Abstract
In this article, we estimate the finite population variance in random nonresponse using simple random sampling, which may be helpful for data analysis in applied and environmental sciences. For the three distinct random nonresponse techniques by Singh and Joarder [25], we have proposed a generalized class of exponential-type estimators that uses an auxiliary variable. Up to the first order of approximation, expressions of the bias and mean square error of the proposed estimators are obtained. The suggested estimators illustrate their superior performances to the current estimators in the comparable strategies in a comparative analysis using the real and simulated datasets.
1. Introduction
In the survey sampling theory, an estimator of the population parameter(s) can be more precise by effectively using auxiliary information. Examples of this context include the ratio estimator, product estimator, and regression estimation method. When there is a strong correlation between the study variable and the auxiliary variable, ratio estimators are frequently employed to estimate the population parameter. The first estimation of the finite population variance with the known population variance and coefficient of variation of the auxiliary variable was carried out by Das and Tripathi [1]. Kadilar and Cingi [2] proposed some ratio-type variance estimators using ratio estimators in a simple and stratified random sampling. Singh and Malik [3] introduced a new family of estimators using auxiliary attributes under population variance in simple random sampling. Panda and Sahoo [4] suggested a family of exponential estimators for estimating the finite population variance using auxiliary information in simple random sampling. Haq et al. [5] developed estimators of the finite population variance using the information on the study variables under stratified random sampling. Many other researchers have also significantly contributed to estimating the finite population variance. Sunday et al. [6] adopted a variable step hybrid block method for the approximation of Kepler Problem by integrating the Lagrange polynomial with limits of integration selected at special points. Juraev et al. [7, 8] used regularization formula and matrix factorization for explicit form of the approximate solutions of the Cauchy Problem.
The issue of nonresponse in human-related surveys affects practically all questionnaire designs. Nonresponse occurs when some survey participants choose only to complete part of the questionnaire or when the interviewers fail to approach survey nonrespondents. In sample surveys, nonresponse is a possible source of errors. The estimation of population parameters exhibits significant variance and nonresponse bias due to the missing data. It is only possible to obtain data from some units in the chosen sample due to nonresponse. Nonresponse diminishes the specified sample on one hand and the estimator’s effectiveness on the other. The existence of nonresponse can occasionally be random. Hansen and Hurwitz [9] were the first to study the nonresponse problem in a postal survey. In addition, Argyros [10], Akbarov et al. [11], and Rubin [12] suggested three different conditions of missingness of response in the sensitive survey: observed at random (OAR), missing at random (MAR), and parameter distinctness (PD). Furthermore, missing at random (MAR) and missing completely at random (MCAR) is prominent, according to Heitjan and Basu [13].
Some researchers such as Singh and Joarder [14] and Ahmed et al. [15] suggested estimating finite population variance under random nonresponse using auxiliary information. Singh et al. [16], Pankov et al. [17], Musaev [18], Noor and Noor [19], and Singh and Khalid [20, 21] proposed a strategy to estimate the population mean and variance in the presence of random nonresponse using two-phase successive sampling. Bhushan and Pratap Pandey [22] presented some ratio and product-type estimators of finite population variance in the presence of random nonresponse using auxiliary information. Khalid and Singh [23] suggested some imputation methods for missing data problems due to random nonresponse in two-occasion successive sampling. Many others have dealt with the issue of random nonresponse using auxiliary information to estimate the finite population variance. Motivated by the abovementioned work and looking at the importance of handling the problems of random nonresponse in survey sampling, we suggest a generalized class of exponential-type estimators of finite population variance in the presence of random nonresponse under three different strategies.
The rest of the article is presented as follows: Section 2 discusses the methodology, and some existing estimators are briefly reviewed in Section 3. Section 4 introduces our proposed generalized class of estimators along with the expressions of biases and mean square errors . Section 5 elaborates the efficiency comparison of existing and proposed estimators. In Sections 6 and 7, the results of our empirical study based on real and simulated data are presented, and finally, Section 8 concludes our study.
2. Methodology and Notations
Let denote a population of units from which a simple random sample of size is drawn without replacement. If denotes the number of sampling units on which information could not be obtained due to a random nonresponse, then the remaining units can be treated as a simple random sample from . We are interested in estimating the finite population variance under random nonresponse and assume . Singh and Joarder [14] assumed the discrete distribution aswhere is the probability of nonresponse, , and represents the total number of ways to obtain nonresponses out of a possible
We consider a finite population of distinct objects and a sample of size is drawn by simple random sampling without replacement (SRSWOR) from a population of distinct objects. Let and be the observations of the study variable and auxiliary variable , respectively, which are correlated with a proper amount of correlation in .
Let and be the population means of the study and auxiliary variables along with their sample means and , respectively. Let and be the response sample variances and are conditionally unbiased estimators of the population variances of and , respectively. Let be the population covariance between and and be the moment ratio. Furthermore, we define and
The maximum-likelihood estimators of and defined by Singh and Joarder [14] are specified as
If , then and if , then thus is an admissible estimator of response probability .
We investigate the impact of random nonresponse of the study and auxiliary variables of the generalized class of variance estimators under the following three strategies as discussed by Singh and Joarder [14]: Strategy I: when and corresponding estimate is used Strategy II: when and corresponding estimate is used Strategy III: when and corresponding estimate is used
To obtain the expressions of biases and mean squared error of the existing and proposed estimators, we consider the following relative error terms: let , and , such that . Also, , , , , and .
3. Existing Estimators
Singh and Joarder [14] presented the following three usual estimator strategies for estimating the finite population variance: Strategy I Strategy II Strategy III
Theorem 1. The mean squared error of is given by
Theorem 2. The mean squared error of is given by
Theorem 3. The mean squared error of is given by
Ahmed et al. [15] suggested the following three strategies for estimating the population variance. Each strategy is based on three different types of estimators. Strategy I Strategy II Strategy III where , , and are suitably chosen constants.
The optimum values of the appropriate constants , , and are and
Theorem 4. Following are the expressions of the minimum mean square error of and , respectively:
Bhushan and Pratap Pandey [22] proposed three different strategies from the outlined two methodologies mentioned above. Each strategy is based on three different types of estimators. Strategy I Strategy II Strategy III
Theorem 5. The following are the expressions for minimum mean square error of the estimators and , respectively:where , and
4. The Proposed Generalized Class of Variance Estimators
Motivated by Yadav et al. [24] and Muneer et al. [25], we propose a generalized class of exponential-type estimators of the finite population variance using auxiliary variable in the presence of random nonresponse. These estimators are used in three different random nonresponse strategies and are defined as follows: Strategy I Strategy II Strategy III where and are known population parameters of an auxiliary variable and also and are the arbitrary constants and .
4.1. Properties of the Proposed Generalized Class of Variance Estimators
The properties of the proposed estimators are given in the following theorems:
Theorem 6. The biases of the estimators are given by
Theorem 7. The mean square error of the estimators is given by
The optimum values for the appropriate constants of and are and The details for strategy I are given in the Appendix.
Theorem 8. The minimum of the estimators is given bywhere for ,
The special cases for the strategies I–III are shown in Table 1.
5. Efficiency Comparison
In this section, the existing and proposed estimators are compared theoretically in terms of minimum mean square errors.(i)From (5) and (25), we have(ii)From (7) and (25), we have(iii)From (8) and (25), we get(iv)From (12) and (25), we derive(v)From (16) and (25), we get(vi)From (17) and (25), we have(vii)From (18) and (25), we get
6. Empirical Study
In this section, we have chosen two populations to exhibit the performances of the estimators, given as follows: Population-I (Cochran [26]; p: 182): let be the number of paralytic polio cases in the placebo group and be the number of placebo children Population-II (Maddala [27]; p: 108): the data present experience and salary structure of the University of Michigan economists from 1983 to 1984. Let be the salary (thousands of dollars) and be the years of experience (defined as years since receiving Ph.D.)
The descriptive statistics for populations I and II are shown in Table 2.
The percentage relative efficiency is given by
Tables 3 and 4 show the percentage relative efficiency of the existing and proposed generalized class of estimators under random nonresponse using population-I and -II, respectively, for the two choices of
Interpretation of the results:
From Tables 3 and 1, we report that(i)The proposed generalized class of estimators possesses more efficiency by using real datasets(ii)The estimators under strategy I are always more efficient than strategy II and strategy III(iii)In Tables 3 and 4, the proposed generalized class of estimators provides notable gain over existing estimators for estimating the finite population variance in the presence of random nonresponse while considering the auxiliary information
7. Simulation Study
A simulation study is conducted using R software to obtain the efficiency of the proposed estimators under simple random sampling. The following steps are used to perform the simulation study:(i)A population size of is generated using a bivariate normal distribution at different values of covariance matrices by assuming a positive correlation between and variables(ii)We take a sample size of and the number of nonrespondents for the population-III and population-IV at (iii)Using the random nonresponse technique, the population is repeated 1000 times to obtain the percentage relative efficiency of the proposed and existing estimators using the samples obtained
The descriptive statistics of population-III and population-IV are shown in Table 5.
Tables 6–9 show the values of percentage relative efficiency of the existing and proposed generalized class of estimators under random nonresponse for population-III at and Moreover, Tables 10–13 show the percentage relative efficiency of the existing and proposed generalized class of estimators under random nonresponse for population-IV at and
The bold values in the tables represent the maximum percentage relative efficiency among the proposed and existing estimators.
Interpretation of the results:
From Tables 6–13, we report that(i)The generalized class of exponential-type estimators perform better than the usual estimators in the respective strategies(ii)The proposed optimal estimators achieve a higher gain in terms of percentage relative efficiencies over the existing optimal estimators(iii)Under strategy I, the estimators provide more efficient results as compared to strategy II and strategy III
8. Conclusion
In our study, we have proposed a generalized class of exponential-type estimators for estimating a finite population variance under random nonresponse using an auxiliary variable. The properties of the proposed classes of estimators have been derived up to the first order of approximation. Based on empirical and simulation studies, the proposed generalized class of estimators is more precise than their existing counterparts. Our study reveals the decrease in mean square error with increased sample size. Thus, the proposed generalized class of estimators is recommended for survey practitioners as it might expand the odds of acquiring progressively efficient results of population variance under random nonresponse conditions. This study can also be extended to incorporate the dual use of auxiliary variables to reduce the survey nonrespondents drastically by using fuzzy statistics.
Appendix
A. Outline of the Derivation of Proposed Strategy I in Section 4
It is possible to write the proposed estimator under the error transformation as follows:
Equation (A.1) error term is binomially expanded up to order two and can be written as follows:
Now, applying the expectation on both sides of equation (A.2) we get the expression of bias of
Squaring both sides of equation (A.2) by considering the error term up to order two, we get
Now, taking expectation on both sides of the equation (A.3), we get the of the estimator as
The optimum value of and can be obtained by differentiating partially equation (A.4) w.r.t. and and then equating it to zero to obtain the minimum of
Data Availability
All relevant data and its supporting information files are included within the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.