An improved estimator of finite population mean when using two auxiliary attributes

https://doi.org/10.1016/j.amc.2014.04.069Get rights and content

Abstract

In this paper, we propose an improved estimator of finite population mean using information on two auxiliary attributes under simple random sampling (SRS) and two-phase sampling schemes. The bias and mean squared error (MSE) expressions of the proposed estimators are obtained up to first order of approximation. It is shown theoretically, that the proposed estimator is more efficient than the ratio-product type estimator, regression estimator, difference type estimator and Malik and Singh (2013) [13] estimator in both SRS and two phase sampling. An empirical study using two data sets is also conducted to support the theoretical findings.

Introduction

In survey sampling, the auxiliary information that is correlated with the study variable is frequently used to increase the precision of the estimator. This auxiliary information can be quantified in form of the auxiliary variables and attributes. For this reason, several authors have exploited use of the auxiliary variables and attributes at the estimation stage to increase efficiency of the estimator. For example, the diameter of a tree can be used as a key auxiliary variable when estimating the average height of trees in a forest. Similarly, the breed of a cow is an important auxiliary attribute when estimating average milk yield. Moreover, to estimate the mean hourly wages earned by the people, the auxiliary information can be used in form of the education, marital status, the region of residence, etc. In these examples, the point bi-serial correlation between the study variable and the auxiliary attribute exists and can lead to more precise estimates. For more details, see Naik and Gupta [16], Jhajj et al. [8], Shabbir and Gupta [18], [19] and references cited therein.

In literature several authors have suggested efficient estimators of finite population mean using information on the auxiliary variables and the auxiliary attributes. Some relevant references include Dalabehara and Sahoo [2], [3], Tracy et al. [25], Kadilar and Cingi [9], Gupta and Shabbir [6], Singh and Vishwakarma [23], Singh et al. [24], Koyuncu and Kadilar [12], Shabbir and Gupta [19], Abd-Elfattah et al. [1], Grover and Kaur [5], Koyuncu [10], [11], Malik and Singh [13], [14], Singh and Malik [15], Singh and Solanki [22], and Haq and Shabbir [7]. In this paper, we use the auxiliary information in form of the auxiliary attribute that is correlated with the study variable.

Consider a finite population U={U1,U2,,UN} of size N. Assume that there is a complete dichotomy in the population depending on the presence and absence of the auxiliary attribute ϕj, for j=1,2. Let yi, ϕi1 and ϕi2 be the observations of the study variable and the two auxiliary attributes associated with the ith unit (i=1,2,,N). Let ϕij=1, if the ith unit in the population possesses the auxiliary attribute ϕj, and ϕij=0 otherwise. A random sample of size n is drawn from U by using simple random sampling (SRS) without replacement (SRSWOR). Let Aj=i=1Nϕij and aj=i=1nϕij be the total number of units in the population and sample respectively that possess an auxiliary attribute ϕj (j=1,2). Similarly, let the corresponding population and sample proportions respectively are Pj=1Ni=1Nϕij=AjN and pj=1ni=1nϕij=ajn, for j=1,2. Let sy2=1n-1i=1n(yi-y)2, sϕj2=1n-1i=1n(ϕij-pj)2 and Sy2=1N-1i=1N(yi-Y)2, Sϕj2=1N-1i=1N(ϕij-Pj)2 respectively be the sample and population variances of the study variable (y) and the auxiliary attribute (ϕj), where y=1ni=1nyi and Y=1Ni=1Nyi. Let syϕj=1n-1i=1n(yi-y)(ϕij-pj) and Syϕj=1N-1i=1N(yi-Y)(ϕij-Pj) be the sample and population point bi-serial covariance between y and ϕj, respectively. Similarly, let ρˆyϕj=syϕjsysϕj and ρyϕj=SyϕjSySϕj be the sample and population bi-serial correlation coefficient between y and ϕj, respectively. Let sϕ1ϕ2=1n-1i=1n(ϕi1-p1)(ϕi2-p2) and ρˆϕ1ϕ2=sϕ1ϕ2sϕ1sϕ2 be the sample phi-covariance and phi-correlation between the two auxiliary attributes ϕ1 and ϕ2, respectively. Similarly, the corresponding population phi-covariance and phi-correlation between ϕ1 and ϕ2 are Sϕ1ϕ2=1N-1i=1N(ϕi1-P1)(ϕi2-P2) and ρϕ1ϕ2=Sϕ1ϕ2Sϕ1Sϕ2, respectively. Let Cy=SyY and Cϕj=SϕjPj be the coefficients of variation of y and ϕj, respectively.

