^{1}

^{2}

In this paper, we propose a class of estimators for estimating the finite population mean of the study variable under Ranked Set Sampling (RSS) when population mean of the auxiliary variable is known. The bias and Mean Squared Error (MSE) of the proposed class of estimators are obtained to first degree of approximation. It is identified that the proposed class of estimators is more efficient as compared to [1] estimator and several other estimators. A simulation study is carried out to judge the performances of the estimators.

The problem of estimation in the finite population mean has been widely considered by many authors in different sampling designs. In application, there may be a situation when the variable of interest cannot be measured easily or is very expensive to do so, but it can be ranked easily at no cost or at very little cost. In view of this situation, [

The auxiliary information plays an important role in increasing efficiency of the estimator. [

In RSS, perfect ranking of elements was considered by [

In this paper, we suggest a class of estimators for the population mean, using known population mean of the auxiliary variable in RSS. It is shown that the proposed class of estimators outperforms as compared to the [

In ranked set sampling (RSS), we select m random samples, each of size m units from the population, and rank the units within each sample with respect to a variable of interest. In order to facilitate the ranking, the design parameter m, is chosen to be small. From the first sample the unit having the lowest rank is selected, from the second sample the unit having second lowest rank is selected and the process is continued until from the last sample the unit having the highest rank is selected. In this way, we obtain m measured units, one from each sample. The cycle may be repeated r times until

Suppose that the variable of interest Y is difficult to measure and to rank, but there is the auxiliary variable X, which is correlated with Y. The variable X may be used to obtain the rank of Y. To perform the sampling procedure, m bivariate random samples, each of size m units are drawn from the population then each sample is ranked with respect to one of the variables Y or X. Here, we assume that the perfect ranking is done on basis of the auxiliary variable X while the ranking of Y is with error. An actual measurement from the first sample is then taken of the unit with the smallest rank of X, together with the variable Y associated with the smallest rank of X. From the second sample of size m the Y associated with the second smallest rank of X is measured. The process is continued until from the mth sample, the Y associated with the highest rank of X is measured. The cycle is repeated r times until

We consider a situation when rank the elements on the auxiliary variable. Let

where

To obtain the bias and

such that

and

where

coefficients of variation of Y and X respectively. It also be noted that the values of

The variance of

[

When population mean (

The bias and MSE of

and

When population mean (

The bias and MSE of

and

[

where

The minimum bias and MSE of

are given by

and

The difference-type estimator for population mean (

where d is a constant.

The minimum variance of

is given by

Following [

where

The bias of

The MSE of

where

We discuss two cases.

Case 1: Sum of weights is unity (i.e.

Solving (17), the optimum value of

Substituting

Case 2: Sum of weights is flexible (i.e.

Solving (17), the optimum values of

and

Substituting the optimum values of

Following [

where

where

Solving (21), we have

Taking expectation of both sides of above equation, we get bias of

Squaring both sides of Equation (22) and ignoring higher order terms of e’s, we have

Taking expectation of both sides of above equation, we obtain the MSE of

where

We discuss two cases.

Case 1: Sum of weights is unity (i.e.

The optimum value of

Thus, the minimum MSE of

Case 2: Sum of weights is flexible (i.e.

For

and

Substituting the optimum values of

Note: It is difficult to make the theoretical comparison due to complexity, therefore we adopt the numerical study.

We use the same data set as earlier used by [

Population (source: [

Y = Number of acres devoted to farms during 1992 (ACRES92).

X = Number of large farms during 1992 (LARGEF92).

