Modified Extreme Ranked Sets Sampling with Auxiliary Variable

Abstract

In this paper, a modified extreme ranked set sampling (MERSS) scheme is presented. The ratio estimator under the modified scheme is discussed, and expression of mean square error is derived. The performance of the ratio estimator under MERSS is compared with simple random sampling (SRS), ranked set sampling (RSS), median ranked set sampling (MRSS), quartile ranked set sampling (QRSS) and extreme ranked set sampling (ERSS). A simulation study is conducted, and the results showed that MERSS provides the efficient results as compared to SRS, RSS, MRSS, QRSS and ERSS.

Appendix

The ratio estimator based on MERSS is given by

$$ \hat{\mu }_{yMh} = \mu_{x} \frac{{\bar{Y}_{\left[ M \right]h} }}{{\bar{X}_{\left( M \right)h} }} $$

(13)

Using general form of Taylor series for a bivariate function

$$ \begin{aligned} & f\left( {\bar{X}_{\left( M \right)h} ,\bar{Y}_{\left[ M \right]h} } \right) \cong f\left( {\mu_{x} , \mu_{y} } \right) + \left( {\bar{X}_{\left( M \right)h} - \mu_{x} } \right)f_{{\bar{X}_{\left( M \right)h} }} \left( {\mu_{x} , \mu_{y} } \right) + \left( {\bar{Y}_{\left[ M \right]h} - \mu_{y} } \right)f_{{\bar{Y}_{\left[ M \right]h} }} \left( {\mu_{x} , \mu_{y} } \right) \\ & \quad + \frac{1}{2!}\left[ {\left( {\bar{X}_{\left( M \right)h} - \mu_{x} } \right)^{2} f_{{\bar{X}_{\left( M \right)h} \bar{X}_{\left( M \right)h} }} \left( {\mu_{x} , \mu_{y} } \right) + \left( {\bar{Y}_{\left[ M \right]h} - \mu_{y} } \right)^{2} f_{{\bar{Y}_{\left[ M \right]h} \bar{Y}_{\left[ M \right]h} }} \left( {\mu_{x} , \mu_{y} } \right) + 2\left( {\bar{X}_{\left( M \right)h} - \mu_{x} } \right)\left( {\bar{Y}_{\left[ M \right]h} - \mu_{y} } \right)f_{{\bar{X}_{\left( M \right)h} \bar{Y}_{\left[ M \right]h} }} \left( {\mu_{x} , \mu_{y} } \right)} \right] + \ldots \\ \end{aligned} $$

(14)

where $f\left( {\bar{X}_{\left( M \right)h} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{Y}_{\left[ M \right]h} } \right) = \hat{\mu }_{yMh}$ and $h = e,o$ denotes the sample size is even or odd. By using general form of Taylor series for bivariate function, we will get the first degree of approximation of the estimator in Eq. (13) is obtained as:

((15))

where , the MSE of $\hat{\mu }_{yMh}$ from (15) is approximated as

((16))