ISRN Probability and Statistics

VolumeΒ 2012, Article IDΒ 657682, 14 pages

http://dx.doi.org/10.5402/2012/657682

## An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys

School of Studies in Statistics, Vikram University, Ujjain 456010, India

Received 28 February 2012; Accepted 29 March 2012

Academic Editors: J.Β Hu and J.Β LΓ³pez-Fidalgo

Copyright Β© 2012 Ramkrishna S. Solanki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper suggests a class of estimators for estimating the finite population mean of the study variable using known population mean of the auxiliary variable . Asymptotic expressions of bias and variance of the suggested class of estimators have been obtained. Asymptotic optimum estimator (AOE) in the class is identified along with its variance formula. It has been shown that the proposed class of estimators is more efficient than usual unbiased, usual ratio, usual product, Bahl and Tuteja (1991), and Kadilar and Cingi (2003) estimators under some realistic conditions. An empirical study is carried out to judge the merits of suggested estimator over other competitors practically.

#### 1. Introduction

The literature on survey sampling describes a great variety of techniques for using auxiliary information to obtain more efficient estimators. Ratio, product, and regression methods of estimation are good examples in this context (see Singh [1]). If the correlation between the study variable and the auxiliary variable is positive (high), the ratio method of estimation is quite effective. On the other hand, if this correlation is negative (high) the product method of estimation envisaged by Robson [2] and rediscovered by Murthy [3] can be employed quite effectively. The classical ratio and product estimators are considered to be the most practicable in many situations, but they have the limitation of having at most the same efficiency as that of the linear regression estimator. In the situation where the relation between the study variable and the auxiliary variable is a straight line and passing through the origin, the ratio and product estimators have efficiencies equal to the usual regression estimator. But in many practical situations, the line does not pass through the origin, and in such circumstances the ratio and product estimators do not perform equally well to the regression estimator. Keeping this fact in view, a large number of authors have paid their attention toward the formulation of modified ratio and product estimators, for instance, see Singh [4, 5], Srivastava [6β8], Reddy [9, 10], Gupta [11], Sahai [12], Sahai and Ray [13], Ray and Sahai [14], Srivenkataramana and Tracy [15, 16], Bandyopadhyay [17], Vos [18], Sisodia and Dwivedi [19], Adhvaryu and Gupta [20], Chaubey et al. [21], Mohanty and Sahoo [22], Singh and Shukla [23], Naik and Gupta [24], Bahl and Tuteja [25], Mohanty and Sahoo [26], Upadhyaya and Singh [27], Singh and Tailor [28], Kadilar and Cingi [29], Singh et al. [30], and others.

In this paper, we have envisaged a new class of estimators for population mean of study variable using information on an auxiliary variable which is highly correlated with the study variable and have shown that the suggested class of estimators is more efficient than some existing estimators.

Consider a finite population of size from which a sample of size is drawn according to simple random sampling without replacement (SRSWOR). Let and be the sample mean estimators of the population means and respectively, of the study variable and the auxiliary variable . Let , , (correlation coefficient between the variables and ) and , where

The remaining part of the paper is organized as follows. Section 2 gives the brief review of some estimators for the population mean of study variable with its properties. In Section 3, a new class of estimators for the population mean is described, and the expressions for the asymptotic bias and variance are obtained. Asymptotic optimum estimator (AOE) in the suggested class is obtained with its variance formula. Section 4 addresses the problem of efficiency comparisons, while in Section 5 an empirical study is carried out to evaluate the performance of different estimators.

#### 2. Reviewing Estimators

It is very well known that the sample mean is an unbiased estimator of population mean , and under (SRSWOR), its variance is given by where .

The usual ratio and product estimators of population mean of the study variable are, respectively, defined as

Bahl and Tuteja [25] suggested exponential ratio-type and product-type estimators for population mean , respectively, as

Kadilar and Cingi [31] suggested a chain ratio-type estimator for population mean as

Following the procedure adopted by Kadilar and Cingi [31], one may define a chain product-type estimator for population mean as To the first degree of approximation, the biases and variances of estimators , , , , , and are, respectively, given by From (2.1) and (2.12)β(2.17), we have made some efficiency comparisons between the estimators , , , , , , and , as shown in Table 1.

#### 3. Suggested Class of Estimators

We define the following class of estimators for the population mean as where are suitable chosen scalars.

It is to be mentioned that for , the class of estimators reduces to the following class of estimators which is due to Sahai and Ray [13].

