Innovative memory-type calibration estimators for better survey accuracy in stratified sampling

Abstract

Calibration methods play a vital role in improving the accuracy of parameter estimates by effectively integrating information from various data sources. In the context of population parameter estimation, memory-type statistics—such as the exponentially weighted moving average (EWMA), extended exponentially weighted moving average (EEWMA), and hybrid exponentially weighted moving average (HEWMA)—leverage both current and historical data. This study proposes new ratio and product estimators within a calibration framework that utilizes these memory-type statistics. A simulation study is conducted to evaluate the performance of the proposed estimators. The mean squared error (MSE) and relative efficiency (RE) are computed, accompanied by graphical representations to illustrate the behavior of the estimators. The performance of the proposed estimators is compared with existing memory-type estimators. Furthermore, a real-world application is presented to validate the effectiveness of the proposed methods. The comparison, based on MSE and RE across different values of the smoothing constant, demonstrates that the calibration-based memory-type estimators outperform the existing approaches. The results indicate that the proposed estimators consistently achieve lower MSE and higher RE, confirming their superiority over traditional methods.

Introduction

Parameter estimation is crucial for heterogeneous populations in survey sampling. Past information is considered highly significant and beneficial in improving the estimation process. When the past (auxiliary) information is positively and linearly correlated with the study variable, the ratio estimator is generally preferred for estimating population parameters. Conversely, if the relationship between the auxiliary variable and the study variable is negatively linear, the product estimator becomes more appropriate. The simple ratio estimator, as proposed by Cochran¹, is given by:

$$\:{\stackrel{-}{Y}}_{r}=\frac{\stackrel{-}{y}}{\stackrel{-}{x}}\stackrel{-}{X}.$$

(1)

The approximate mean square error (MSE) expression for ratio estimator is presented as

$$\:MSE{(\stackrel{-}{Y}}_{r})\approx\:\theta\:\left[{{C}_{x}}^{2}+{{C}_{y}}^{2}-2\rho\:{C}_{y}{C}_{x}\right].$$

(2)

The simple product estimator, proposed by Robson², is given by:

$$\:{\stackrel{-}{Y}}_{p}=\frac{\stackrel{-}{y}}{\stackrel{-}{X}}\stackrel{-}{x}.$$

(3)

The MSE for product estimator has the form

$$\:MSE{(\stackrel{-}{Y}}_{p})\approx\:\theta\:\left[{{C}_{x}}^{2}+{{C}_{y}}^{2}+2\rho\:{C}_{y}{C}_{x}\right].$$

(4)

By incorporating both past and current information, the exponentially weighted moving average (EWMA) statistic was first introduced by Roberts³ to enhance the efficiency of estimators. The EWMA statistic is defined as follows:

$$\:{Z}_{t}=\lambda\:\stackrel{-}{{y}_{t}}+\left(1-\lambda\:\right){Z}_{t-1},$$

(5)

where $\:t>0$. As the weight parameter $\:\lambda\:$ varies from 0 to 1, greater emphasis is placed on the current study information, while the influence of past (auxiliary) data correspondingly decreases. The EWMA statistic, as defined in Eq. (5), reduces to the simple sample average when the smoothing constant $\:\lambda\:=1$. In this context, $\:{Z}_{t-1}$​ in Eq. (5) represents the previous value of the EWMA statistic. The initial value $\:{Z}_{0}$​ is typically set as the expected average. The expressions for the mean and variance of the EWMA statistic are given as follows:

$$\:E\left({Z}_{t}\right)=\stackrel{-}{Y}$$

(6)

and

$$\:Var\left({Z}_{t}\right)=\frac{{{S}^{2}}_{y}}{n}\left[\frac{\lambda\:}{2-\lambda\:}\left\{1-{\left(1-\lambda\:\right)}^{2t}\right\}\right].$$

(7)

As time t approaches to infinity (t → ∞), the variance of $\:{Z}_{t}$ reduces to

$$\:Var\left({Z}_{t}\right)=\frac{{{S}^{2}}_{y}}{n}\left[\frac{\lambda\:}{2-\lambda\:}\right].$$

