Simulation based new method for population variance using auxiliary information

The problem of effective estimation of population variance takes up one of the leading positions in sampling, as it is one of the basic issues that quantify the dispersion, variability, and uncertainty that is present in the real-world population. Contrary to population mean, variance parameters are nonlinearly entered into inferential computations and are extremely liable to sampling variations, accurate estimation of which is not only difficult theoretically but also very important in practice. The traditional sample variance is an unbiased estimator of the population variance under the conditions of simple random sampling without replacement (SRSWOR); although its efficiency is in many cases poor, especially where there are small sample sizes or non-homogeneous populations involved. In order to overcome these shortcomings, there has been growing interest in the survey sampling theory on the integration of complementary information in order to optimize the performance of estimators. Strongly correlated auxiliary variables whose population parameters are known, offer a very effective method of reducing variability of sampling. With appropriate use of auxiliary information, it is possible to build variance estimators that dominate a standard unbiased estimator in a fairly broad sense, that is, mean squared error (MSE).

The efficiency of estimators can be greatly contributed with the use of auxiliary variables which in effect minimizes sampling variability. Specifically, in case where they are strongly correlation between the study and auxiliary variables, then their inclusion in the estimation can yield more believable and sound results. Variance of population is a central factor in the theory of sampling. It is applied in calculating the necessary sample size to reach a certain level of precision, constructions of confidence intervals, and carrying out the tests of hypotheses. Precise prediction of the variance is even more crucial when funding is limited such that it is impractical to take large samples. More reliable conclusions can be achieved in situations where they want to enhance the efficiencies of estimators but with no augmentation on the size of a sample. A preliminary comprehensive discourse on population variance estimate was presented in¹. To estimate population variance using auxiliary information², developed exponential estimators using ratio-product types estimators. Reference³ introduced several estimators aimed at enhancing the estimation of population variance. Several researchers have proposed several estimators for determining the variance of a population. Many authors including^{3,4,5,6,7,8,9,10,11,12,13,14}, have frequently used auxiliary information for estimation of population variance.

Research gap

Concise estimation of population variance is a key component of an effective survey sampling that grounds the quality of inference and sound decision making in the various fields of practice including agriculture science, economics, and social sciences. The uniform estimators such as the unbiased sample variance, which is another variant of estimators in simple random sampling and no replacement, are consistent and unbiased, but their efficiency is not always optimal especially with finite samples or heterogeneous populations. Even though the auxiliary information has been widely utilized in improving the estimation of mean of a population, its systematic use in estimating the amount of variation is relatively immature.

The currently available auxiliary-variable-based estimators of variance including ratio, product and regression-type variance estimators are largely based on linear approximations and asymptotic properties, thus constraining their usage in situations with moderate correlation, nonlinear relationship, or convoluted population structure. In addition, the behaviour of these estimators has rarely been verified by extensive simulation studies, necessary to reflect finite-sample behaviour as well as to evaluate strength in different circumstances.

Accordingly, a literature gap can be observed in the development of simulation-based approaches that have stringent methods of incorporating auxiliary information to formulate extremely efficient and flexible estimators of population variance. To fill this gap, the current study proposed a new estimator which incorporates both the theoretical derivation and the evaluation via simulation, which thus gives a sound framework of estimating the variance, which is both practically feasible and sound in theory. This contribution is an important development in the methods of survey sampling, especially when the traditional estimators are inefficient or when there is auxiliary information that is underutilized.

Novelty of the work

1. (1)

   As much as auxiliary information has been widely used in enhancing the estimation of population mean, the use of auxiliary information in population variance estimation is still minimal. The current research proposes a new estimator to estimate population variance, the first attempt to use auxiliary information in such a situation through simulation based approach.

2. (2)

   The proposed methodology is an improvement over classical approaches, which only use theoretical approximations to estimate performance of an estimator over a variety of sample sizes, correlation structures and population settings. This makes it possible to have a more flexible and robust evaluation of efficiency, especially in finite samples.

3. (3)

   The suggested estimator proves to be much more effective than all the considered existing estimators. The methodology also states viable recommendations on how best the auxiliary information can be used to estimate the variance.

4. (4)

   The suggested approach is universal and flexible, and can be a powerful aid in estimating variance in agriculture, economics, social sciences, and other areas where ancillary data exists, which fills a gap between theory and practice in a survey sampling methodology.