We set

Here

and

where

To find the possible values of the ratio

a | b | g | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

−1.5 | −1.5 | −1 | 0.1 | 140.6 | 103.2 | 160.9 | 161.4 | 153.2 | 164.5 | 0.00573 | 0.00574 | 0.00573 |

−1.5 | −1.5 | −1 | 0.5 | 139.3 | 103.2 | 159.9 | 160.5 | 163.8 | 164.4 | 0.00590 | 0.00604 | 0.00596 |

−1.5 | −1.5 | −1 | 0.9 | 148.1 | 103.4 | 167.1 | 167.5 | 165.5 | 171.8 | 0.00462 | 0.00404 | 0.00431 |

−1.5 | −1.5 | 1 | 0.1 | 144.5 | 103.3 | 164.1 | 164.8 | 157.3 | 167.8 | 0.00516 | 0.00485 | 0.00499 |

−1.5 | −1.5 | 1 | 0.5 | 132.4 | 103.0 | 154.5 | 156.8 | 157.5 | 158.7 | 0.00689 | 0.00764 | 0.00725 |

−1.5 | −1.5 | 1 | 0.9 | 144.6 | 103.3 | 164.2 | 168.6 | 163.4 | 168.8 | 0.00514 | 0.00482 | 0.00497 |

−1.5 | 0 | −1 | 0.1 | 136.9 | 103.1 | 158.0 | 158.6 | 148.0 | 161.5 | 0.00625 | 0.00658 | 0.00641 |

−1.5 | 0 | −1 | 0.5 | 137.6 | 103.1 | 158.6 | 159.2 | 162.0 | 163.0 | 0.00615 | 0.00642 | 0.00628 |

−1.5 | 0 | −1 | 0.9 | 142.5 | 103.3 | 162.4 | 162.9 | 162.7 | 167.0 | 0.00546 | 0.00530 | 0.00538 |

−1.5 | 0 | 1 | 0.1 | 130.1 | 103.0 | 152.8 | 153.7 | 141.0 | 156.1 | 0.00520 | 0.00816 | 0.00766 |

−1.5 | 0 | 1 | 0.5 | 140.9 | 103.2 | 161.2 | 163.5 | 164.9 | 165.7 | 0.00568 | 0.00567 | 0.00567 |

−1.5 | 0 | 1 | 0.9 | 137.2 | 103.1 | 158.3 | 162.7 | 159.5 | 162.9 | 0.00620 | 0.00651 | 0.00635 |

−1.5 | 1.5 | −1 | 0.1 | 140.8 | 103.2 | 161.1 | 161.6 | 150.5 | 164.8 | 0.00569 | 0.00570 | 0.00569 |

−1.5 | 1.5 | −1 | 0.5 | 140.3 | 103.2 | 160.7 | 161.2 | 164.1 | 165.3 | 0.00576 | 0.00582 | 0.00578 |

−1.5 | 1.5 | −1 | 0.9 | 135.2 | 103.1 | 156.7 | 157.3 | 158.7 | 161.1 | 0.00649 | 0.00697 | 0.00673 |

−1.5 | 1.5 | 1 | 0.1 | 138.2 | 103.2 | 159.1 | 159.9 | 147.8 | 162.7 | 0.00605 | 0.00629 | 0.00616 |

−1.5 | 1.5 | 1 | 0.5 | 139.2 | 103.2 | 159.8 | 162.2 | 163.0 | 164.3 | 0.00592 | 0.00602 | 0.00598 |

−1.5 | 1.5 | 1 | 0.9 | 143.3 | 103.3 | 163.1 | 168.0 | 163.9 | 168.2 | 0.00533 | 0.00513 | 0.00522 |

1.5 | −1.5 | −1 | 0.1 | 133.4 | 103.1 | 155.4 | 156.0 | 142.9 | 158.8 | 0.00672 | 0.00743 | 0.00706 |

1.5 | −1.5 | −1 | 0.5 | 140.8 | 103.2 | 161.1 | 161.6 | 164.5 | 165.7 | 0.00569 | 0.00578 | 0.00576 |

1.5 | −1.5 | −1 | 0.9 | 140.3 | 103.2 | 160.8 | 161.3 | 162.0 | 165.4 | 0.00575 | 0.00578 | 0.00576 |

1.5 | −1.5 | 1 | 0.1 | 142.3 | 103.2 | 162.4 | 163.1 | 152.1 | 166.1 | 0.00546 | 0.00540 | 0.00541 |