In order to obtain the bias and mean squared error (MSE) of the estimators, we define the following relative error terms.

Let ξ0=y-YY and ξj=pj-PjPj, such that E(ξ0)=E(ξj)=0 for j=1,2; Eξ02=λ1Cy2, Eξ12=λ1Cyϕ12, Eξ22=λ1Cyϕ22, E(ξ0ξ1)=λ1ρyϕ1CyCϕ1, E(ξ0ξ2)=λ1ρyϕ2CyCϕ2, E(ξ1ξ2)=λ1ρϕ1ϕ2Cϕ1Cϕ2, where λ1=(1/n)-(1/N).

The outline of the paper is as follows: In Section 2, we overview several existing estimators of Y in SRS. In Section 3, an improved estimator of Y using information on two auxiliary attributes is proposed. The expressions for bias and MSE are obtained under first order of approximation. In Section 4, the existing and proposed estimators are considered under two-phase sampling. In Section 5, we provide theoretical comparison to evaluate the performances of the estimators under both sampling schemes considered here. An empirical study is conducted in Section 6, and concluding remarks are given in Section 7.

Section snippets

Existing estimators in SRS

In this section, we consider several estimators of finite population mean.

Proposed estimator in simple random sampling (SRS)

In this section, an improved estimator of Y using information on two auxiliary attributes is proposed. The bias and MSE of the proposed estimator are obtained to first order of approximation.

The traditional ratio and product estimators of Y based on single auxiliary attribute (ϕ1), respectively, are given byYˆR=yP1p1andYˆP=yp1P1.

Following Singh and Espejo [21], average of both ratio and product estimators of Y, is given byYˆRP1=y2P1p1+p1P1.

Similarly, by using information on second

Estimators in two-phase sampling

In Section 3, we proposed an improved estimator of the finite population mean by assuming that the population means of the auxiliary attributes are known. In application there exists a situation when complete auxiliary information or attribute is not available. In that case, a method of double sampling or two-phase sampling can be used to obtain the estimates of the unknown population parameters. The two-phase sampling scheme involves drawing a large random sample of size n1 using SRSWOR, to

Comparison of estimators

In this section, we compare the proposed estimator with the existing estimators considered in Sections 2 Existing estimators in SRS, 3 Proposed estimator in simple random sampling (SRS), 4 Estimators in two-phase sampling in both SRS and two-phase sampling schemes.

Empirical study

In this section, we consider two data sets in order to numerically evaluate the performances of the estimators in both SRS and two-phase sampling schemes.

Population I

Source: Singh and Chaudhary [20], p. 177.

The population comprises of 34 wheat farms in 34 villages in a certain region of India.

Let y be the area under wheat crop (in acres) during the year 1974, p1 be the proportion of farms under wheat crop which have more than 500 acres land during the year 1971, and p2 be the proportion of farms under wheat crop which have more than 100 acres

Conclusion

In this paper, we proposed an improved estimator of the finite population mean by utilizing information on two auxiliary attributes in both SRS and two-phase sampling schemes. Bias and MSE expressions of proposed estimators, YˆHS and YˆHS, are obtained under first order of approximation. It is worth mentioning that, both theoretically and numerically, the proposed estimator always performs better than all estimators considered here under SRS. Also the proposed estimator outperforms the Malik

Acknowledgements

The authors are thankful to the editor-in-chief and the two anonymous referees for their valuable comments and suggestions that led to an improved version of the article.

References (25)

  • Government of Pakistan, Crops Area Production by Districts. Ministry of Food, Agriculture and Livestock Division,...
  • H.S. Jhajj et al.

    A family of estimators of population mean using information on auxiliary attribute

    Pak. J. Stat.

    (2006)
  • View full text