Inferences on a Scale Parameter of Bivariate Rayleigh Distribution by Ranked Set Sampling

In this paper, we obtain several estimators of a scale parameter of Morgenstern type bivariate Rayleigh distribution based on the observations made on the units of the ranked set sampling regarding the study variable which is correlated with the auxiliary variable. We also compare the efficiency of these estimators. Finally, we illustrate the methods developed by using a real data set.


Introduction
) defined a class of bivariate distributions, and Farlie (1960) extend it to the multivariate case. This class of distributions is known as Farlie-Gumbel-Morgenstern (FGM) distribution. Some well-known marginal distributions are considered and studied in literature: For example logistic (Gumbel, 1961), gamma (D'Este, 1981 ; , uniform (Bairamov and Bekci, 1999 ;Tahmasebi and Jafari, 2012;Singh and Mehta, 2015), exponential (Gumbel, 1960, Balasubramanian andBeg, 1997;Thomas, 2008, 2011) and generalized exponential Jafari, 2014, 2015)  We consider several unbiased estimators of parameter 2  using ranked set sampling (RSS). This technique of sampling was first proposed by McIntyre (1952) and has a more efficient sampling method than simple random sampling (SRS) method for estimating the population mean. Some modifications of RSS are presented in literatures: For example  in MTBRD based on the RSS, and compared the efficiency of these estimators and the estimator based on SRS. In Section 3, we obtain different estimators for 2  in MTBRD by using ERSS and MERSS methods. Also, the efficiency of all estimators are evaluated. In Section 4, we obtain unbiased estimator for 2  in MTBRD by DRERSS method. In Section 5, we illustrate the proposed methods using a real data set.

Estimating based on RSS
In the RSS technique, the sample selection procedure is composed of two stages. At the first stage of sample selection, n simple random samples of size n are drawn from an infinite population and each sample is called a set. Then, each of units is ranked from the smallest to the largest. At the second stage, the r th observation unit from the r th ranked set is taken. Ranking of the units is done with a low-level measurement such as using previous experiences, visual measurement or using a concomitant variable. Stokes (1977) described the procedure of RSS for bivariate random variable ) , ( Y X , where X is the variable of interest and Y is a concomitant variable that is not of direct interest but is relatively easy to measure, as follows: Step 1. Randomly select n independent bivariate samples, each of size n .
Step  We can also provided a ranked set sample of size n by each sample measurement of Y which is taken on the unit that has the maximum value for the X variable. Let

Inferences on a Scale Parameter of Bivariate Rayleigh Distribution by Ranked Set Sampling
Pak.j.stat.oper.res. Vol.XIII No.1 2017 pp1-16 5 Remark 2.1 Our assumption is that  is known, but sometimes  may not be known.

We know that the correlation coefficient between X and Y in MTBRD is
So by using the sample correlation coefficient q of the RSS observations Step 1. Select n random samples each of size n bivariate units from the population.
Step 2. If the sample size n is even, then select from 2 n samples the smallest ranked unit X together with the associated Y and from the other 2 n samples the largest ranked unit Step 3. If n is odd then select from     Step 1. Select n samples each of size n from MTBRD using SRS. Identify by judgment the minimum of each sample with respect to the variable X .
Step 2. Repeat step 1, but for the maximum.  Finally, the efficiency of 2  The results are given in Table 2

Estimating based on DRERSS
In this section, first we obtain different estimators for 2  based on DRERSS method with concomitant variable. This method introduced by Al-Omari (2011) and can be described as follows: Step 1. Select 2 n random samples each of size n bivariate units from the population.
Step 2. Select the coefficient

and ]
[x is the largest integer value less than or equal to x .  Step 3. If n is even, from the first 2 2 n samples select the  Proof. The proof is obvious.

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The efficiency of The results are given in Table 3

An application
A reappraisal of caloric requirements in healthy women are done by Owen et al. (1986).
The results of this study show that the body weight of women was highly related to the resting metabolic rate (RMR) of the women.
We considered a bivariate data set from the 44 women data such that the first component X represents the body weight(kg), and the second components Y represents resting metabolic rate (RMR) (kcal/24 hr). Clearly, the the body weight(kg) can be measured very easily but the RMR is difficult to measure. We selected 6 random samples with size 6 from 44 women data and ranked the sampling units of each sample according to the X variate (body weight). We measureed the ranked set sample observations