International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 378

ISSN 2229-5518

H.S. Jhajj1 and H. K. Bhangu2

For estimating the median of the population, we have proposed two estimators using linear transformations using the information on median of the auxiliary variable. The expressions for biases, mean square errors and their minimum values have been obtained. It has been shown that proposed estimators are always efficient than the ratio estimator and equally efficient to the other estimators derived from a different approach respectively defined by Kuk and Mak (1989). The comparison of estimators among the proposed estimators with respect to their biases has also been done.The results have been illustrated by carrying out the simulation study.

In survey sampling, statisticians have given more attention to the estimation of population mean, total, variance etc. but median is regarded as a more appropriate measure of location than mean when the distribution of variables such as income, expenditure etc is highly skewed. In such situations, it is necessary to estimate median. First of all some statisticians such as Gross(1980), Sedransk and Meyer(1978), Smith and Sedransk(1983) have considered the problem of estimating the median by dealing exclusively with variable under study Y only.

1. Prof. & Head, Department of Statistics, Punjabi University, Patiala-147002, India. Email: drhsjhajj@yahoo.co.in

2. Department of Community Medicine, SGRDIMSAR, Amritsar-143501, India. Email: harpreet3182@gmail.com

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Kuk and Mak (1989) are the first to introduce the estimation of median of study variable Y by using information of the values on the auxiliary variable X highly correlated with Y for the units in the sample and its known median M X for the whole population. Later problem of estimation of median was discussed by various authors such as Chambers and Dunstan(1986), Rao et al.(1990), Mak and Kuk(1993), Rueda et al.(2001),Arcos et al.(2005), Garcia and Cebrian(2001), Meeden(1995), and Singh, S. et al(2007).

Using known value of population median MX of the auxiliary variable X, Kuk and Mak (1989)

suggested an estimator for the population median MY of study variable Y under simple random sampling similar to ratio estimator of its population mean as

*M*ˆ YR = *M*ˆ *M *X

ˆ

X

(1.1)

where, ˆ

and ˆ

are the estimators of MY and MX respectively based on a simple random

sample of size n drawn from the population.

Let 𝑌𝑖 and 𝑋𝑖 denote the values on the ith unit of the population i = 1, 2, 3, …, N for the study variable Y and auxiliary variable X respectively and corresponding small letters denote the

values in the sample.

Suppose that Y(1), Y(2),…,Y(n) are the values of Y on the sample units in ascending order. Further, let t be an integer such that Y(t) ≤ MY ≤ Y(t+1) and let p = t/n be the proportion of Y

values in the sample that are less than or equal to the median value MY , an unknown population

parameter. If pˆ is a predictor of p, the sample median ˆ Y

as Qˆ ( pˆ ), where pˆ = 0.5.

can be written in terms of quantiles

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Kuk and Mak (1989) define a matrix of proportions (Pij, (i,j=1,2 )) of units in the population

as

X ≤ MX | X > MX | Total | |

Y ≤ MY | P11 | P12 | P1. |

Y > MY | P21 | P22 | P2. |

Total | P.1 | P.2 | 1 |

Where for instance, P11 denotes the proportion of the units in the population with Y≤ 𝑀𝑌 and

X ≤ 𝑀𝑋 . In practice, the Pij are usually unknown but can be estimated by 𝑝𝑖𝑗 based on a similar

cross-classification of the sample. Thus, 𝑝11 , for instance, represents the proportion of units in

the sample with Y≤ 𝑀𝑌 and X ≤ 𝑀𝑋 . For estimating the population median MY of study variable

Y, Kuk and Mak(1989) has also proposed two other estimators, position estimator 𝑀�𝑌𝑃 and

stratification estimator 𝑀�𝑌𝑆 respectively derived from a different approach.

𝑀�𝑌𝑃 = 𝑄�𝑌 (𝑝1 ) (1.2)

where, 𝑝1 = 2/𝑛{𝑛𝑋 𝑝11 + (𝑛 − 𝑛𝑋 )(

− 𝑝11 )}

where, 𝑛𝑋 be the number of units in the sample with X ≤ MX.

𝑀�𝑌𝑆 = inf{𝑦: 𝐹�𝑌 (𝑦) > 1/2} (1.3)

where, 𝐹�𝑌 (𝑦) ≅

{𝐹�𝑌1(𝑦) + 𝐹�𝑌2(𝑦)} and for any value of y, let 𝐹�𝑌1(𝑦) be the proportion

among those units in the sample with X ≤ 𝑀𝑋 that have Y values less than or equal to y.