The EWMA statistic in Eq. (5) was used by Noor-ul-Amin⁴ to advise the memory type estimators. The EWMA statistic for study variable $\:y$ in stratified sampling is denoted by $\:{Z}_{st}$ as follows:

$$\:{Z}_{st}=\lambda\:{\stackrel{-}{y}}_{st}+\left(1-\lambda\:\right){Z}_{s\left(t-1\right)}.$$

(8)

The EWMA statistic for the auxiliary variable $\:x$, as proposed by Aslam et al.⁵, is denoted by $\:{Q}_{st}$ and is given by:

$$\:{Q}_{st}=\lambda\:{\stackrel{-}{x}}_{st}+\left(1-\lambda\:\right){Q}_{s\left(t-1\right)}.$$

(9)

The projected memory type EWMA ratio and product estimators under stratified sampling can be written as:

$$\:{{\stackrel{-}{y}}^{M}}_{rst}=\frac{{Z}_{st}}{{Q}_{st}}\stackrel{-}{X}$$

(10)

and

$$\:{{\stackrel{-}{y}}^{M}}_{pst}=\frac{{Z}_{st}}{\stackrel{-}{X}}{Q}_{st}.$$

(11)

The estimated MSE expressions for the stratified memory-type ratio and product estimators are given, respectively, as follows:

$$\:MSE\left({{\stackrel{-}{y}}^{M}}_{rst}\right)\approx\:\frac{\lambda\:}{\lambda\:-2}\sum\:_{h=1}^{L}{W}_{h}^{2}{\theta\:}_{h}\left[{S}_{yh}^{2}+{R}^{2}{S}_{xh}^{2}-2R{S}_{yxh}\right]$$

(12)

and

$$\:MSE\left({{\stackrel{-}{y}}^{M}}_{pst}\right)\approx\:\frac{\lambda\:}{\lambda\:-2}\sum\:_{h=1}^{L}{W}_{h}^{2}{\theta\:}_{h}\left[{S}_{yh}^{2}+{R}^{2}{S}_{xh}^{2}+2R{S}_{yxh}\right].$$

(13)

The extended exponentially weighted moving average (EEWMA) statistic for the study variable $\:y$, as proposed by Naveed et al.⁶, is denoted by $\:{Z}_{te}$​ and is defined as follows:

$$\:{Z}_{te}={\lambda\:}_{1}\stackrel{-}{{y}_{t}}{-\lambda\:}_{2}{\stackrel{-}{y}}_{t-1}+\left(1-{\lambda\:}_{1}+{\lambda\:}_{2}\right){Z}_{te-1};\:0<{\lambda\:}_{1}\le\:1\:\&\:0\le\:{\lambda\:}_{2}\le\:{\lambda\:}_{1}.$$

(14)

The EEWMA statistic for the auxiliary variable $\:x,$ symbolized by $\:{Q}_{te}$, takes the form

$$\:{Q}_{te}={\lambda\:}_{1}{\stackrel{-}{x}}_{te}{-\lambda\:}_{2}{\stackrel{-}{x}}_{t-1}+\left(1-{\lambda\:}_{1}+{\lambda\:}_{2}\right){Q}_{te-1}.$$

.

The variance for EEWMA statistic is presented as

$$\:var\left({Z}_{te}\right)={\sigma\:}_{y}^{2}\phi\:.$$

(15)

The memory-type ratio and product estimators based on the EEWMA statistic, as developed by Zahid et al.⁷, are expressed as follows:

$$\:{\widehat{t}}_{ermit}=\frac{{Z}_{te}}{{Q}_{te}}{\mu\:}_{x}$$

(16)

and

$$\:{\widehat{t}}_{epmit}=\frac{{Z}_{te}}{{\mu\:}_{x}}{Q}_{te}.$$

(17)

The MSEs for both memory type ratio and product estimators can be presented as

$$\:MSE\left({\widehat{t}}_{ermit}\right)\approx\:\theta\:\phi\:\left[{C}_{y}^{2}+{C}_{x}^{2}-2\rho\:{C}_{y}{C}_{y}\right]$$

