An improved estimator of finite population mean when using two auxiliary attributes
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 of size . Assume that there is a complete dichotomy in the population depending on the presence and absence of the auxiliary attribute , for . Let , and be the observations of the study variable and the two auxiliary attributes associated with the unit (). Let , if the unit in the population possesses the auxiliary attribute , and otherwise. A random sample of size is drawn from by using simple random sampling (SRS) without replacement (SRSWOR). Let and be the total number of units in the population and sample respectively that possess an auxiliary attribute . Similarly, let the corresponding population and sample proportions respectively are and , for . Let , and , respectively be the sample and population variances of the study variable and the auxiliary attribute , where and . Let and be the sample and population point bi-serial covariance between and , respectively. Similarly, let and be the sample and population bi-serial correlation coefficient between and , respectively. Let and be the sample phi-covariance and phi-correlation between the two auxiliary attributes and , respectively. Similarly, the corresponding population phi-covariance and phi-correlation between and are and , respectively. Let and be the coefficients of variation of and , respectively.
In order to obtain the bias and mean squared error (MSE) of the estimators, we define the following relative error terms.
Let and , such that for ; , , , , , , where .
The outline of the paper is as follows: In Section 2, we overview several existing estimators of in SRS. In Section 3, an improved estimator of 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 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 based on single auxiliary attribute , respectively, are given by
Following Singh and Espejo [21], average of both ratio and product estimators of , is given by
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 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 The population comprises of 34 wheat farms in 34 villages in a certain region of India.Source: Singh and Chaudhary [20], p. 177.
Let be the area under wheat crop (in acres) during the year 1974, be the proportion of farms under wheat crop which have more than 500 acres land during the year 1971, and 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, and , 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.
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