Int. J. Agricult. Stat. Sci., Vol. 10, No. 2, pp. 333-334, 2014
ISSN : 0973-1903
ORIGINAL ARTICLE
MODIFIED RATIO ESTIMATOR USING QUARTILES OF AUXILIARY VARIABLE IN RANKED SET SAMPLING S. Maqbool and Shakeel Javaid*1 1
Division of Agricultural Statistics, SKUAST- Kashmir - 190 025, India. Department of Statistics and O.R., Aligarh Muslim University, Aligarh - 202 002, India. E-mail:
[email protected]
Abstract : A method of sampling based on ranked sets is an efficient alternative to simple random sampling, which uses measurements on selected subsets of the primary sample. It can be applied in many studies where exact measurement of an element is very difficult, but the variable of interest can be relatively easily ranked. In this paper, modified ratio estimator using quartiles of an auxiliary variable is developed for population mean in ranked set sampling. Mean square error of the proposed estimator is obtained and compared with traditional ratio rank set sampling. Key words : Ranked Set Sampling, Ratio estimator, Mean square error, Quartiles, Auxiliary variable.
1. Introduction Ranked set sampling (RSS) is an innovative sampling design originally developed by McIntyre (1952) for situations where taking the actual measurements for sample units is difficult, but there are mechanisms readily available for either informally or formally ranking a set of sample units. RSS is a method of collecting data that improves estimation by utilizing the sampler’s judgement or auxiliary information about the relative sizes of the sampling units. Prior to quantifying the data, the researcher samples from the population, then ranks the sample units based on his or her judgement about their relative sizes of the variable of interest. Kadilar et al. (2009) used this technique to improve ratio estimator, whereas Maqbool et al. (2012) suggested some modified ratio and chain ratio estimators. Recently, Ranka and Mandowara (2013) proposed modified ratio-cum-product estimator for finite population mean using information on coefficient of variation. In this paper, we suggest modified ratio estimators of the population mean of the study variable using quartile deviation (QD) and mean of the auxiliary variable.
2. The Proposed Estimator Let yi(n) and xi(n) be the ith judgement ordering in *Author for correspondence.
the ith set for the study variable and ith set for the auxiliary variable, respectively.
R$ smrss =
yi b n g − bQD
Where, b =
(1)
xi b ng
Sxi b n g yi b n g Si xi2b n g
and QD = (Q3 – Q1) / 2
Here, Si x 2i b n g is the sample variance of the auxiliary variable and Sx ib n g y ib n g is the covariance between auxiliary variable and the variable of interest.
1 n 1 n Let y b n g = ∑ y i and xb n g = ∑ x i are the rank n i=1 n i=1 set sample means for the variables Y and X. Mean square error of the proposed estimator in (1) can be obtained using Taylor series method which is defined as
e
g ex − Xj j c h ∂hb∂c,d bg c ∂hbc,d g + ey − Yj
h x ib n g , y ib n g ≅ h X, Y +
∂d
Received April 12, 2014 Revised September 16, 2014
i n
bg
i n
(2)
Accepted October 05, 2014
334
S. Maqbool and Shakeel Javaid
Where,
e j hc X,Y h = R
h x i b n g , y i b n g ≅ R$ smrss
and
−
U| V| W
(3)
These can be applied to proposed estimator in order to obtain MSE.
F y b g − bQD I ∂G JK xb g H −R≅ ex
X
R$ smrss
∂x i b n g
ib n g − X
F y b g − bQD I ∂G JK xb g H + ey
The proposed estimator is always more efficient than the traditional estimator, if
d
i
∂y ib n g
LM MN
ρ2 QD 2 C 2x bn g 2ρQDC x C y 1− f 2 2 R C ix b n g + − n X2 X2
j
−2 R 2 ρC x C y + 2 RQDρ2 C 2yb n g + R 2 C 2yb n g
b g − Yj
(4)
i n
LM y b g bQD OP MN x b g − x b g PQe x b g − Xj 1 + e y − Yj xb g b g 2 i n
ρ≤
i n
2 i n
(5)
i n
i n
LM ey b g − bQDj OP MM X PP varex b g j N Q 2
d
Ε R$ smrss − R
i
1− f ≅ nX 2
i n
QDC 2ix b n g + 2RXC 2y b n g
which is true, if (9)
When the condition (9) is satisfied, proposed estimator is more efficient than the traditional rank set ratio estimator.
i n
2
Acknowledgements
L y b g − bQD OP covar x , y + var y (6) −2 M MN x b g PQ e b g b g j e b g j b QD S b g 1− f L M Εd R$ − Ri ≅ R S b g+ nX M X N O 2bQDS b g − − 2 RρS b g S b g P X PQ i n
i n
2QDRC x C y
ρ≤0
Squaring and taking expectation of (5), we get 2
1− f 2 2 R C xb n g − 2 R 2ρC x C y + R 2 C2ybn g n After simplification, we get ≤
i n
R$ smrss − R ≅ −
d i
Var R$ smrss ≤ Var R$ rss
i n
i n
− 2R2ρCx Cy + 2RQDρ2C2ybn g + R2 C2ybn g (8)
2
3. Efficiency Comparison
i n
i n
OP Q
2ρQDCxCy
i n
i n
The authors wish to thank the editor for his comments and suggestions which have helped to improve the presentation of this article.
i n
2
2
2
smrss
ix n
2
2 2 ix n 2
2 ix n
x n
2
y n
1− f (7) 2QDρ2 S2yb n g + S2yb n g 2 nX After simplification, MSE of the proposed estimator is given as +
d
Ε R$ smrss
ρ QD C b g 1− f L M − Ri ≅ R C b g+ n M X N 2
2
2
2 ix n
2
2
2 x n
References Kadilar, C., Y. Unyazia and H. Cingi (2009). Ratio estimator for the population mean using rank set sampling. Statist. Papers, 50, 301-309. McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 3, 385-390. Maqbool, S., A. H. Mir and Iqbal Jeelani (2012). On some aspects of ratio estimator and chain ratio type estimators using rank set sampling. Proceedings of the VII International Conference on Optimization and Statistics (ISOS-2012) & III National Conference on Statistical Inference, Sampling Techniques and Related areas, Dec. 21-23, A.M.U., Aligarh, pp. 123-130. Ranka, Nita Mehta and V. L. Mandowara (2013). A modified ratiocum-product estimator of finite population mean using ranked set sampling. Comm. in Statistics, Theory and Methods, 47, 3284-3289.