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Similarly, 𝐹�𝑌2(𝑦) is the proportion among those with X > 𝑀𝑋 .

Defining

𝑒0 =

𝑀�𝑌

𝑀𝑌

− 1 , 𝑒 = 𝑀�𝑋 − 1

𝑀𝑋

such that E(ek ) ≅ 0 and | ek | < 1 for k = 0,1

Using results of Kuk and Mak (1989) up to the first order of approximation, we have

E(𝑒 2) = (1 − 𝑓)(4𝑛)−1

[𝑀𝑌 𝑓𝑌 (𝑀𝑌 )]−2

(1.4)

E(𝑒 2) = (1 − 𝑓)(4𝑛)−1

[𝑀𝑋 𝑓𝑋 (𝑀𝑋 )]−2

(1.5)

𝐸(𝑒0𝑒1)=(1 − 𝑓)(4𝑛)−1[4𝑃11(𝑋, 𝑌) − 1][𝑀𝑋 𝑀𝑌 𝑓𝑋 (𝑀𝑋 )𝑓𝑌 (𝑀𝑌 )]−1 (1.6)

where it is being assumed that as N→∞, the distribution of the bivariate variable (X,Y)

approaches to a continuous distribution with marginal densities 𝑓𝑋 (𝑥) and 𝑓𝑌 (𝑦) for X and Y

respectively. This assumption holds in particular under a superpopulation model framework,

treating the values of (X,Y) in the population as a realization of N independent observations from

a continuous distribution. We also assume that 𝑓𝑋 (𝑥) and 𝑓𝑌 (𝑦) are positive.

When the median 𝑀𝑋 of the auxiliary variable X is known, we propose following estimators of population median using linear transformation under the simple random sampling design as

𝑀�𝐻1 =

𝑀�𝑌

𝑀�𝑋

[𝑀�𝑋 + 𝛼�𝑀𝑋 − 𝑀�𝑋 �] (2.1)

𝑀�𝐻2 =

𝑀�𝑌

[𝑀𝑋 + 𝑣�𝑀� ′ − 𝑀

�] (2.2)

𝑀𝑋

𝑋 𝑋

where 𝑀� ′ = 𝑁𝑀𝑋 −𝑛𝑀�𝑋

𝑋 𝑁−𝑛

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where 0 ≤ 𝛼 ≤ 1 , 0 ≤ 𝑣 ≤ 1

Assuming that sample size is large enough such that terms involving 𝑒𝑖 ’s more than second

degree are negligible in the expansions of estimators 𝑀�𝐻1𝑎𝑛𝑑 𝑀�𝐻2 in terms of 𝑒𝑖 ’s while

obtaining biases and mean squared errors.

Using results (1.2) – (1.4), the biases and MSE’s of 𝑀�𝐻1 𝑎𝑛𝑑 𝑀�𝐻2 , up to first order of approximation are

Bias (𝑀�𝐻1 ) = 𝑀𝑌 𝛼(1 − 𝑓)(4𝑛)−1 [ {𝑀𝑋 𝑓𝑋 (𝑀𝑋 )}−2

−{4𝑃11 (𝑋, 𝑌) − 1}{𝑀𝑋 𝑀𝑌 𝑓𝑋 (𝑀𝑋 )𝑓𝑌 (𝑀𝑌 )}−1 ] (2.3)

Bias (𝑀�𝐻2 ) = −𝑀𝑌

𝑛𝑣

(𝑁−𝑛)

(1 − 𝑓)(4𝑛)−1{4𝑃11 (𝑋, 𝑌) − 1}{𝑀𝑋 𝑀𝑌 𝑓𝑋 (𝑀𝑋 )𝑓𝑌 (𝑀𝑌 )}−1 (2.4)

MSE (𝑀�𝐻1 ) = (1 − 𝑓)(4𝑛)−1 [{ 𝑓𝑌 (𝑀𝑌 )}−2 + 𝛼 �

𝑀𝑌 �2

𝑀𝑋

{𝑓𝑋 (𝑀𝑋 )}−2 (𝛼 − 2𝐶) ] (2.5)

where C = [4𝑃11 (𝑋,𝑌)−1]𝑀𝑋 𝑓𝑋 (𝑀𝑋 )

𝑀𝑌 𝑓𝑌 (𝑀𝑌 )

(2.6)

From (2.5), we note that MSE of 𝑀�𝐻1 decreases with the decrease in the value of 𝛼 provided

C ≤ 𝛼/2.

