in-Chief for his valuable suggestions for improvement of the paper. The Author also wishes to thank Mr. A.A. Basumatary, Lecturer, Department of Mathematics, St. Anthony's College, Shillong for doing computational work.