Exponential logarithmic approach for estimation of population parameter in two-phase sampling

Introduction

Survey sampling in statistical terminology describes the process of choosing a sample from an interest group that is the target population. A brief interview or discussion with people may be referred to as a survey. Depending on the desired approach and the goals to be met, there are numerous ways in which a survey can be conducted. In survey sampling, missing data or missing values occur when a chosen respondent refuses to participate in a survey for various reasons. Various techniques are available to handle the problems of missing data. One of the best techniques to deal with missing data is imputation. It is a familiar and special method of handling non-response in which various options are used to impute missing values for one or more study variables. The approach of the current study is to identify certain classes of estimators that are more effective than other pertinent estimators and are typically in practice in two-phase sampling. An imputation technique that solves missing item values in an efficient manner, disallowing incompleteness and making its analysis simple. Recently, many authors have given notable attention to the study of imputation techniques. In case of observing missing values of the variable under study and its estimation there upon using two auxiliary variables in two phase sampling in a predictive approach with unknown population mean of the first auxiliary variable has been studied [1] . Using an auxiliary variable, an effective class of estimators for a finite population in stratified random sampling has been suggested [2] . A study has been attempted modestly to offer some effective imputation techniques of the difference and ratio types when there are missing values under rank set sampling [3] . A factor-type exponential ratio estimator as a technique for estimating population mean has been addressed, where the estimators make use of additional data for an auxiliary variable, including kurtosis and coefficient of variation [4] . To estimate the population mean of the study variable in stratified two-phase sampling while taking measurement error and non-response into account concurrently, three classes of more efficient and generalized combined regression-cum-ratio estimators have been introduced [5] .

Using imputation of missing data under two-phase sampling, an efficient exponential chain ratio estimator for estimating the finite population mean of the study variable has been proposed [6] . To handle the missing data, a few new classes of exponential-logarithmic imputation approaches have been given [7] . A family of logarithmic cum exponential estimators using two-phase sampling for population mean with risk analysis has been suggested [8] . In two phase sampling, an efficient class of estimators for finite population mean estimation has been addressed when both measurement errors and non-response occur at the same time [9] . Efficient estimation combining exponential and ln functions under two-phase sampling has been suggested [10] . During survey sampling, there is a chance that the population mean of the auxiliary variable will be unknown to the practitioners. The solution to this problem of an unknown value is two-phase sampling, often known as double sampling. This research aims to explore the properties of the Das and Singh (2023) estimators and propose double sampling variants of them [7] . The simple random sampling without replacement (SRSWOR) scheme has been taken into consideration throughout the work. To show how well the recommended estimators work in comparison to alternative estimators, an empirical study is conducted. Let in a finite population  

Methodology

Several widely available and conventional techniques for imputation include the following classical methods:

Mean Methods

In this method each non-responding data is replaced with the mean of responded data, Here

The unit estimator of the population mean under this imputation method, 𝒀 ‾ is given by

Compromised Method

The compromised method of imputation is suggested by Singh and Horn . (1 ) (1

Where  is a chosen constant.

The unit estimator of this imputation method is   Bias and MSE of the unit estimator

The optimum mean square error (MSE) of

Proposed Imputation Techniques and Corresponding unit Estimators

Three imputation techniques with their corresponding unit estimators are suggested as follows (1)

The unit estimator of this proposed method of imputation is 11 e x p lo g 1

The unit estimator of this proposed method of imputation is

Properties of the proposed estimators Consider the following approximations to find the bias and MSE of the proposed estimators

from the equation (2.4) and taking expectations on both sides, we will get

Putting the corresponding values of        

from the equation (2.5) and taking expectations on both sides, we will get

Using equation (2.5) to construct and keeping the terms up to the second-degree of approximations, we will obtain

from the equation (2.6) and taking expectations on both sides, we will get

Using equation (2.6) to construct and keeping the terms up to the second-degree of approximations, we will obtain

(2.12)  for equating it to zero

Optimum Mean Square errors of

Putting the optimum value of 1  i.e.  

Differentiating equation (2.11) in respect of 2  for equating it to zero

Substituting the optimum value of 2  i.e.  

Differentiating equation (2.12) in respect of 3  for equating it to zero

Substituting the optimum value of 3  i.e.

Results and Discussion

Efficiency Comparison

This section presents a theoretical and empirical demonstration of the performance of the suggested methods of imputation and its resulting estimators over the mean, ratio and compromised methods of imputation.

