Regression Estimators in Ranked Set, Median Ranked Set and Neoteric Ranked Set Sampling

This paper proposes new regression estimators in me dian ranked set and neoteric ranked set sampling us i one and two auxiliary variables. The proposed estim ators have large gain in presicion compared to the classical ranked set sampling (RSS) design. A simul ation study is designed to see the performance of suggested estimators. A real data set example is al so used. In this data set, we have examined a rare endemic annual plant species which is grown in Turk ey. We have found that suggested estimators are highly efficient that existing estimators.


Introduction
Ranked set sampling (RSS) is first proposed by McIntyre (1952). He discussed this sampling design is often more efficient than simple random sampling of the same size.  (NRSS). In this study we have proposed regression estimators in median ranked set sampling (MRSS) and neoteric ranked set sampling. To compare the efficiency we have conducted simulation study with simulated and real data sets.

Ranked Set Sampling
Ranked Set Sampling (RSS) design can be described as follows: 1.
Select a simple random sample of size 2 n units from the target finite population and divide them into n samples each of size n.

2.
Rank the units within each sample in increasing magnitude by using personal judgment, eye inspection or based on a concomitant variable.

3.
Select the ith ranked unit from the ith ( ) We introduce regression estimators using one and two auxiliary information, respectively, as: ( )

Median Ranked Set Sampling
Al-Omari (2012) has introduced (MRSS) as follows: 1. Select n random samples each of size n bivariate units from the finite population of interest.

2.
The units within each sample are ranked by visual inspection or any other cost free method with respect to a variable of interest.

3.
If n is odd, select the 1 2 n +       th-smallest ranked unit X together with the associated Y from each set, i.e., the median of each set. If n is even, from the first 2 n sets select the 2 n       th ranked unit X together with the associated Y and from the other 2 n sets select the 2 2

4.
The whole process can be repeated m times if needed to obtain a sample of size nm units.
… , For odd and even sample sizes the units measured using MRSS are denoted by MRSSO and MRSSE, respectively. For odd sample size let 1 1 1  1  1  1  2  2  2 , Z , be the sample mean of X , Z and Y respectively in (MRSS).
For even sample size let , Z , , Z , , Z , be the sample mean of X , Z and Y respectively.
In MRSS we can re-write estimators in defined in section two as given by, respectively ( )

Neoteric Ranked Set Sampling
Zamanzade and Al-Omari (2016) have defined a new neoteric ranked set sampling (NRSS). The NRSS scheme can be described as follows: 1.
Select a simple random sample of size 2 n units from the target finite population.

2.
Ranked the 2 n selected units in an increasing magnitude based on a concomitant variable, personal judgment or any inexpensive method.

3.
If n is an odd, then select the ( ) n 1 i 1 n 2 ( ) (8) be the sample means of study and auxiliary variables in (NRSS). And now we can define regression estimators in (NRSS) as follows: ( )

Simulation Study
In this section, we conducted a simulation study to investigate the properties of proposed estimators. In the simulation study, we consider finite populations of size 10000 N = generated from a bivariate normal distribution. The samples were generated from a bivariate normal distribution using mvrnorm function in R programme. In the simulation, . We have computed mean square errors (MSEs) and percent relative efficiencies (PREs) of estimators with respect to RSS y for 3, 4,5, 6, 7 = n on the basis of 60.000 replications and displayed in Table1. From the Table1 we can say that regression estimators are more efficient in (NRSS) and (MRSS) compared to (RSS). We can also see that when the sample size is odd, the regression estimators in (NRSS) are more efficient than regression estimators in (MRSS).

Real Data Set
In this data set, we have examined a rare endemic annual plant species which is grown in Ankara-Turkey. 896 seeds of endemic plant's weight, height and papus are measured. Using this data set as a population by setting height as study variable (Y) and weight (X) and papus (Z) as auxiliary variables. We have selected 10000 sample under median ranked set and neoteric ranked set sampling design. From each sample we have estimated the height of plant with suggested estimators and calculated mean square error. According to mean square error value we have decided that which sampling plan is suitable for this data set. Descriptive statistics of data set: . The result of real data set is summarized in Table2.

Conclusion
In this study, we have introduced regression estimators in median and neoteric ranked set sampling designs which are recently introduced by Al-Omari (2012) and Zamanzade and Al-Omari (2016). We have compared the efficiency of our suggested estimators with Al-Omari (2012) and Zamanzade and Al-Omari (2016) estimators. We have found that our suggested regression estimators are highly efficient than existing estimators.