Overall, the originality is that the researchers designed a theoretically correct, practically sound, and universally applicable estimation of population variance based on a simulation and auxiliary information.

Objective of the study

This research aims to provide a new improved estimator for estimating population variance using auxiliary information under simple random sampling. In particular, the study will:

1. 1.

   Express the theoretical bias and MSE in the case of SRSWOR.

2. 2.

   In order to acquire the precise expression of an estimator based on known parameters of auxiliary variables.

3. 3.

   To study the bias and MSE of the suggested estimator up to the first order of first-order approximation.

4. 4.

   To check the performance of the suggested estimator compared to existing estimators, we use real data sets and a simulation study.

Methodology

Let a population consists of N units i.e. (:gamma) = (:left{{gamma}_{1},:::{gamma}_{2},::dots:,::{gamma}_{N}:right}). A small part is taken from (:gamma) with the help of simple random sampling without replacement. The study variable is represented by (Y), and the auxiliary is of (X). Consider a population variances for Y and X are denoted by (:{S}_{y}^{2}) and (:{S}_{x}^{2}). Let a sample variances for y and x are denoted by (:{s}_{y}^{2}) and (:{s}_{x}^{2}).

Let

To derive the properties of estimators, we used the following error terms:

Where

Here (:{Lambda}_{40}=:{beta:}_{2left(yright)}) and (:{Lambda}_{04}={beta:}_{2left(xright)}) are the population coefficient of kurtosis.

Here we study some well-known estimators for estimation of population variance, which are given by:

1. (i)

   The usual adopted variance estimator is specified by;

1. (ii)

   The ratio estimator developed by¹ is specified by:

1. (iii)

   The adopted regression estimator for variance is given by:

where b= (:left[frac{{s}_{y}^{2}{Lambda}_{22}^{*}:}{{s}_{x}^{2}{Lambda}_{04}^{*}}right]).

Solving (4), up to first order of approximation

.

Apply expectation on both sides (10), we have:

As we know that

.

where

.

1. (iv)

   The following estimator developed by² is given by:

1. (v)

   The following improved estimator is developed by³ is given by:

where

.

1. (vi)

   The following three adopted estimator for variance is given by⁴:

The properties of (:{:widehat{S}}_{KCingi1}^{2}), (:{:widehat{S}}_{KCingi2}^{2})and (:{:widehat{S}}_{KCingi3}^{2})are given by:

where

where

where

.

Proper estimation of population variance is a fundamental aspect of survey sampling, which directly reflects on the accuracy of statistical evaluation, efficiency of estimators and accuracy of decision-making in different sectors, such as agriculture, economics and social sciences. Traditional estimators of variance under simple random sampling, though unbiased, are usually not efficient especially in small samples or non-homogeneous populations. In real survey application situations, it is often the case that there is auxiliary information available, which can be of great benefit in understanding how variable is varied, although it has not been extensively used to estimate the variance compared to its common use in estimating means. In addition, the conventional auxiliary variable based estimators of variance, which include the ratio, product, and regression-type estimators, are mostly based on the asymptotic approximations and linearity, and thus they are not as flexible and robust in the real world sampling situations. It is obvious that there is a requirement of using methodologies that are capable of working with heterogeneous populations and moderate correlations and finite sample requirements and making full use of the auxiliary information. The simulation-based methods provide a strong solution, as they allow a thorough assessment of the estimator behaviour in a vast variety of controlled conditions and also offer a practical advice to apply in practice.

Inspired by these shortcomings, and take motivation from¹⁵, we develop an estimator for population variance by incorporating auxiliary information in order to enhance efficiency, minimize mean squared error, and offer a powerful and flexible instrument to the researcher. The work intends on filling the gap existing between theoretical developments in survey sampling and its practical application with regard to methodological rigor as well as practicality in real world situations of data-driven decision-making. The recommended estimator is given by:

Simplify (17), in terms of error, we obtain:

Now take expectations on both sides f (18), we have:

Now to obtain MSE of (:{:widehat{S}}_{Prop}^{2}), squaring (18) we have:

By taking expectancy on both sides:

Differentiate Eq. (21) w.r.t (:{U}_{1}), (:{U}_{2}):

Putting (:{U}_{1left(Optright)}) and (:{U}_{2left(Optright)}) in (2):