While for , it reduces to the new transformed class of estimators defined as

To obtain the bias and variance of suggested class of estimators , we write

such that Expressing (3.1) in terms of βs, we have We assume that , so that , , and are expandable. Now expanding the right-hand side of (3.6), we have Neglecting the terms of βs having greater than two in (3.7), we have

or Taking expectation of both sides of (3.9), we get the bias of the class of estimators to the first degree of approximation as Squaring both sides of (3.9) and neglecting terms βs having power greater than two, we have Taking expectation of both sides of (3.11), we get the variance of to the first degree of approximation as which is minimum when Thus, the resulting minimum variance of is obtained as The minimum variance of equals to the approximate variance of the usual linear regression estimator defined as where , and .

It is interesting to note that if we set and in (3.12), we get the variances of the classes of estimators and , respectively, as If we assume the value of is specified by (say), then the variances of and are, respectively, given by If the value of is specified , then the variances of the estimators and are, respectively, given by To illustrate our general results, we consider a particular case of the proposed class of estimators with its properties. If we set in (3.1), we get an estimator of population mean as Putting in (3.10) and (3.12), we get the bias and variance of , to the first degree of approximation, respectively, as Expression (3.20) clearly indicates that the proposed estimator is better than conventional unbiased estimator if , a condition which is usually met in survey situations.

#### 4. Efficiency Comparisons

From (3.17), we have or equivalently From (3.18), we have or equivalently

From (2.1), (2.12)β(2.17), and (3.12), we have(i) or equivalently(ii) or equivalently(iii) or equivalently(iv) or equivalently(v) or equivalently(vi) or equivalently(vii) or equivalently From (2.1), (2.12), (2.14), (2.16), and (3.21), we have made some efficiency comparisons between the estimators , , , , and , as shown in Table 2.

It is clearly indicated from Table 2 that the estimator is more efficient than estimators , , , and if

#### 5. Empirical Study

To judge the merits of the suggested estimator over usual unbiased estimator , usual ratio estimator , Bahl and Tuteja [25] exponential ratio-type estimator and Kadilar and Cingi [31] chain ratio-type estimator we have considered three populations. Descriptions of the populations are given below.

*Population I (source: Srivastava [7], page 406)*

fiber per plant in jute fiber crops, height, , , , .

*Population II (source: Gupta and Shabbir [32])*

the level of apple production amount (1 unit = 100 tones), the number of apple trees (1 unit = 100 trees), , , , .

*Population III (Source: Kadilar and Cingi [33])*

the level of apple production amount, the number of apple trees, , , , .

We have computed the percent relative efficiencies (PREs) of different estimators , , ,ββ, and of the population mean with respect to by using the following formula:
and results are summarized in Table 3.

Table 3 exhibits that the proposed estimator is more efficient than usual unbiased estimator , usual ratio estimator , Bahl and Tuteja [25] estimator , and Kadilar and Cingi [31] estimator in the sense of having the largest PRE in all three populations. In populations II and III, the proposed estimator has the largest gain in efficiency over all the estimators, while in population I, there is marginal gain in efficiency as compared to . It is further noted that the condition has been satisfied in population I (), II (), and III (). Thus, we recommend the proposed estimator for its use in practice wherever the condition is satisfied.