(18)

and

$$\:MSE\left({\widehat{t}}_{epmit}\right)\approx\:\theta\:\phi\:\left[{C}_{y}^{2}+{C}_{x}^{2}+2\rho\:{C}_{y}{C}_{y}\right].$$

(19)

Further, the hybrid exponentially weighted moving average (HEWMA) statistic by Haq⁸ is defined as:

Furthermore, the hybrid exponentially weighted moving average (HEWMA) statistic, introduced by Haq⁸, is defined as follows:

$$\:{E}_{t}={\lambda\:}_{2}{\stackrel{-}{X}}_{t}+\left(1-{\lambda\:}_{2}\right){E}_{t-1}$$

and

$$\:H{E}_{t}=\left(1-{\lambda\:}_{1}\right){HE}_{t-1}+{\lambda\:}_{1}{E}_{t}.$$

The MSE expressions for the memory-type ratio and product estimators, as presented by Noor-ul-Amin²¹, are given as follows:

$$\:\left.\begin{array}{c}MSE\left({\widehat{\stackrel{-}{Y}}}_{rmt}\right)\approx\:\theta\:{\mu\:}_{y}^{2}\frac{{\left({\lambda\:}_{2}{\lambda\:}_{1}\right)}^{2}}{{\left({\lambda\:}_{2}-{\lambda\:}_{1}\right)}^{2}}\gamma\:\left[{C}_{y}^{2}+{C}_{x}^{2}-2\rho\:{C}_{y}{C}_{y}\right]\\\:MSE\left({\widehat{\stackrel{-}{Y}}}_{pmt}\right)\approx\:\theta\:{\mu\:}_{y}^{2}\frac{{\left({\lambda\:}_{2}{\lambda\:}_{1}\right)}^{2}}{{({\lambda\:}_{2}-{\lambda\:}_{1})}^{2}}\gamma\:\left[{C}_{y}^{2}+{C}_{x}^{2}+2\rho\:{C}_{y}{C}_{y}\right]\end{array}\right\}.$$

(20)

Survey sampling has established that the linear regression estimator performs efficiently when the regression line of the study variable ($\:Y$) passes through the origin and the variance of the study variable is proportional to the auxiliary variable ($\:X$). Under these conditions, it often outperforms both ratio and product estimators⁹.

Bhushan et al.^10,11 proposed effective combined and separate-type estimators for population mean estimation under a stratified sampling framework, supported by real data applications. In a related development, Song and Kawai¹² introduced an adaptive approach within stratified sampling for estimating failure probabilities. Additionally, Pandey et al.¹³ developed a calibration estimator for population variance under two-phase stratified sampling by incorporating non-sampling errors through calibrated weights.

The use of auxiliary information has been widely recognized in the literature for improving the efficiency of point estimators. Several researchers have contributed to this area. For instance, Koyuncu and Kadilar¹⁵ extended a general class of estimators to the stratified random sampling technique. Yasmeen et al.¹⁶ proposed an exponential estimator for estimating the population mean using a transmuted ancillary variable, while Malik and Singh¹⁷ introduced exponential-type estimators utilizing two auxiliary variables in a stratified context.

Kumar et al.¹⁸ studied the estimation of the population mean in the presence of non-response and measurement error, using two auxiliary variables. Raza et al.¹⁹ proposed a ratio-type regression estimator that is robust to outliers through the use of re-descending M-estimators. Saini and Kumar²⁰ applied stratified and ranked set sampling techniques to improve population mean estimation under stratification.

Noor-ul-Amin²¹ was among the first to propose memory-type ratio and product estimators based on the exponentially weighted moving average (EWMA) and hybrid EWMA (HEWMA) statistics. Kumar et al.²² later extended this approach by introducing generalized ratio and product estimators using the HEWMA statistic under simple random sampling (SRS) without replacement. Bhushan et al.¹¹ addressed the estimation of the population mean using EWMA statistics under SRS and proposed memory-type logarithmic estimators, while Bhushan et al.²³ further evaluated the performance of these log memory-type estimators under the same sampling scheme.

