<p>The estimation of population variance is essential in survey sampling. Variance estimation is important for decision-making in almost every field of science, medical, agriculture etc., because for decision-making the variation in characteristics under study is an important measure. For example, the dose of the medicine is decided based on variation in the body temperature. Variance estimation becomes more efficient when suitable auxiliary information is used. This study introduces a New Extended Exponentially Weighted Moving Average (NEEWMA) statistic with three smoothing parameters to construct improved estimators for the population variance. Using this framework, a new extended ratio, product, regression, and generalized class of memory-type estimators are proposed. The bias and mean squared error (MSE) expressions of the proposed new extended memory-type estimators are derived up to the first order of approximation. To evaluate the efficiency of estimators, an empirical study is performed using a real dataset of the daily Air Quality Index (AQI) of Ajmer city and two simulation studies are conducted under symmetric (normal) and asymmetric (gamma) distributions. Additionally, the performance of new extended memory-type estimators over traditional and memory-type estimators is compared. Simulation results under symmetric distribution demonstrate that the proposed new extended generalized class of memory-type estimator (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\vartheta _{m}^{nee}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ϑ</mi> <mrow> <mi>m</mi> </mrow> <mrow> <mi mathvariant="italic">nee</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>) achieves reduction in MSE by approximately <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$76.37\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>76.37</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$88.32\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>88.32</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> , and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$87.96\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>87.96</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> relative to the traditional ratio (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\vartheta _{r}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϑ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>), product (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\vartheta _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϑ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>), and regression (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\vartheta _{reg}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϑ</mi> <mrow> <mi mathvariant="italic">reg</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>), respectively, while under asymmetric distribution, the MSE of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\vartheta _{m}^{nee}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ϑ</mi> <mrow> <mi>m</mi> </mrow> <mrow> <mi mathvariant="italic">nee</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> reduces approximately by <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$77.01\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>77.01</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$89.30\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>89.30</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> , and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$88.16\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>88.16</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> relative to <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\vartheta _{r}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϑ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\vartheta _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϑ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> , and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\vartheta _{reg}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϑ</mi> <mrow> <mi mathvariant="italic">reg</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> respectively, by taking the average over all the values of correlation coefficient (<InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\rho$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>) and sample size (<i>n</i>).</p>

Variance estimation using New Extended EWMA: a simulation study under symmetric and non-symmetric distributions

期刊论文
发表日期：

2026年4月20日

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Prayas Sharma,
Mamta Kumari

摘要

The estimation of population variance is essential in survey sampling. Variance estimation is important for decision-making in almost every field of science, medical, agriculture etc., because for decision-making the variation in characteristics under study is an important measure. For example, the dose of the medicine is decided based on variation in the body temperature. Variance estimation becomes more efficient when suitable auxiliary information is used. This study introduces a New Extended Exponentially Weighted Moving Average (NEEWMA) statistic with three smoothing parameters to construct improved estimators for the population variance. Using this framework, a new extended ratio, product, regression, and generalized class of memory-type estimators are proposed. The bias and mean squared error (MSE) expressions of the proposed new extended memory-type estimators are derived up to the first order of approximation. To evaluate the efficiency of estimators, an empirical study is performed using a real dataset of the daily Air Quality Index (AQI) of Ajmer city and two simulation studies are conducted under symmetric (normal) and asymmetric (gamma) distributions. Additionally, the performance of new extended memory-type estimators over traditional and memory-type estimators is compared. Simulation results under symmetric distribution demonstrate that the proposed new extended generalized class of memory-type estimator ( $\vartheta _{m}^{nee}$ $ϑ_{m}^{nee}$ ) achieves reduction in MSE by approximately $76.37\%$ $76.37 %$ , $88.32\%$ $88.32 %$ , and $87.96\%$ $87.96 %$ relative to the traditional ratio ( $\vartheta _{r}$ $ϑ_{r}$ ), product ( $\vartheta _{p}$ $ϑ_{p}$ ), and regression ( $\vartheta _{reg}$ $ϑ_{reg}$ ), respectively, while under asymmetric distribution, the MSE of $\vartheta _{m}^{nee}$ $ϑ_{m}^{nee}$ reduces approximately by $77.01\%$ $77.01 %$ , $89.30\%$ $89.30 %$ , and $88.16\%$ $88.16 %$ relative to $\vartheta _{r}$ $ϑ_{r}$ , $\vartheta _{p}$ $ϑ_{p}$ , and $\vartheta _{reg}$ $ϑ_{reg}$ respectively, by taking the average over all the values of correlation coefficient ( $\rho$ $ρ$ ) and sample size (n).