Theoretical Comparison

Here, we derive the conditions under which the suggested estimators perform better than the mentioned traditional estimators under both the designs. Theoretical comparisons under design I are as follows:

Comparison with mean method:

So, the proposed estimator 𝑦 ‾ 𝐵𝐺3 is better than 𝑦 ‾ 𝑟 under both the designs.

Comparison with ratio method

So, the proposed estimator

So, the proposed estimator

So, the proposed estimator

Comparison with compromised method

So, the proposed estimator

So, the proposed estimator

Empirical study

To analyse the efficacy of the proposed estimators, we have selected a population data sets. The specifics of these data sets are delineated below.

Population A Source: District Census Handbook, Orissa (1981) The demographic information pertaining to 109 Villages/ Towns/Wards within the urban area under the jurisdiction of police station Baria, Tahasil-Champua, Orissa has been incorporated for examination. In this context, X denotes the mean number of non-workers within the village, while Y signifies the mean number of literate individuals residing in the village. Here, we have considered the sample sizes 𝑛 ′ = 50 fixed and subsample (n) between 10% to 45% and the response rate (r) between the 75% to 95% with different correlation coefficients.

The percentage relative efficiencies (PREs) of the proposed estimators and exiting estimators with respect to the mean method of imputation r y have been calculated and presented in Tables 1 to 4. The expressions of PREs are given as

We have computed MSE and the Percentage Relative Efficiencies (PRE) of different estimators under this population data set.

Simulation Analysis

To determine how well the suggested imputation techniques perform in comparison to mean and ratio approaches of imputation, we conduct simulation research in this part utilizing statistical computational software [11] . We used the 'genCorGen' function in the package 'simstudy' to generate data sets for this manuscript [12] . We created data sets from the normal distribution using the specified parameters and correlation coefficients for the study and auxiliary variables. To create a dataset of the variables (X, Y) and determine the necessary parameters and correlation coefficient, we take into account a population size of N = 8500 and we consider one simulation combination for 2   .

For data from Normal distribution: Data= (X , Y ) .

X: ( 3 2 1 .0 0 2 9 , 0 .9 9 8 6 ) ;: ( 4 5 7 .0 0 1 8 , 0 .9 9 2 5 )

Y

The following steps have been used for the simulation of the required population:

Step 1: The 'genCorGen' function is used to create a data set with the normal distribution of variables X and Y of size N = 8500 using the statistical computer program [12] Software.

Step 2: The parameters have been calculated.

Step 3: A sample of sizes ' 2 5 0 0 , n  n = 340, 765, 1020, 1445, 1700, and 1870 (i.e., sample sizes lie between 4% and 22%) was chosen from this artificial data set with response rates r = (265, 282, 296, 313), (597,635, 666, 704), (796, 847, 887, 939), (1127, 1199, 1257, 1329), (1326, 1411, 1479, 1564), and (1459,552, 1627, 1720), respectively (i.e., the response rate lies between 78% and 92% of the total).

Step 4: Sample statistics i.e. sample mean, sample variance, and the values of the proposed and existing estimators are calculated for this sample under imputation techniques.

Step 5: Steps (3) and ( 4) are repeated 50,000 times.

Step 6: The MSE of every estimator is calculated through the formula of the MSE given by     50000 2 1 1 50000

MSE of the existing and proposed estimators under design I

Conclusion

In this paper, we have proposed three imputation methods along with their point estimators to estimate the population mean of study variable. The bias, mean square error and minimum mean square error have been derived. Empirical and simulation studies have been carried out to examine the efficiency of the proposed estimators. The current study allows for an inference of the following conclusions. From Tables 1 to 4, for all cases where empirical studies have been conducted under two-phase sampling design. We generated a data set utilizing the statistical software R, derived from a normal distribution, and conducted a simulation to analyse the efficiencies of the proposed estimators. Our examination of tables (3-4) indicates that the efficiency outcomes of the proposed estimators exhibit almost a similar trend to that presented in tables (1-2). The MSE of proposed estimators for the proposed estimators in maximum cases. This demonstrates how the suggested imputation methods and its resulting estimators are better than the traditional imputation methods. This method improves the estimator's robustness and accuracy, which makes it a useful tool in contemporary data science and statistical analysis. Survey practitioners may be encouraged for their practical applications after observing promising behaviour of the suggested imputation methods, if non-response is not ignorable in the survey data.

References

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