Similarly, up to the first order of approximation, we get

MSE(𝑀�𝐻2 ) = (1 − 𝑓)(4𝑛)−1[ {𝑓𝑌 (𝑀𝑌 )}−2 + 𝜃 �

𝑀𝑌 �2

𝑀𝑋

{𝑓𝑋 (𝑀𝑋 )}−2𝑣(𝜃𝑣 − 2𝐶) ] (2.7)

where 𝜃 = 𝑛

𝑁−𝑛

In (2.7), we note note that MSE of 𝑀�𝐻2 decreases with decrease in the value of 𝑣 provided

C ≤ 𝜃𝑣/2.

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MSE (𝑀�𝐻1 ) minimizes for

𝛼 = [4𝑃11 (𝑋,𝑌)−1]𝑀𝑋 𝑓𝑋 (𝑀𝑋 ) = C (2.8)

𝑀𝑌 𝑓𝑌 (𝑀𝑌 )

and its minimum value is given by

𝑀𝑆𝐸𝑚𝑖𝑛 (𝑀�𝐻1 ) = (1 − 𝑓)(4𝑛)−1 {𝑓𝑌 (𝑀𝑌 )}−2 {1 − (4𝑃11(𝑋, 𝑌) − 1)2} (2.9) Bias of optimum estimator 𝑀�𝐻1 is given by

Bias (𝑀�𝐻1 ) = (1 − 𝑓)(4𝑛)−1{4𝑃11 (𝑋, 𝑌) − 1}[ {𝑀𝑋 𝑓𝑋 (𝑀𝑋 )𝑓𝑌 (𝑀𝑌 )}−1

−{4𝑃11 (𝑋, 𝑌) − 1}𝑀𝑌 −1 {𝑓𝑌 (𝑀𝑌 )}−2 ] (2.10)

Similarly, MSE(𝑀�𝐻2 ) minimizes for

𝑣 = (𝑁−𝑛)[4𝑃11 (𝑋,𝑌)−1]𝑀𝑋 𝑓𝑋 (𝑀𝑋 ) = (𝑁−𝑛) 𝐶 (2.11)

𝑛𝑀𝑌 𝑓𝑌 (𝑀𝑌 ) 𝑛

and its minimum value is given by

𝑀𝑆𝐸𝑚𝑖𝑛 (𝑀�𝐻2 ) = (1 − 𝑓)(4𝑛)−1 {𝑓𝑌 (𝑀𝑌 )}−2 [1 − {4𝑃11(𝑋, 𝑌) − 1}2] = 𝑀𝑆𝐸𝑚𝑖𝑛 (𝑀�𝐻1 ) (2.12)

and bias of optimum estimator of 𝑀�𝐻2 is given by

Bias (𝑀�𝐻2 ) = −(1 − 𝑓)(4𝑛)−1{4𝑃11(𝑋, 𝑌) − 1}2 𝑀𝑌 −1{𝑓𝑌 (𝑀𝑌 )}−2 (2.13)

Using the expressions (2.10) and (2.13), we have

|Bias (𝑀�𝐻1 )| = � 𝑀𝑌 𝑓𝑌 (𝑀𝑌 )

� (2.14)

|Bias (𝑀�𝐻2 )|

− 1

[4𝑃11 (𝑋,𝑌)−1]𝑀𝑋 𝑓𝑋 (𝑀𝑋 )

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Expression (2.14) shows that bias of (𝑀�𝐻2 ) is smaller than the bias of (𝑀�𝐻1)

1 𝑀𝑌 𝑓𝑌 (𝑀𝑌 )

if 𝜌𝑐 <

2 𝑀𝑋

𝑓𝑋

(𝑀𝑋

(2.15)

)

where 𝜌𝑐 = [4𝑃11(𝑋, 𝑌) − 1] ,the correlation coefficient between the variables X and Y,goes

from -1 to 1 as 𝑃11(𝑋, 𝑌) increases from 0 to 1/2 which implies that 𝜌𝑐 is negative and positive

for 𝑃11(𝑋, 𝑌) belongs to [0

1) and (1

4 4

1] respectively.