However, in certain situations, ratio and product-type estimators may not outperform the traditional linear regression estimator. As a result, these estimators are often limited by the requirement that their efficiency should not be lower than that of the regression estimator. In much of the previous literature, it has been observed that the performance of such estimators depends on specific conditions that must be satisfied to achieve improved estimation.

To address these limitations, a widely adopted approach known as calibration has been introduced. Calibration techniques are commonly used in survey sampling to enhance the estimation of population parameters. Deville and Särndal²⁴ were the first to propose the calibration estimator, which has since been extended by numerous researchers. The calibration method relies on minimizing a distance function to reduce the discrepancy between the original sampling weights and the new calibrated weights, while satisfying a set of calibration constraints.

Furthermore, Wu and Sitter²⁵ generalized the calibration estimation framework by developing a pseudo-empirical likelihood method that incorporates model-based calibration. This approach provides efficient estimators for quadratic and other second-order finite population functions.

Furthermore, Hidiroglou and Särndal²⁶ applied the calibration procedure within a two-phase sampling framework using a two-step approach, and discussed its application for estimating variances and domain-specific parameters. To estimate the parameters of linear equations, particularly in cases where the equations have no exact solution, Berge²⁷ employed the calibration technique as an alternative method. Additionally, Harms and Duchesne²⁸ constructed an estimator based on population quantiles in the presence of auxiliary data and compared its performance with other quantile-based calibration estimators.

Calibration techniques are frequently used to incorporate historical data, enhancing the accuracy of population parameter estimation, as noted by Kim et al.²⁹. Koyuncu and Kadilar³⁰ also proposed a novel estimator within the calibration framework for estimating the population mean. Sud et al.³¹ introduced an estimator for the population total under the assumption that the auxiliary and study variables are negatively correlated. More recently, Jabeen et al.³² developed a new strategy under the randomized response technique, integrating a calibration scheme that minimizes the distance between the original and calibrated weights.

Further contributions to calibration methodology under stratified sampling were made by Singh et al.³³, followed by several other researchers, including^{34,35,36,37,38}. The latter proposed two calibrated estimators within the stratified sampling framework that incorporate both the expected value and the coefficient of variation information for each stratum. The main objective of defining calibrated weights is to minimize the chi-squared distance measure, subject to newly defined calibration constraints. In stratified random sampling, the calibration approach is commonly used to determine optimal weights across strata. Additionally, Clement et al.³⁹ proposed estimators for domain totals using the calibration technique under a stratified sampling design.

The primary objective of this study is to develop new memory-type calibrated estimators for estimating the population mean, extending the methodologies introduced by Noor-ul-Amin^4,5,21. Specifically, the study proposes memory-type ratio and product estimators within the framework of calibration estimation, employing a chi-square distance function and subject to calibration constraints. To assess the efficiency of the proposed estimators, a comprehensive simulation study is carried out, comparing their performance with existing estimators using MSE and RE, supported by graphical illustrations. Additionally, a real-life dataset is analyzed to demonstrate the practical applicability and effectiveness of the proposed estimators.

Simple ratio estimator in stratified sampling

This section provides an overview of existing ratio estimators in stratified random sampling. The sample means of the study and auxiliary variables in stratified random sampling, as defined by Kadilar and Cingi⁴⁰, are given by:

$$\:{\stackrel{-}{y}}_{st}=\sum\:_{h=1}^{k}{W}_{h}{\stackrel{-}{y}}_{h}$$

and

$$\:{\stackrel{-}{x}}_{st}=\sum\:_{h=1}^{k}{W}_{h}{\stackrel{-}{x}}_{h}.$$

where k denotes the number of strata. The ratio estimator in stratification is given by

$$\:{\stackrel{-}{y}}_{sre}=\frac{{\stackrel{-}{y}}_{st}}{{\stackrel{-}{x}}_{st}}\stackrel{-}{X}.$$

(21)

