improved ratio estimator using non- conventional location parameter ...

Int. J. Agricult. Stat. Sci. Vol. 13, No. 2, pp. 453-455, 2017

ISSN : 0973-1903

ORIGINAL ARTICLE

IMPROVED RATIO ESTIMATOR USING NON- CONVENTIONAL LOCATION PARAMETER IN RANKED SET SAMPLING Shakeel Javaid* and S. Maqbool1 1

Department of Statistics & O.R., A. M. U., Aligarh - 202 002, India. Division of Agricultural Statistics and Economics, FOA, Wadura, SKUAST- Kashmir - 190 025, India. E-mail: [email protected]

Abstract : In most sample surveys, a reasonable number of the sampling units (eg. households, businesses, etc.) can be ordered fairly accurately with respect to a variable of interest without actual measurement, and at little cost. However, exact measurement of these units may be very tedious and/or expensive. Ranked Set Sampling (RSS), as proposed by McIntyre (1952), provides an alternative to simple random sampling (SRS) in these situations. The feature of RSS is that it combines SRS with other sources of information such as professional knowledge, judgment, auxiliary information, etc., to yield more representative measurements from the population. Ranked set sampling (RSS) is a method of collecting data that improves estimation by utilizing the sampler’s judgment or auxiliary information about the relative sizes of the sampling units. 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, improved ratio estimator using non-conventional location parameter 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, Non-Conventional Location Parameter.

1. Introduction The concept of ranked set sampling (RSS) was first introduced by McIntyre (1952) as a process of improving the precision of the sample mean as an estimator of the population mean. Ranked set sampling as described by McIntyre (1952) is applicable whenever ranking of a set of sampling units can be done easily by judgment method. The advantage of using this sampling procedure over simple random sampling (SRS) procedure is well represented in the available literature. The RSS can be applied in many areas such as forest, agriculture, animal science, medicine etc. Dell and Clutter (1972) demonstrated that for comparable sample sizes, the RSS procedure results in more accurate parameter estimators than simple random sampling (SRS). Equivalently, RSS requires fewer measured observations than SRS to attain the same level of precision. The improvement in precision comes about because RSS adds structure to the data, in the form of the sampler’s ranking, that is absent in SRS. Kadilar *Author for correspondence

Received May 21, 2017

et al. (2009) used this technique to improve ratio estimator, whereas Maqbool et al. (2012) suggested some modified ratio and chain ratio estimators. Mehta Nita and Mandowara (2013) proposed modified ratiocum-product estimator for finite population mean using information on coefficient of variation. In this paper, we suggest improved ratio estimators of the population mean of the study variable using Tri-Mean estimator (TM) of the auxiliary variable. Maqbool, Raja and Javaid (2016) have developed a new ratio type estimator and obtained their MSE equations. The main advantage of the TM is that it is robust against outliers.

2. The Proposed Estimator In RSS, the sampler first selects m independent simple random samples from the population of interest. Each sample is of size m and is drawn without replacement. Thus, the total initial sample size is m2. We call each SRS a set. Within each of the m sets, the sampled items are ranked based on the researcher’s judgment of their relative sizes. This ranking is Revised September 17, 2017 Accepted October 09, 2017

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Shakeel Javaid and S. Maqbool

performed prior to measuring the variable of interest. Therefore, the researcher must have some method for estimating relative sizes of the variable of interest. The researcher could use visual inspection of the items to form rankings or the value of an auxiliary variable correlated with the variable of interest. After ranking the m items in each of the m sets, a subsample is drawn for measurement. This subsample consists of the smallest ranked unit from the first set, the secondsmallest ranked unit from the second set and so forth, so that the subsample contains m items, each representing a different rank from the sets. The variable of interest is then measured for the subsample. Let yi(n) and xi(n) be the ith judgement ordering in the ith set for the study variable and ith set for the auxiliary variable, respectively.