2

Using the additional knowledge of C in addition to known value of population median 𝑀𝑋 of

auxiliary variable X, we can construct from (2.1) and (2.2) efficient estimators of population median 𝑀𝑌 of study variate Y.

To compare the proposed estimators with 𝑀�𝑌𝑅 given by Kuk and Mak(1989) and usual sample median 𝑀�𝑌 , we first write the expressions of MSEs of estimators 𝑀�𝑌𝑅 and 𝑀�𝑌 of population

median up to the first order of approximation as

MSE (𝑀�𝑌𝑅 ) = (1 − 𝑓)(4𝑛)−1 [ {𝑓𝑌 (𝑀𝑌 )}−2 + �

𝑀𝑌 �2

𝑀𝑋

{𝑓𝑋 (𝑀𝑋 )}−2

−2{4𝑃11(𝑋, 𝑌) − 1}(

𝑀𝑌

𝑀𝑋

){𝑓𝑋 (𝑀𝑋 )𝑓𝑌 (𝑀𝑌 )}−1 ] (3.1)

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MSE (𝑀�𝑌 ) = (1 − 𝑓)(4𝑛)−1 {𝑓𝑌 (𝑀𝑌 )}−2 (3.2)

Using (2.9) & (3.1), we have

MSE (𝑀�𝑌𝑅 ) - MSE (𝑀�𝐻1 ) = [

𝑀𝑌

𝑀𝑋

{𝑓𝑋 (𝑀𝑋 )}−1 – {𝑓𝑌 (𝑀𝑌 )}−1 {4𝑃11(𝑋, 𝑌) − 1}]2

≥ 0, which is always true. (3.3)

Similarly, using (2.9) & (3.2), we have

MSE (𝑀�𝑌 ) - MSE (𝑀�𝐻1 ) = {4𝑃11(𝑋, 𝑌) − 1 }2

≥ 0, which is always true. (3.4)

From (3.3) and (3.4), we note that the estimator 𝑀�𝐻1 is always efficient than the estimator 𝑀�𝑌𝑅

defined by Kuk and Mak (1989) and usual sample median 𝑀�𝑌 but it is equally efficient to the

other two estimators proposed by KUK and Mak (1989).

To obtain the rough idea about the efficiencies of proposed estimators over the existing ones, simulation study has been carried out using R software in which we drew 10,00,000 repeated samples from a bivariate normal population for different correlation coefficient values

with different samples sizes having Medians : 𝑀𝑌 = 4, 𝑀𝑋 = 3, Means: 𝜇𝑋 = 4, 𝜇𝑌 = 3 and

Standard deviations : 𝜎𝑌 = 3, 𝜎𝑋 = 1. Numerical values of results are given in table 4.1

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Table 4.1 Biases of different estimators

Correlation coefficient (𝜌𝑐 ) | Sample size(n) | Bias | |||||

Correlation coefficient (𝜌𝑐 ) | Sample size(n) | 𝑀�𝐻1 | 𝑀�𝐻2 | 𝑀�𝑌 | 𝑀�𝑌𝑅 | 𝑀�𝑌𝑃 | 𝑀�𝑌𝑆 |

0.3 | 3 | 0.049 | 0.045 | 1.89 | 0.154 | 0.151 | 0.152 |

0.3 | 5 | 0.032 | 0.029 | 1.15 | 0.094 | 0.093 | 0.093 |

0.3 | 7 | 0.022 | 0.019 | 0.84 | 0.066 | 0.058 | 0.057 |

0.3 | 9 | 0.018 | 0.015 | 0.67 | 0.053 | 0.049 | 0.049 |

0.5 | 3 | 0.055 | 0.051 | 1.89 | 0.097 | 0.088 | 0.087 |

0.5 | 5 | 0.037 | 0.032 | 1.15 | 0.062 | 0.058 | 0.057 |

0.5 | 7 | 0.026 | 0.023 | 0.84 | 0.044 | 0.036 | 0.034 |

0.5 | 9 | 0.022 | 0.018 | 0.67 | 0.037 | 0.030 | 0.030 |

0.7 | 3 | 0.025 | 0.022 | 1.89 | 0.029 | 0.028 | 0.027 |

0.7 | 5 | 0.021 | 0.019 | 1.15 | 0.026 | 0.026 | 0.026 |

0.7 | 7 | 0.017 | 0.013 | 0.84 | 0.021 | 0.019 | 0.017 |

0.7 | 9 | 0.015 | 0.012 | 0.67 | 0.017 | 0.016 | 0.014 |

0.9 | 3 | 0.058 | 0.046 | 1.89 | 0.059 | 0.812 | 0.811 |

0.9 | 5 | 0.036 | 0.01 | 1.15 | 0.029 | 0.486 | 0.484 |

0.9 | 7 | 0.021 | 0.011 | 0.84 | 0.017 | 0.355 | 0.358 |

0.9 | 9 | 0.014 | 0.005 | 0.67 | 0.011 | 0.273 | 0.269 |

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Table 4.2 Comparison of efficiencies of estimators

