Applying the Shrinkage Technique for Estimating the Scale Parameter of Weighted Rayleigh Distribution

This paper includes the estimation of the scale parameter of weighted Rayleigh distribution using well-known methods of estimation (classical and Bayesian). The proposed estimators were compared using Monte Carlo simulation based on mean squared error (MSE) criteria. Then, all the results of simulation and comparisons were demonstrated in tables.


Introduction
Rayleigh distribution is one of the failure distributions which have been used in reliability, lifetesting, and survival analysis. Being first presented by Lord Rayleigh [1], these statistical properties were originally derived in connection with a problem in acoustics. More details on the Rayleigh distribution can be found in Johnson et al. [2].
Weighted distributions are applied in research associated with reliability meta-analysis, biomedicine, econometrics , renewal processes , physics, ecology, and branching processes. Research on such applications can be found in Zelen and Feinleib [3], Patil and Ord [4], Patil and Rao [5], Gupta and Keating [6], Gupta and Kirmani [7], and Oluyede [8]. The weighted Rayleigh distribution was published by Reshi et al. [9]. They presented a new class of sizebiased generalized Rayleigh distribution and investigated the various structural and characterizing properties of that model, as also performed by Das and Roy [10]. Rashwan [11] introduced the double weighted Rayleigh distribution properties and estimation. AL-Kadim and Hussein [12] published a research on the estimation of the reliability of weighted Rayleigh distribution through five methods, using simulation and a comparison between the proposed estimators. Ahmed and Ahmed [13] presented the characterization and estimation of double weighted Rayleigh distribution. Salman and Ameen [14] estimated the shape
The aim of this paper is to estimate the scale parameter of weighted Rayleigh distribution using shrinkage estimation methods that depend on classical estimators, namely the moment (mom) and maximum likelihood (ML) methods.

Weighted Rayleigh Distribution (WRD)
To present the concept of a weighted distribution, suppose that T is a nonnegative random variable that follows Rayleigh distribution wi h one p me e { )}, then the (pdf) of T is given by f (t , ) = ; > 0 , > 0 … 1) and w ( t ) a non-negative weight function satisfying the condition µ w = E [w(t)] . Then the r v T w, which is defined on the interval (L,U), is having pdf as below : where, E ) ) = ∫ From equations (1) and (2), we get that the probability density function of the random variable Tw will be f (t;) = ; t > 0 , > 1 … 3) and the cumulative distribution function ( cdf ) of Tw will be F (t; ) = 1-… 4) By pu ing -1 = θ>0 in equations (3) we get f w (t;) = ; t > 0 ,  > 0 … 5) and the cumulative distribution function (cdf) of T W will be F W (t; ) = 1-… 6) Accordingly, the reliability and hazard functions will be respectively as follows:

Methods Estimation Maximum Likelihood Estimation (MLE)
Firstly , we find the likelihood func ion L 1, 2, 3,…, n; ) based on the following =L(t 1 ,t 2 ,t 3 ,…, n; Method of Moments Le 1, 2,…, n efe o ndom s mple of size n from the WRD with pdf (5). Then the moment estimator  of is obtained by setting the mean of the distribution to be equal to the sample mean , i.e., E(T K )= ∑ The moment estimator  of  is obtained as below

Shrinkage Method
Thompson [16] studied the problem of the shrinkage of a usual estimator  of he p me e θ, depending on the observations of random samples along with prior studies and previous experiences. This was performed through merging the usual estimator  and the initial estimate θo s linear mixture, via the shrinkage weight factor Ø(  ,  . The resulted estimator is so called the shrinkage estimator which has the form below .. (11) where Ø (  denotes the trust of  and ( Ø (  signifies the trust of θ , which might be constant or a function of  function of sample size. It could be also found by reducing the mean square error for  . Thompson referred to the following significant reasons to use the initial value. 1 -Supposing that he ini i l v lue θo is near to the true value, then it is essential to use it. 2 -If he ini i l v lue θo is ne o he c u l v lue of he p me e θ , then we reach a bad situation [1,5,15,17,18,19]. In this state, there is no doubt to take the moment method as an initial value. Consequently, equation (11) becomes :

The shrinkage weight function ( sh1)
In this section, we claim the shrinkage weight factor as a function of sample size n, as below.

Modified Thompson type shrinkage weight function (sh3)
In this subsection, we consider the modified shrinkage weight factor introduced by Thompson, as follows.
Hence, the shrinkage estimator of θ becomes:

Simulation Study
In this section, Monte Carlo simulation study was applied to compare the performance of the considered estimators for the sc le p me e θ, which were obtained using different sample sizes (n=10,30,50,100), based on 1000 replications through MSE criteria, as in the steps below [6] .
Step 2: Transform the uniform random samples to random samples following the WRD and using the cumulative distribution function (c.d.f), as follows:
Step 6: Based on (L=1000) replication, the MSE for all proposed estimation methods of θ is utilized by: where θ denotes the suggested estimation method in iterative i for real θ.

Numerical results
We demonstrate all the results in the tables below.  2 -For n=10 (small sample size), the mean squared error (MSE) of the scale parameter  3 is lower than the other estimators, followed by  2 and  1. Hence, the best estimator in this case is  3 for all  = 2, 3, 4, 5.
3 -For n = 30, 50, 100 (medium sample size), the MSE of the scale parameter  3 has a lower value than the other estimators, followed by  2 and  1. Consequently, the best estimator in this case is  3 for all  = 2, 3, 4, 5. 4 -For all n, the MSE for all proposed estimators is approximately fixed with respect to .

Conclusions
From the results of the analysis, the maximum likelihood method was the best because it showed minimum MSE fo ll v lues of θ when he sizes of sample were 30 , 50 , and 100.