Modified Exponential Ratio and Product Estimators for Finite Population Mean in Double Sampling

This paper presents exponential ratio and product estimators for estimating finite population mean using auxiliary information in double sampling and analyzes their properties. These estimators are compared for their precision with simple mean per unit, usual double sampling ratio and product estimators. An empirical study is also carried out to judge the merits of the suggested estimators. Zusammenfassung:Diese Arbeit pr̈ asentiert exponentielle Verh ältnisund Produktscḧatzer zur Scḧ atzung des Mittels einer endlichen Population unter Verwendung zus̈ atzlicher Information beidouble samplingund analysiert deren Eigenschaften. Diese Sch ätzer werden auf ihre Pr äzision mit dem einfachen Mittel und den geẅ ohnlichendouble samplingVerhältnisund Produktscḧatzern verglichen. Eine empirische Studie wurde auch durchgef ü rt, um die Vorteile der vorgeschlagenen Sch ätzer zu pr̈ ufen.


Introduction
It is well known that the use of an auxiliary variable X at the estimation stage improves the precision of an estimate of the population mean of a character Y under study. Out of many ratio, product and regression methods of estimation are good examples in this context. When the correlation between study variable Y and auxiliary variable X is positive high, the classical ratio estimator is considered to be the most practicable. The product estimator of Robson (1957), which is rediscovered by Murthy (1964), is employed quite effectively in the case of high negative correlation between study variable Y and auxiliary variable X. Further if the relation between Y and X is a straight line passing through the neighborhood of the origin and the variance of Y about this line is proportional to auxiliary variable X, the ratio estimator is as good as regression estimator.
Let in a finite population U = {U 1 , . . . , U N } of size N the value of the variables on the ith unit x i /N be the population means of the study variable y and the auxiliary variable x, respectively.

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Austrian Journal of Statistics, Vol. 36 (2007), No. 3, 217-225 For estimating the population meanȲ of y a simple random sample of size n is drawn without replacement from the population U . Letȳ = n i=1 y i /n andx = n i=1 x i /n be the unbiased estimators of population meansȲ andX, respectively. Then the classical ratio estimator is defined byȳ and the product estimator is given byȳ whereX, the population mean of the auxiliary variable x is known. With known population meanX, Bahl and Tuteja (1991) suggested the exponential ratio-type estimator and the exponential product-type estimator for the population meanȲ .
If the population meanX of the auxiliary variable x is not known before start of the survey, then it may be more efficient to do the sampling in two-phase (or double sampling). It is a powerful and cost effective (economical) procedure and hence has eminent role to play in survey sampling, see Hidiroglou and Sarndal (1998);Hidiroglou (2001). It is usually employed when the number of units required to give the desired precision on different items is widely different. This procedure is also useful when it is proposed to use the information gathered in the first phase as auxiliary information in order to increase the precision of the information to be gathered in the second phase. Thus, in a survey to estimate the production of lime crop based on orchards as sampling units, a comparatively larger sample is taken to determine the acreage under the crop while the yield rate is determined from only a sub-sample of the orchards selected for determining acreage, see Sukhatme (1962). As another example, suppose it is considered desirable to select a sample of agricultural holdings with probability proportionate to area, but information on area is not available. We may then decide to take an initial random sample of holdings and collect information on their areas (say, by asking the holders) and then take a sub-sample of holding with probability proportionate to area and collect information on the characters under study from this sub-sample, see Raj (1968). Double sampling is also used if the value of auxiliary variable x is obtained by performing a non destructive experiment where as to obtain the value of study variable y of a unit a destructive experiment has to be performed, see Unnikrishan and Kunte (1995, p.104). Neyman (1938) was the first to formulate double sampling (or two-phase sampling) in connection with collecting information on the strata sizes in a stratified sampling.
The objective of this paper is to propose double sampling versions of Bahl and Tuteja (1991) estimators and study their properties. Throughout the paper simple random sampling without replacement (SRSWOR) scheme has been considered. An empirical study is carried out to demonstrate the performance of the suggested estimators over others.

Proposed Ratio and Product Estimators
When the population meanX of the auxiliary variable x is unknown, a first-phase sample of size n is drawn from the population on which only the auxiliary variable x is observed. Then a second phase sample of size n is drawn on which both study variable y and auxiliary variable x are observed. Letȳ = n i=1 y i /n andx = n i=1 x i /n denote the sample means of variables y and x, respectively, obtained from the second sample of size n and x = n i=1 x i /n those obtained from the first sample of size n . Then the double sampling version of the ratioȳ Rd and productȳ P d estimators of population meanȲ are given bȳ It is to be mentioned that the estimatorȳ Rd is due to Sukhatme (1962).
In double (or two-phase) sampling, we suggest the following modified exponential ratio and product estimators forȲ , respectively, aŝ It is easily observed thatȳ Rd ,ȳ P d ,Ŷ ReM d , andŶ P eM d are biased estimators, but the bias being of the order n −1 , can be assumed negligible in large samples. It is assumed that the sample size n is large enough so that the biases of the estimatorsȳ Rd ,ȳ P d ,Ŷ ReM d and Y P eM d are negligible and the variances of these estimators are obtained up to the terms of order n −1 , see Srivastava (1970). The following two cases will be discussed: Case-I when the second phase sample of size n is a sub-sample of the first-phase sample of size n , and Case-II when the second phase sample of size n is drawn independently of the first-phase sample of size n , see Bose (1943).

Case-I
To obtain the variance of the estimatorsŶ ReM d andŶ P eM d , we writeȳ =Ȳ (1 + e 0 ), where λ = 1/n − 1/N , λ = 1/n − 1/N , C y = S y /Ȳ , C x = S x /X, and ρ = S yx /(S y S x ) is the correlation coefficient between y and x, S 2 Expanding the right hand sides, multiplying out and neglecting the terms of e s greater than or equal to two, we get Squaring both sides, taking expectations and using (1), we get their variances to the first degree of approximation as where var I (·) stands for the variance in Case-I, λ * = 1/n−1/n = λ−λ , and a = C x /C y .
To the first degree of approximation the variances ofȳ Rd andȳ P d are and the variance of the usual unbiased estimatorȳ under the SRSWOR scheme is

Efficiency Comparison
From (4) and (8) From (4) and (6) we have var Now combining (9) and (10), we observe that exponential ratio estimatorŶ ReM d is more efficient than the usual unbiased estimatorȳ and the double sampling ratio estimatorȳ Rd , if 1/4 < ρ/a < 3/4, a condition which is usually met in practice.
Conditions (23) and (24) are usually met in survey situations.
We have computed the percent relative efficiencies ofȳ,ȳ Rd ,ȳ P d ,Ŷ ReM d , andŶ P eM d with respect toȳ in Case-I and Case-II and the findings are given in Table 1.

Conclusion
Table 1 clearly indicates that the ratio and product estimatorŶ ReM d andŶ P eM d are more efficient thanȳ,ȳ Rd , andȳ P d . It is also observed that in Case-I the performances of the proposed estimators are better than in Case-II except in population V. Thus, the use of the suggested estimators should be preferred in practice.