BAYESIAN ESTIMATION OF THE PARAMETER OF THE p-DIMENSIONAL SIZE-BIASED RAYLEIGH DISTRIBUTION

In this paper, we have derived the probability density function of the size-biased p-dimensional Rayleigh distribution and studied its properties. Its suitability as a survival model has been discussed by obtaining its survival and hazard functions. We also discussed Bayesian estimation of the parameter of the sizebiased p-dimensional Rayleigh distribution. Bayes estimators have been obtained by taking quasi-prior. The loss functions used are squared error and precautionary.


Introduction
The Rayleigh distribution is frequently employed by engineers, physicists and other scientists as a model for the analysis of data resulting from investigations involving wave propagation, radiation and related inquiries.It was first derived by Lord Rayleigh [3] in connection with a study of acoustic problems.The p.d.f. of the p-dimensional Rayleigh distribution is given as h(x; p, θ) = The rth moment about the origin is ) . Therefore ) ) Although the general form of the Rayleigh distribution with p >3 might have limited applications.The Rayleigh distribution with p.d.f. with p =1 is sometimes called the folded Gaussian, the folded normal, or the half-normal distribution.
In the Bayesian approach it is assumed that the parameter θ is itself a random variable.In this paper, we consider the Bayesian analysis (estimation) problem of the scale parameter of a p-dimensional size-biased Rayleigh distribution using the squared error and precautionary loss functions under quasi-prior.
When observation is selected with probability proportional to its size, the resulting distribution is called sizebiased.Statistical analysis based on size-biased samples has been studies in detail since the early 70's.The concept of size-biased sampling was mainly developed by Rao and Zelen & Feinleib [6].The size-biased distribution occurs naturally for some sampling plans in biometry, wildlife studies and survival analysis, among other.
Consider the p-dimensional Rayleigh distribution [1] whose p.d.f. is given by equation (1).Now using the relationship , we get the p.d.f. of size-biased p-dimensional Rayleigh distribution as ) ) . Now ) ) ) ) . Therefore and .

Cumulative distribution function
The cumulative distribution function of a size-biased p-dimensional Rayleigh distribution is

Survival function
The survival function of a size-biased p-dimensional Rayleigh distribution is )

Hazard function
The hazard function of a size-biased p-dimensional Rayleigh distribution is

Estimation of the Parameter (a) Maximum Likelihood Estimation (MLE)
The estimation of the parameter of the size-biased p-dimensional Rayleigh distribution is obtained by the method of MLE using equation (2).The likelihood function of the size-biased p-dimensional Rayleigh distribution is as follows: )

Now the log likelihood function is given by log
) ) .

(b) Bayesian Estimation
Under Bayesian analysis the fundamental problems are those of the choice of a prior distribution g() and a loss function l.Let us suppose that very little information is available about the parameter (the suitable prior for this case is given in [4].Assuming independence among the parameters, consider a quasi prior

Loss function
Let θ be an unknown parameter of some distribution ( | ) and suppose we estimate θ by some statistic  ̂.Let ( ̂, ) represent the loss incurred when the true value of the parameter is θ and we are estimating θ by the statistic  ̂.

Precautionary Loss Function
Norstrom [2] introduced an alternative asymmetric precautionary loss function and also presented a general class of precautionary loss functions with quadratic loss function as a special case [5].A very useful and simple asymmetric precautionary loss function is given as

)
Differentiating equation (3) w.r.t. and setting the results equal to zero, we have under this precautionary loss function is denoted by P   , and is obtained by solving the following equation:  ̂P = Eπ() 2 .The joint density (i.e., likelihood) function of the size-biased p-dimensional Rayleigh distribution is given by f (|) ' theorem, the joint density function (6) along with the prior (4), we obtain the following joint posterior density function of the size-biased p-dimensional Rayleigh distribution f(θ | ) = (|)() ∫ (|)() ∞ 0 , which on substituting the value of () and (|) gives f(θ |) = under squared error loss function is the posterior mean given by  ̂ = ∫  (|) of f(θ |) from equation (7) in equation (8) and solving it, we get  ̂ = ∫  ( equation (9), we obtain