Correlation coefficient (𝜌𝑐 ) | Sample size(n) | MSE | Relative efficiencies | |||||

Correlation coefficient (𝜌𝑐 ) | Sample size(n) | 𝑀�𝑌 | 𝑀�𝑌𝑅 | 𝑀�𝐻1 | 𝑀�𝐻2 | 𝑀�𝑌 | 𝑀�𝑌𝑅 | 𝑀�𝐻1 |

0.3 | 3 | 1.89 | 3.516 | 1.88 | 1.88 | 100 | 53.75 | 100.5 |

0.3 | 5 | 1.15 | 2.124 | 1.13 | 1.13 | 100 | 54.14 | 101.8 |

0.3 | 7 | 0.84 | 1.491 | 0.82 | 0.82 | 100 | 56.33 | 102.4 |

0.3 | 9 | 0.67 | 1.122 | 0.65 | 0.65 | 100 | 59.82 | 103.1 |

0.5 | 3 | 1.89 | 2.114 | 1.71 | 1.71 | 100 | 89.40 | 110.5 |

0.5 | 5 | 1.15 | 1.258 | 1.01 | 1.01 | 100 | 91.41 | 113.8 |

0.5 | 7 | 0.84 | 0.901 | 0.72 | 0.72 | 100 | 93.22 | 116.7 |

0.5 | 9 | 0.67 | 0.698 | 0.56 | 0.56 | 100 | 95.98 | 119.6 |

0.7 | 3 | 1.89 | 1.457 | 1.39 | 1.39 | 100 | 129.7 | 136.0 |

0.7 | 5 | 1.15 | 0.881 | 0.82 | 0.82 | 100 | 130.5 | 140.2 |

0.7 | 7 | 0.84 | 0.639 | 0.59 | 0.59 | 100 | 131.4 | 142.4 |

0.7 | 9 | 0.67 | 0.505 | 0.46 | 0.46 | 100 | 132.7 | 145.7 |

0.9 | 3 | 1.89 | 0.835 | 0.81 | 0.81 | 100 | 226.4 | 233.3 |

0.9 | 5 | 1.15 | 0.501 | 0.48 | 0.48 | 100 | 229.5 | 239.6 |

0.9 | 7 | 0.84 | 0.365 | 0.35 | 0.35 | 100 | 230.1 | 240.0 |

0.9 | 9 | 0.67 | 0.287 | 0.27 | 0.27 | 100 | 233.4 | 248.2 |

From Table 4.1, we see that proposed estimators have lower bias than all the three estimators proposed by

Kuk and Mak(1989) and usual sample median estimator. Also the bias of estimator 𝑀�𝐻2 is lower than

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estimator 𝑀�𝐻1 . From table 4.2, it is clear that proposed estimators always have higher efficiency

than the ratio estimator defined by Kuk and Mak(1989) and usual sample median estimator.

The theoretical study shows that proposed estimators are always more efficient than ratio estimator defined by Kuk and Mak (1989) as well as usual sample median estimator for all the

situations. It has also been shown that both the estimators 𝑀�𝐻1 and 𝑀�𝐻2 are equally efficient but

in spite of exact bias of 𝑀�𝐻2 as compared to the bias of 𝑀�𝐻1 taken up to first order of

approximation is smaller than 𝑀�𝐻1 . Biases of the proposed estimators are less than the estimators

proposed by Kuk and Mak(1989).It is also shown that efficient estimators can be constructed by choosing the values of 𝛼 and 𝑣 in the proposed estimators corresponding to given values of C or

range of C. Numerical results given in table 4.2 by using simulation also show that the proposed estimators are always efficient than ratio estimator defined by Kuk and Mak (1989) as well as usual sample median estimator and table 4.1 shows that bias of proposed estimators is less than the estimators proposed by Kuk and Mak(1989) and usual sample median estimator.

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