#### References

- S. Singh,
*Advanced Sampling Theory with Applications. Vol. I, II*, Kluwer Academic, Dordrecht, The Netherlands, 2003. - D. S. Robson, βApplications of multivariate polykays to the theory of unbiased ratio-type estimation,β
*Journal of the American Statistical Association*, vol. 52, pp. 511β522, 1957. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. N. Murthy, βProduct method of estimation,β
*Sankhya*, vol. 26, pp. 69β74, 1964. View at Google Scholar Β· View at Zentralblatt MATH - M. P. Singh, βOn the estimation of ratio and product of the population parameters,β
*Sankhya*, vol. 27, pp. 321β328, 1965. View at Google Scholar - M. P. Singh, βRatio cum product method of estimation,β
*Metrika*, vol. 12, pp. 34β42, 1967. View at Google Scholar Β· View at Zentralblatt MATH - S. K. Srivastava, βAn estimator using auxiliary information in sample surveys,β
*Bulletin/Calcutta Statistical Association Bulletin*, vol. 16, pp. 121β132, 1967. View at Google Scholar - S. K. Srivastava, βA generalized estimator for the mean of a finite population using multi auxiliary information,β
*Journal of the American Statistical Association*, vol. 66, pp. 404β407, 1971. View at Google Scholar - S. K. Srivastava, βA class of estimators using auxiliary information in sample surveys,β
*The Canadian Journal of Statistics*, vol. 8, no. 2, pp. 253β254, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - V. N. Reddy, βOn ratio and product methods of estimation,β
*Sankhya B*, vol. 35, no. 3, pp. 307β316, 1973. View at Google Scholar - V. N. Reddy, βOn transformed ratio method of estimation,β
*Sankhya C*, vol. 36, pp. 59β70, 1974. View at Google Scholar - P. C. Gupta, βOn some quadratic and higher degree ratio and product estimators,β
*Journal of the Indian Society of Agricultural Statistics*, vol. 30, no. 2, pp. 71β80, 1978. View at Google Scholar - A. Sahai, βAn efficient variant of the product and ratio estimators,β
*Statistica Neerlandica*, vol. 33, no. 1, pp. 27β35, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Sahai and S. K. Ray, βAn efficient estimator using auxiliary information,β
*Metrika*, vol. 27, no. 4, pp. 271β275, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. K. Ray and A. Sahai, βEfficient families of ratio and product-type estimators,β
*Biometrika*, vol. 67, no. 1, pp. 211β215, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Srivenkataramana and D. S. Tracy, βAn alternative to ratio method in sample surveys,β
*Annals of the Institute of Statistical Mathematics*, vol. 32, no. 1, pp. 111β120, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Srivenkataramana and D. S. Tracy, βExtending product method of estimation to positive correlation case in surveys,β
*The Australian Journal of Statistics*, vol. 23, no. 1, pp. 93β100, 1981. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Bandyopadhyay, βImproved ratio and product estimators,β
*Sankhya*, vol. 42, no. 1-2, pp. 45β49, 1980. View at Google Scholar - J. W. E. Vos, βMixing of direct, ratio, and product method estimators,β
*Statistica Neerlandica*, vol. 34, no. 4, pp. 209β218, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. V. S. Sisodia and V. K. Dwivedi, βA modified ratio estimator using coefficient of an auxiliary variable,β
*Journal of the Indian Society of Agricultural Statistics*, vol. 33, no. 2, pp. 13β18, 1981. View at Google Scholar - D. Adhvaryu and P. C. Gupta, βOn some alternative sampling strategies using auxiliary information,β
*Metrika*, vol. 30, no. 4, pp. 217β226, 1983. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. P. Chaubey, T. D. Dwivedi, and M. Singh, βAn efficiency comparison of product and ratio estimator,β
*Communications in Statistics. A. Theory and Methods*, vol. 13, no. 6, pp. 699β709, 1984. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Mohanty and L. N. Sahoo, βA class of estimators based on mean per unit and ratio estimators,β
*Statistica*, vol. 47, no. 3, pp. 473β477, 1987. View at Google Scholar Β· View at Zentralblatt MATH - V. K. Singh and D. Shukla, βOne-parameter family of factor type ratio estimators,β
*Metron*, vol. 45, no. 1-2, pp. 273β283, 1987. View at Google Scholar - V. D. Naik and P. C. Gupta, βA general class of estimators for estimating population mean using auxiliary information,β
*Metrika*, vol. 38, no. 1, pp. 11β17, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Bahl and R. K. Tuteja, βRatio and product type exponential estimators,β
*Journal of Information & Optimization Sciences*, vol. 12, no. 1, pp. 159β164, 1991. View at Google Scholar Β· View at Zentralblatt MATH - S. Mohanty and J. Sahoo, βA note on improving the ratio method of estimation through linear transformation using certain known population parameters,β
*Sankhya*, vol. 57, no. 1, pp. 93β102, 1995. View at Google Scholar Β· View at Zentralblatt MATH - L. N. Upadhyaya and H. P. Singh, βUse of transformed auxiliary variable in estimating the finite population mean,β
*Biometrical Journal*, vol. 41, no. 5, pp. 627β636, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. P. Singh and R. Tailor, βUse of known correlation coefficient in estimating the finite population mean,β
*Statist Transaction*, vol. 6, no. 4, pp. 555β560, 2003. View at Google Scholar - C. Kadilar and H. Cingi, βAn improvement in estimating the population mean by using the correlation coefficient,β
*Hacettepe Journal of Mathematics and Statistics*, vol. 35, no. 1, pp. 103β109, 2006. View at Google Scholar Β· View at Zentralblatt MATH - H. P. Singh, R. Tailor, and R. Tailor, βOn ratio and product methods with certain known population parameters of auxiliary variable in sample surveys,β
*Statistics and Operations Research Transactions*, vol. 34, no. 2, pp. 157β180, 2010. View at Google Scholar Β· View at Zentralblatt MATH - C. Kadilar and H. Cingi, βA study on the chain ratio-type estimator,β
*Hacettepe Journal of Mathematics and Statistics*, vol. 32, pp. 105β108, 2003. View at Google Scholar Β· View at Zentralblatt MATH - S. Gupta and J. Shabbir, βVariance estimation in simple random sampling using auxiliary information,β
*Hacettepe Journal of Mathematics and Statistics*, vol. 37, no. 1, pp. 57β67, 2008. View at Google Scholar Β· View at Zentralblatt MATH - C. Kadilar and H. Cingi, βRatio estimators in stratified random sampling,β
*Biometrical Journal*, vol. 45, no. 2, pp. 218β225, 2003. View at Publisher Β· View at Google Scholar