Rˆ JS 

y i ( n )  bTM





yi and x n  

i 1

1 n

i

be the rank

i 1

set sample means for the variables Y and X, respectively. Mean square error of the proposed estimator in Equation (1) can be obtained by using Taylor series method, which is defined as

h ( x i ( n ) , y i ( n ) )  h( X , Y ) 

h(c, d ) ( xi ( n )  X ) c

h(c, d )  ( yi( n)  Y ) d

(2)







    x

i n   X 

 yi  n   Y 

(5)



  yi  n   bTM   2  cov ar xi  n  , yi n    var  yi n   (6) xi n   



 2 RS x n  S y  n   

1 f 2TM 2 S y2 n   S y2n  nX 2





(7)

After simplification, MSE of the proposed estimator is given as 2  2TM 2 C x2 n  2bTMC x C y 1 f  2 2 E Rˆ JS  R    R Cix n   n  X2 X2





 2 R 2C x C y  2 RTM 2 C y2n   R 2C y2n 

(3)



(8)

3. Efficiency Comparison The proposed estimator is always more efficient than the traditional estimator if

 

These can be applied to proposed estimator in order to obtain MSE.  yi  n   bTM  xi  n  ˆ JS  R   R xi  n 

xi n 

 

Var Rˆ JS  Var Rˆ rss

where, hxi n  , yi  n    Rˆ JS and h X ,Y  R .

1

2



n

x

(4)

2   ˆ JS  R  1  f   yi  n   bTM   var xi n   ER 2 2 nX  X 

variable and Sx i(n)Y i(n) is the covariance between auxiliary variable and the variable of interest. n

Y 

2 b 2TM 2 S ix2 n  2bTMSix2 n  1 f  2 E RˆJS  R  R S    ix  n  nX 2  X2 X2

Q  2Q2  Q3 Sxi n  yi n  TM  1 , 4 Si xi2 n 

1 n

in 

Squaring and using expectation to (5) yields,

Here, S i xi2n  is the sample variance of the auxiliary

Let y n  

    y

  ˆ JS  R    yi n   bTM  xi  n   X  R 2 2  xi  n  xi  n  

(1)

xi ( n )

where, b 

 yi  n   bTM  xi  n    yi n 

1 f n

 2 2  2TM 2C x2n  2TMC x C y   R Cix n   X2 X2 

 2 R 2 C x C y  2 RTM  2 C y2n   R 2 C y2 n 





1 f 2 2 R C x n   2 R 2 C x C y  R 2 C y2n  n





Improved Ratio Estimator using Non-Conventional Location Parameter in Ranked Set Sampling

References

After simplification, we get the condition

ρ

Dell, T. R. and J. L. Clutter (1972). Ranked set sampling theory with order statistics background. Biometrics, 28, 545-555.

2TMRC x C y TMC ix2 ( n )  2 RXC y2( n)

Hettmansperger, T. P. and J. W. McKean (1998). Robust Nonparametric Statistical Methods. Wiley, New York.

Which is true, if  0

455

(9)

When the condition (9) is satisfied, proposed estimator is more efficient than the traditional ranked set ratio estimator.

4. Conclusion Ranked set sampling plays a vital role in all kinds of discipline. The availability of auxiliary information enhances the efficiency of the estimator. Improved ratio estimator using Tri-Mean measure is introduced and mean square error of the proposed estimator is obtained. Hence the proposed estimator may be used for better and stable results and can be preferred over the existing estimators for the practical applications.

Kadilar, C., Y. Unyazia and H. Cingi (2009). Ratio estimator for the population mean using rank set sampling. Statist. Papers, 50, 301-309. 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. Maqbool, S., T. A. Raja and Shakeel Javaid (2016). Generalized Modified Ratio Estimator using non-conventional Location Parameter. Int. J. Agricult. Stat. Sci., 12(1), 9597.

Acknowledgement

McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 3, 385-390.

The first author is thankful to UGC for proving the funds under UGC-BSR Research start-up grant [No. F.30-90/2015(BSR)], FD Diary No. 11767, dated 26-03-2015 for carrying out this research work.

Mehta Nita, Ranka and V. L. Mandowara (2013). A modified ratio-cum-product estimator of finite population mean using ranked set sampling. Comm. in Statistics, theory and Methods, 47, 3284-3289.