1.5 | −1.5 | 1 | 0.5 | 145.3 | 103.3 | 164.7 | 167.1 | 168.7 | 169.5 | 0.00504 | 0.00467 | 0.00484 |

1.5 | −1.5 | 1 | 0.9 | 139.1 | 103.2 | 159.9 | 164.3 | 161.1 | 164.6 | 0.00592 | 0.00605 | 0.00598 |

1.5 | 0 | −1 | 0.1 | 133.4 | 103.0 | 155.4 | 156.0 | 144.4 | 158.8 | 0.00672 | 0.00743 | 0.00658 |

1.5 | 0 | −1 | 0.5 | 140.8 | 103.2 | 161.1 | 161.6 | 164.8 | 165.6 | 0.00569 | 0.00568 | 0.00566 |

1.5 | 0 | −1 | 0.9 | 145.9 | 103.3 | 165.2 | 165.6 | 164.8 | 169.6 | 0.00496 | 0.00453 | 0.00473 |

1.5 | 0 | 1 | 0.1 | 142.3 | 103.3 | 162.4 | 163.1 | 153.6 | 166.1 | 0.00545 | 0.00540 | 0.00540 |

1.5 | 0 | 1 | 0.5 | 141.6 | 103.2 | 161.8 | 164.1 | 165.6 | 166.3 | 0.00557 | 0.00551 | 0.00553 |

1.5 | 0 | 1 | 0.9 | 140.3 | 103.2 | 160.7 | 165.1 | 161.4 | 165.3 | 0.00576 | 0.00582 | 0.00578 |

1.5 | 1.5 | −1 | 0.1 | 139.2 | 103.2 | 159.9 | 160.4 | 151.8 | 163.5 | 0.00591 | 0.00605 | 0.00597 |

1.5 | 1.5 | −1 | 0.5 | 133.0 | 103.0 | 155.1 | 155.7 | 158.1 | 159.2 | 0.00679 | 0.00749 | 0.00713 |

1.5 | 1.5 | −1 | 0.9 | 137.3 | 103.1 | 158.4 | 158.9 | 159.0 | 162.7 | 0.00619 | 0.00650 | 0.00634 |

1.5 | 1.5 | 1 | 0.1 | 141.7 | 103.2 | 161.9 | 162.4 | 154.4 | 165.5 | 0.00555 | 0.00551 | 0.00520 |

1.5 | 1.5 | 1 | 0.5 | 142.3 | 103.3 | 162.3 | 164.6 | 166.4 | 166.8 | 0.00548 | 0.00534 | 0.00540 |

1.5 | 1.5 | 1 | 0.9 | 135.2 | 103.1 | 156.8 | 161.0 | 157.8 | 161.2 | 0.00648 | 0.00701 | 0.00672 |

We investigate the percentage relative efficiency (PRE) of ratio estimator

The PREs of our proposed estimator and other existing estimators with respect to conventional estimator are given in

Since

Therefore, the proposed class of estimators can be preferred over its competitive estimators in application under RSS.

The authors wish to thank the editor and the anonymous referees for their suggestions which led to improvement in the earlier version of the manuscript.

Lakhkar Khan,Javid Shabbir, (2016) An Efficient Class of Estimators for the Finite Population Mean in Ranked Set Sampling. Open Journal of Statistics,06,426-435. doi: 10.4236/ojs.2016.63038

Estimator | Remarks | |||||
---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 1 | Usual RSS mean estimator | |

1 | 0 | 0 | 1 | 0 | Usual RSS ratio estimaotr | |

0 | 0 | 1 | 0 | Kadilar et al. (2009) ratio type estimator | ||

1 | 0 | 0 | 1 | Regression type estimator | ||

1 | 0 | 0 | 1 | Difference type estimator | ||

1 | 0 | 1 | 0 | Difference-ratio estimator | ||

0 | 1 | 0 | Generalied difference-ratio estimator | |||

1 | 0 | 1 | 0 | Regression-ratio estimator | ||

1 | 0 | 1 | 1 | 0 | Exponential type estimator | |

1 | 1 | 1 | 0 | Regression-exponential type estimator |