Exploring auxiliary information for estimating finite population mean by an improved class of estimators

Introduction

Within the field of statistics, the estimation of population parameters has proven to be a difficult task. Sampling methods have played an important role in developing various estimators suitable for different situations. The fact is that sampling is the modern approach towards all kinds of data driven inferences. It is the art and science of obtaining any needed information at the lowest possible cost and achieving an objective index of accuracy. Drawing a sample using simple random sampling from a homogeneous population with regard to the study's features has shown to be an effective technique for drawing inferences about population parameters. In most of circumstances, the statistician focus on estimation of the population mean or total. For instance, we might be interested in estimating the average or total annual income of a household in a locality. The estimation can be improved by utilizing some auxiliary variables that are highly correlated with the variable for which the estimate is needed. The use of supplementary information in sampling procedures is as old as survey sampling itself. It is a common practice to employ knowledge about auxiliary variables to create better designs and more effective estimators. This information may be used in the planning phase of the survey, in the estimation process, or at both stages simultaneously. The literature on survey sampling outlines a wide range of methods for utilizing auxiliary information (see, e.g., Cochran, 1977; Murthy, 1967) [1,2] . Cochran (1977) [1] used auxiliary information to develop the ratio estimator for estimating the population mean. In the ratio estimator, the study variable and the auxiliary variable had a strong positive correlation, and the regression line should be intersected at the origin. When there is a strong negative correlation between the research variable and the auxiliary variable, Murthy (1964) rediscovers the Robson (1957) product estimator, which is used quite successful in these situations. Further, if the regression line of the study variable on auxiliary variable is a straight line passing through origin and the variance of study variable about this line is proportional to the magnitude of auxiliary variable, the ratio estimator is just as accurate as the regression estimator. However, in many situations the regression line does not pass through the origin. The ratio estimator does not perform as well in these cases as the regression estimator does. Consequently, authors such as Srivastava (1967) [3] , Reddy (1973, 1974), Walsh (1970) [4] , Chakrabarty (1979) [5] , Bandyopadhyay (1980) [6] , Srivenkataramana (1980) [7] , Upadhyaya et al. (1999) [8] , Singh et al. (2003) [9] to suggested modified ratio estimators in order to provide better alternatives. In recent years, new estimators have also emerged, in addition to many other contributions seen in sampling literature. Here we mention some estimators suggested by Choudhury et al. (2012) [10] , Swain (2013), Jerajuddin et al. (2016) [11] , Gupta et al. (2018) [12] , Muili and Audu (2019), Dar et al. (2022), Lall et al. (2024) [13] , etc. The product method of estimation has also been used by some authors when the study variable and auxiliary variable are negatively correlated. The ratio and product estimators have been combined by some authors to propose some more efficient estimators. The estimator proposed by Singh et al. (2003) [9] , Perri (2007), Sifawa et al. (2021) [14] combines ratio and product estimators and has a smaller mean square error than the classical ratio estimator. In recent years, some work has been done to propose efficient families of estimators and one can go through Kadilar and Cingi (2004), Subzar et al. (2018), Singh et al. (2020) [15] , etc.

Summary of some existing estimators

The proposed estimator

Motivated by Lall et al. (2024) [13] estimator of population mean we propose a generalized estimator of the population mean by using the auxiliary information as,

where 𝛼 is any real constant to be defined such that the mean squared error of the proposed is minimum.

In order to find the Bias and mean square error of these estimators, let us define,

Expanding the right-hand side of equation (3.2) retaining the terms up to the second order of 𝜀 ′ 𝑠, so we can have,

and taking expectation on both side we get bias of the estimator 𝑡 as

The proposed class of estimator up to first order will be unbiased for

So, when 𝛽 ≈ 𝑅, i.e., the regression line of 𝑦 𝑜𝑛 𝑥 is linear passing through origin then the above unbiasedness condition reduces to 1 + 2( 𝑁 -𝑛 𝑛 ).So, when the sample size increases, then ( 𝑁 -𝑛 𝑛 ) → 0, hence 𝛼 → 1. In this case, our proposed class of estimator reduces to

which is an unbiased estimator of Ȳ. It utilizes the information as

, where 𝐸[𝐼(𝑥)] = 1. As we increase the sample size, 𝐼(𝑥) → 1 i.e. the difference between the population total 𝑁 X and the sample total 𝑛 x is minimized. So when the sample selected based on the auxiliary characteristic is representative then the proposed class of estimator provides a better estimate of Ȳ. Again, from Equation (3.3), we have

Squaring both sides of (3.7) and retaining the terms up to first order approximation, we have

Theoretical comparison

We first compare the mean square error (MSE) of the proposed class of estimators 𝑡 with the different competing estimators /class of estimators proposed by different authors. Again, the classes are compared using the bias criterion. For 𝐶 𝑦 ≈ 𝐶 𝑥 , it reduces to

Comparison of efficiency

Conclusion

In this paper an effort has been made to propose a generalized estimator of the population mean by using the auxiliary information Motivated by Lall et al. (2024) [13] estimator of population mean. Expressions for bias and mean square error (MSE) of proposed a class of estimators up to the first order approximation have been obtained using SRSWOR. Again, our proposed class of estimators is compared with several other estimators namely simple mean estimator (𝑡 0 ), ratio estimator (𝑡 1 ), product estimator ( 𝑡 2 ) and the different classes of estimators due to Srivastava ( 𝑡 3 ) Chakrabarty (𝑡 4 ) Srivenkataramana (𝑡 5 ), Singh and Espejo (𝑡 6 ) Swain (𝑡 7 ), Jerajuddin and Jai Kishun (𝑡 8 ), Gupta and Yadav (𝑡 9 ), Muili and Audu(𝑡 10 ), Singh and Nigam (𝑡 11 ) and Lall and Dash (𝑡 12 ) both theoretically and numerically by considering all possible samples of sizes 5 of 6 natural populations available in different text books. Again, the study reveals that our proposed class of estimators is asymptotically unbiased with a high percent relative efficiency and the sampling distribution is symmetrical and mesokurtic. The conditions under which our proposed class of estimators is preferable than others are also derived. From both the theoretical and simulation study favours the proposed class of estimators than other competing estimators.

Acknowledgement

AcknowledgmentsThe authors are highly thankful to the reviewers and the editors for their valuable suggestions for improving the paper

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