In this section we compared the mean squared error of the existing and suggested estimators, which are given in detail:

1. (i)

   By comparing (2) and (22).

or

(:{lambda:{S}_{y}^{4}Lambda}_{40}^{*}-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (ii)

   By comparing (4) and (22).

or

(Lambda:{S}_{y}^{4}) (:left({Lambda}_{40}^{*}+{Lambda}_{04}^{*}-2{Lambda}_{22}^{*}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (iii)

   By comparing (6) and (22).

or

(Lambda:{S}_{y}^{4}{Lambda}_{40}^{*}left(1-{rho:}^{*2}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (iv)

   By comparing (8) and (22).

or

.

(Lambda:{S}_{y}^{4}) (:left({Lambda}_{40}^{*}+frac{{Lambda}_{04}^{*}}{4}-{Lambda}_{22}^{*}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (v)

   By comparing (10) and (22).

or

.

(Lambda:{text{S}}_{text{y}}^{4}) (:left({{Lambda}}_{40}^{text{*}}+{{text{G}}_{0}^{2}{Lambda}}_{04}^{text{*}}-2{text{G}}_{0}{{Lambda}}_{22}^{text{*}}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (vi)

   By comparing (14) and (22).

or

(Lambda:{S}_{y}^{4}) (:left({Lambda}_{40}^{*}+{{G}_{1}^{2}Lambda}_{04}^{*}-2{G}_{1}{Lambda}_{22}^{*}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (vii)

   By comparing (15) and (22).

or

.

(Lambda:{S}_{y}^{4}) (:left({Lambda}_{40}^{*}+{{G}_{1}^{2}Lambda}_{04}^{*}-2{G}_{2}{Lambda}_{22}^{*}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

1. (viii)

   By comparing (16) and (22).

or

(Lambda:{S}_{y}^{4}) (:left({Lambda}_{40}^{*}+{{G}_{1}^{2}Lambda}_{04}^{*}-2{G}_{2}{Lambda}_{22}^{*}right)-) (:frac{lambda:{S}_{y}^{4}:left{{Lambda}_{04}^{*}-8left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right)left(-2+{Lambda}_{04}^{*}right){Lambda}_{40}^{*}right}::}{16left[-1+lambda:left(-1+{rho:}_{{S}_{y}^{2}{S}_{x}^{2}:}right){Lambda}_{40}^{*}:right]}ge:)0, which is always true.

Here, we used some real data sets, to assess efficiency of estimators. The summary statistic is given by:

Data-I: [Source:⁷]

Y=Food cost of family employs.

X=Weekly income of family.

Data-II: [Source:⁶]

Y= Number of teachers in institution.

X= Number of students.

Data-III: [Source:⁵]

Y = In the non-choice packet test, adults of the drugstone beetle species Stegobium paniceum.

X= Peppermint packets that were either microwaved for one, two, or 3 min.

To evaluate the performance of all considered estimators by generating synthetic data from a bivariate normal distribution and evaluating the estimators MSEs and PREs. Populations of size 1000, have been generated using Bivariate normal distribution. We select sample of size n = 200, from each population.

Population-I:

and

Population-II:

and

Population-III:

and

In Table 1, we include summary statistics for all real data sets. Table 2, include the mean squared error of all the considered estimators using real data sets. We visualize the numerical result of real data sets with the help of line and distribution graph, and shown in Figs. 1 and 2. The numerical result of MSEs and PREs of all considered estimators using simulation study are given in Tables 4 and 5. The result from the simulation study are illustrated via graphs; Figs. 3 and 4, displays the mean squared error of all estimators, while Figs. 5 and 6, presents the PREs derived from the simulation analysis.

The mean squared error (MSE) of the existing and the proposed estimators of population variance are given in Table 2, with respect to three population. As it can be seen in the table, the suggested estimator demonstrates the minimum MSE in all three cases, which means that it is more efficient than the standard variance estimator and other estimators based on auxiliary information that can be considered in this article. In Population-I, the suggested estimator represents significant decrease in MSE compared to the ratio, regression, and difference-cum-exponential estimators, which show its greater accuracy. In Population-II, the suggested estimator has the lowest MSE in spite of the large magnitude of variability showing its numerical stability. In Population-III, again the proposed estimator is superior to all the competing estimators, and this proves that it is robust to a wide range of population structures. These findings are clear indications that the estimator proposed is effective within the utilization of auxiliary information and are more efficient in variance estimation as compared to the current methods.

