Improved Ratio-type Estimators of Population Mean in Ranked Set Sampling Using Two Concomitant Variables

In this paper, we propose an efficient class of ratio-in-exponential-type estimators with two concomitant variables using Ranked Set Sampling (RSS) scheme which improves the available estimators. The biases and mean square errors (MSEs) of the proposed estimators are obtained up to first degree approximation. Comparisons among the proposed and competitor estimators are made both theoretically and through simulation study. It turned out that when the variable of interest and the concomitant variables jointly followed a trivariate Gamma distribution, the proposed class of estimators dominates all other competitor estimators.


Introduction
Ranked set sampling (RSS) is a sampling technique which is used to reduce cost and increase efficiency in that situation where the measurement of survey variable is costly and time consuming, but it can be ranked easily at no cost or at very little cost.The technique of RSS was first introduced by Mclntyre (1952) to increase efficiency of the estimator of the population mean.The general method of a RSS can be described as follows: First, m subsamples, each of size m, are drawn at random from a population.Next, for each sub sample, the elements in the subsample are ranked relating to the concomitant variables and then one and only one element of the subsample ranked is measured.The procedure produces a sample of n measurement of independent order statistics.Takahasi and Wakimoto (1968) proved the mathematical theory that the sample mean under RSS is an unbiased estimator of the finite population mean and more precise than the sample mean estimator under simple random sampling (SRS).In some situations, ranking may not be done perfectly.To tackle this problem, Stokes (1977) considered the case where the ranking of elements is done on basis of the auxiliary variable X instead of judgment.Singh et al. (2014) proposed an estimator for population mean and ranking of the elements is observed on the basis of auxiliary variable.The use of the auxiliary information plays an important role in increasing efficiency of the estimators.Samawi and muttlak (1996) have suggested an estimator for population ratio in RSS and showed that it has less variance as compared to usual ratio estimator in SRS.Khan and Shabbir (2015) suggested a class of Hartley-Ross type unbiased estimator in RSS.Khan and Shabbir (2016) have also suggested Hartley-Ross type unbiased estimators in RSS and stratified ranked set sampling (SRSS).Khan et al. (2016) proposed unbiased ratio estimator of finite population mean in SRSS.Munoz and Rueda (2009) used relative bias (RB) and the relative root mean square error (RRMSE), for the comparison of different estimators.For the more detail see Chambers and Dunstan (1986), Rao et al. (1990), Silva and Skinner (1995) and Harms and Duchesne (2006).
In this article, we investigate the properties of the usual mean estimator in RSS and propose an efficient class of the ratio-in-exponential type estimators using two concomitant variables under RSS scheme.

Ranked set sampling procedure with two concomitant variables
In ranked set sampling m independent random samples each of size m are chosen and the items in each sample are selected with equal probability and without replacement from a finite population of size N.The items of each random sample are ranked with respect to the characteristic of the study variable or concomitant variable.Let Y be the study variable and X and Z are the two concomitant variables.Then randomly select 2 m trivariate sample elements from the population and allocate them into m sets, each of size m.Each sample is ranked with respect to one of the concomitant variable X or Z.Here, ranking is done on the basis of the concomitant variable X.An actual measurement from the first sample is then taken on the item with the smallest rank of X, together with variable Y and Z associated with smallest rank of X. From the second sample of size m , the variables Y and Z associated with the second smallest rank of X are measured.By this way, this procedure is continued until, the Y and Z values associated with the highest rank of X are measured from the th m sample.This completes one cycle of sampling process.The procedure is repeated r times to obtain a sample of size n mr  items.Thus in a RSS scheme, a total of 2  mr items have been drawn from the population and only mr of them are selected for analysis.To estimate the population mean,   , Y in RSS using ratio-type estimator with two concomitant variables, the procedure of selecting n ranked set samples can be summarized as follows: Step 1: Randomly select 2 m trivariate sample items from the population.
Step 2: Allocate these 2 m items into m sets, each of size m .
Step3: Each set is ranked with respect to the concomitant variable X.
Step 4: Select the th i ranked item from the th i  


depends on order statistics from some specific distribution (see Arnold et al, 1992).

Proposed class of estimators in RSS
We propose a class of ratio-in-exponential type estimators having two concomitant variables X and Z in RSS as  and 2  are unknown constant whose values are to be determined so that MSE of () is minimized and k is a scalar quantity which can take 0 or 1 values.Also X and Z are the population means of X and Z respectively.
To find the bias and MSE of the estimators, we define the following error terms:  z i m  depend on order statistics from some specific distributions.
In terms of , es up to first order of approximation, we have The bias of () Taking square of Eq. ( 4) and then expectations, the MSE of () The optimum values of 1 2 ( 1) and 1) It is remarked that for different values of 1  and 2  in Eq (3), we can get various exponential ratio-type estimators from the proposed family of estimators () . Some are given below as: The bias and MSE of 1( ) ,

RSS y
are given respectively and (ii) For 1 0 The bias and MSE of 2( )

RSS y
, are given respectively The bias and MSE of The bias and MSE of The bias and MSE of 5( )

RSS y
, are given respectively The optimum values of  .( 1) (vi) For 0 k  , Eq.( 3) becomes The bias and MSE of 6( )

RSS y
, are given respectively The optimum values of  1)

Conclusion
In Tables 1 and 3, we see that the proposed ratio-in-exponential type estimators () have less MSE and RRMSE values as compared to () RSS y .Also, MSE and RRMSE decrease with increase in the sample size.The simulation result of Table 4 indicate that the proposed estimators have reasonable biases, since the values of percentage RB are all less than 2% in absolute terms.Also, the value of percentage RB decreases with increase in sample size .n mr  So, we conclude that the proposed ratio-in-exponential type estimators are preferable than the usual mean estimator under RSS scheme.
coefficients of variation of X, Y and Z respectively.The values of[ : ]    x i m  and[ : ]

Table 1 : The Simulated MSE of Different Estimators
MSE, RRMSE and RB for different estimators are computed using ranked set sampling scheme as described in Section 2. Estimators are then compared in the term of MSE, RB, RRMSE and percentage relative efficiency (PRE).We have computed the PRE of different ratio-in-exponential type estimators of population mean   Y with respect to usual unbiased mean estimator () RSS y for different values of m and r .The results are shown in Tables 1, 2, 3 and 4. The findings indicate that with increase in sample size, MSEs, RB, RRMSEs decrease which are expected results.We used the following expressions to obtain the MSE, RB, RRMSE and PRE: