Bayes Estimation of the Mean of Normal Distribution Using Moving Extreme Ranked Set Sampling

Moving extreme ranked set sampling (MERSS) is one of useful modifications of ranked set sampling (RSS). The method has been investigated by many authors and proved to be a good competitor to simple random sampling (SRS). In this paper, Bayes estimation of the mean of normal distribution based on MERSS is considered and compared with SRS counterpart. The suggested estimators are found to be more efficient than that from SRS.


Introduction
There are some studies investigated ranked set sampling (RSS) from a Bayesian point of view.Al-Saleh and Muttlak (1998) investigated Bayesian estimators of the mean of the exponential distribution.Lavine (1999) studied some aspects of Bayesian RSS.Kim and Arnold (1999) considered Bayesian estimation under both balanced and generalized RSS.Al-Saleh and Muttlak (2000) considered Bayesian estimation using RSS and found a Bayes estimator of exponential distribution under conjugate prior and gave an application of real data.Al-Salehand Abu Hawwas (2002) used the notion of multiple imputations to characterize the Bayes estimators using RSS.This characterization was used to approximate complicated Bayesian estimators and was applied to the case of normal distribution with conjugate prior.For more details about RSS see Koyuncu and Kadilar (2009), Li and Balakrishnan (2008), Al-Omari and Jaber (2008), and Al-Omari, et al. (2009).
Moving extreme ranked set sampling was proposed by Al-Odat and Al-Saleh (2001).This method uses only extremes with varied set size to reduce error in ranking.Al-Saleh and Al-Hadhrami (2003a,b) investigated the method parametrically and found maximum likelihood estimators (MLE) of some parameters of normal, exponential and uniform distributions.Based on MERSS, several ratio, product and chain type estimators were studied by Al-Hadhrami (2007).Abu-Dayyeh and Al-Sawi (2007) investigated inference about the mean of the exponential distribution using MERSS.Estimation of the mean using concomitant variable was studied by Al-Saleh and Al-Ananbeh (2007).Al-Hadhrami et al. (2009) found the MLE of variance of the normal distribution and investigated its properties.This MLE is unbiased estimator and more efficient than the competitor estimator from SRS. Bayesian inference on the variance of normal distribution using MERSS was considered by Al-Hadhrami and Al-Omari (2010).Generalized MLE, confidence intervals, and different testing hypotheses were considered.It was shown that the modified inferences using MERSS are more efficient than their counterparts based on SRS.Al-Saleh and Samawi (2010) considered the estimation of the odds, F/(1 − F) based on MERSS, and showed that the estimator based on MERSS have some advantages over that based on SRS.
In this paper, Bayesian estimator of the population mean of normal distribution is considered and compared with estimators based on simple random sampling.
The remaining part of this paper is organized as follows: Bayes estimator of the mean based on MERSS is given in Section 2. In Section 3, Bayes estimators of the mean of normal distribution are considered based on constant and conjugate priors.An approximation of ˆMERSS θ is given in Section 4. In Section 5 a simulation study is presented to evaluate the performance of the suggested estimators.Finally, conclusions are given in Section 6.

Bayes estimator of the mean based on MERSS
The MERSS can be summarized as follows: Step 1: Select m random samples of size 1, 2, 3,…, m , respectively.
Step 2: Identify the maximum of each set, visually or by any cost free method without actual measurement of the variable of interest.
Step 3: Measure accurately the selected judgment identified maxima.
Step 4: Repeat Steps 1, 2, 3 but for minimum of each set.
Step 5: Repeat the above steps r times until the desired sample size, rm n 2 = is obtained.For one cycle, the following MERSS sample is obtained , where : X is the element of rank i from a set of size .j For simplicity, use i X for Then, the sample of one cycle is given by The notion of multiple imputations used by Al-Salehand Abu Hawwas (2002) can be also applied for MERSS with similar formula developed for RSS.Assume that x * Now, let us introduce the following identity based on MERSS that is similar to one provided by Rubin (1978).

Bayes estimator of the mean of normal distribution
Assume that f is the density of a normal random variable with mean θ and variance 1 .
θ θ as a prior density.Thus, the posterior density is given by The Bayes estimator using square error loss function, which is the mean of posterior distribution is In the following we will introduce the Bayes estimators of the θ based on constant and conjugate priors.

Bayes estimator of the mean of normal distribution with constant prior
Assuming that ( ) 1 The following lemma summarizes some properties of this estimator.

ˆ( ( , )) ( , )
MERSS MERSS Consider the sample , x a + y a + .Then, the estimator based on these observations is given by x a x a y a y a d x a x a y a θ θ θ MERSS MERSS x a y a x y a

2.
Consider the sample -,x y .Then, the estimator based on these observations is , and since -x has the same distribution as y, we got ( )

Bayes estimator of the mean of normal distribution with conjugate prior
is a SRS from the normal distribution N(θ ,1), then the Bayes estimator using N(0,1) as a prior distribution for θ is ˆ/(1 ) m + .Also, the Bayes estimator ˆSRS δ θ based on constant prioris X with risk function1/ m .
If * X is the average of SRS of size ( 1) m m + (full data), and * ˆ( ) SRS x θ is the Bayes estimator forθ based on the full data, then, from Formula (2) we have Some properties of the Bayes estimator of the mean of normal distribution using conjugate prior are stated in the next lemma.
( ) 2. The Bayes risk is given by ( Proof: 1.To prove (1), take the expectation of the both sides of , where w and * X are the averages based on 2m units using MERSS and SRS, respectively.Thus, )

Simulation Study
In order to evaluate the behavior of the Bayes estimator, a simulation was conducted.Some values of θ was generated from (0,1) N and samples are selected from ( ,1) N θ using MERSS.Then, 1 θ was calculated from the sample and the risk function was obtained and compared with that from simple random sample.The results are reported in Table 1.  1, the efficiency is greater than 1 for the set sizes considered in this study and is increasing with .m This indicates that MERSS estimator is more efficient than SRS estimator when estimating the mean of normal distribution.

Conclusions
The Bayes estimation of the mean of normal distribution using MERSS is considered and compared with the estimators based SRS.It is recommended to use MERSS for estimating the population mean of the normal distribution using Bayes estimation.
MERSS of one cycle from this distribution obtained using a full data of ( 1) m m + observations.Let ( ) be a prior density of and ( ) density of θ given .

3 .
Since the Bayes risk of ˆSRS θ with size a sample of size 2mis has the smallest Bayes risk among all other estimators,