Table 3, shows the percentage relative efficiency (PREs) of the existing and proposed estimators of population variance, where the standard estimation of variance is used. In each of the three population scenarios (I-III), the proposed estimator has the largest PRE, meaning it is much more efficient than the traditional estimator of variance and other estimators based on other forms of auxiliary information. In Population-I, the suggested estimator has a PRE of 243.82, which is nearly 2 times the efficiency of the standard estimator of variance, whereas the existing estimators have a PRE of about 126–143. On the same note under Population -II, the proposed estimator has the greatest PRE of 1319.03, which is better than all other estimators such as regression- and ratio-type estimators. It shows the highest PRE (148.29%), which proves its strength in the context of other population structures, in Population-III. These findings illustratively show that the proposed estimator is not only effective in improving efficiency of estimation but also is good at making use of auxiliary information to bring about better results in different conditions.

Table 4, gives the mean squared errors (MSEs) of the existing and proposed estimators using the simulation study in three population situations. It is clear that the suggested estimator provide lowest MSE as compared to all existing estimators, which proves its better performance in practice. In Population-I, the proposed estimator mean squared error is 0.04177, which is a smaller than the regression estimator (0.04207) and is considerably small as compared to the classical variance estimator (0.22918) and other auxiliary-based estimators. The proposed estimator in Population-II, also achieves the lowest MSE (0.71512), which is better than the ratio, regression, and difference-type estimators, which proves its robustness even in situations with higher variability. In Population-III, the suggested estimator is also the most efficient, and the MSE is 0.10180, which is slightly less than the rival methods. On the whole, the findings of the simulation indicate the strong support of the theoretical findings as they show that the proposed estimator makes efficient use of auxiliary information to provide a more accurate variance estimation and has, on average, lower mean squared error values when compared to the existing estimators.

In Table 5, we provide the percentage relative efficiency (PREs) of the existing and proposed estimators using simulation study. In all three population (I-III), the proposed estimator is the one that maximizes the PRE and is therefore highly efficient. In Population-I, the suggested estimator achieves a PRE of 548.62 which is significantly more than the ratio, regression, and other estimators that use auxiliary information which implies an impressive increase in accuracy. In Population -II, the efficiencies of certain existing estimators fluctuate, but the given estimator has the highest PRE (128.24%), which proves its strength even in the situations where the population characteristics are different. In Population- III, the proposed estimator once again has the greatest PRE (141.65%) and it is slightly better that the regression and the US estimators. In general, the simulation study validates the claim that the suggested estimator makes effective use of auxiliary information to offer a more efficient estimation of variance on a regular basis as compared to the current approaches.

The suggested estimators yield the minimum mean squared error and the higher percentage relative efficiency across all data sets, compared to existing estimators. The conclusion to be drawn is that the suggested estimator gives an effective, efficient, and theoretically viable solution of estimating population variance better when using simple random sampling. It connects the gap between sampling methodology and real world data challenges and can allow more accurate and more cost effective decision-making in many domains of surveys.

In this article, we suggested improved estimator for population variance using auxiliary information under simple random sampling. The numerical findings based on real data sets and a simulation study highlight great utility of the inclusion of auxiliary information in variance estimation especially in instances where traditional approaches yield lower accuracy, The numerical findings revealed that the suggested estimator provided significant extensions to estimation accuracy and decreasing sampling error in various situations and population settings. Future research may investigate how the proposed estimator may be generalized to stratified, systematic or cluster sampling designs, where the estimation of variance is necessarily more demanding. The methodology can be generalized to other multivariate auxiliary information, which can also be more efficient in high-dimensional survey applications. Model-assisted approaches could be considered as a more flexible and robust variance estimator by incorporating nonlinear relationship between the study and auxiliary variables or both. Using the suggested methodology on huge survey data in economics, social sciences, and environmental studies will aid in assessing its practical usefulness as well as give some guidance to practitioners. Future studies can be aimed at optimization of the simulation parameters, including sample size and the number of iterations to balance between effectiveness and accuracy of the estimators. Overall, the research offers a methodologically sound and practically strong framework of estimating the variance, and it has several avenues to be followed by future research in order to increase its applicability and efficiency.