Considering that each stratum may have a different population mean, the ratio estimator in stratified random sampling can be expressed as:

$$\:{\stackrel{-}{y}}_{srs}=\frac{\sum\:_{h=1}^{k}{W}_{h}{\stackrel{-}{y}}_{h}}{\sum\:_{h=1}^{k}{W}_{h}{\stackrel{-}{x}}_{h}}\sum\:_{h=1}^{k}{W}_{h}{\stackrel{-}{X}}_{h}.$$

(22)

The mean of the estimator presented in Eq. (22) is given by

$$\:E\left({\stackrel{-}{y}}_{srs}\right)=\stackrel{-}{Y}.$$

The expression for variance can be written as

$$\:var\left({\stackrel{-}{y}}_{srs}\right)=\frac{\sum\:_{h=1}^{k}{{W}_{h}}^{2}\:\left(\frac{{N}_{h}-{n}_{h}}{{N}_{h}{n}_{h}}{{.\:\:s}^{2}}_{hy}\right)}{\sum\:_{h=1}^{k}{{W}_{h}}^{2}\:\left(\frac{{N}_{h}-{n}_{h}}{{N}_{h}{n}_{h}}.\:\:{{s}^{2}}_{hx}\right)}.{\left(\sum\:_{h=1}^{k}{W}_{h}{\stackrel{-}{X}}_{h}\right)}^{2}.$$

(23)

Stratified memory type estimators

The memory type stratified ratio and product estimators were suggested by Aslam et al.⁵ to estimate the population mean. The simple memory type ratio and product estimators by Noor-ul-Amin⁴ are defined as

$$\:\left.\begin{array}{c}{{\stackrel{-}{y}}^{M}}_{rt}=\frac{{Z}_{t}}{{Q}_{t}}{\mu\:}_{x}\\\:{{\stackrel{-}{y}}^{M}}_{pt}=\frac{{Z}_{t}}{{\mu\:}_{x}}{Q}_{t}\end{array}\right\}.$$

(24)

The stratified memory type ratio estimator can be written as

$$\:{y}^{*}=\frac{\sum\:_{h=1}^{k}{W}_{h}{Z}_{th}}{\sum\:_{h=1}^{k}{W}_{h}{Q}_{th}}.\sum\:_{h=1}^{k}{W}_{h}{\mu\:}_{xh}$$

(25)

where$\:\:{Z}_{th}=\lambda\:{\stackrel{-}{y}}_{st}+(1-\lambda\:){Z}_{th-1}$ and $\:{Q}_{th}=\lambda\:{\stackrel{-}{x}}_{st}+(1-\lambda\:){Q}_{th-1}$ are the EWMA statistic for under study and ancillary variables respectively. The expected value of the stratified estimator can be obtained as

$$\:E\left({y}^{*}\right)=\stackrel{-}{Y}.$$

(26)

The MSE for the estimator $\:{y}^{*}$ is given by

$$\:MSE\left({y}^{*}\right)={\left(\sum\:_{h=1}^{k}{W}_{h}{\mu\:}_{xh}\right)}^{2}\frac{\lambda\:}{2-\lambda\:}\sum\:_{h=1}^{k}{{W}_{h}}^{2}{\theta\:}_{h}\left({{S}^{2}}_{yth}+{{S}^{2}}_{xth}-2{R}_{h}{S}_{yxth}\right).$$

(27)

Likewise, the MSE of stratified product memory type estimator is defined by

$$\:MSE\left({y}^{**}\right)={\left(\sum\:_{h=1}^{k}{W}_{h}{\mu\:}_{xh}\right)}^{2}\frac{\lambda\:}{2-\lambda\:}\sum\:_{h=1}^{k}{{W}_{h}}^{2}{\theta\:}_{h}\left({{S}^{2}}_{yth}+{{S}^{2}}_{xth}+2{R}_{h}{S}_{yxth}\right).$$

(28)

Proposed calibration estimation of memory type estimators under stratified sampling design

The discrepancy between the calibrated and original weights, as well as the associated distance measure, can be minimized through the calibration process. The primary objective of the calibration method is to provide more precise and accurate estimates compared to conventional estimators. By leveraging various relationships between the study and auxiliary variables, we develop calibrated memory-type ratio and product estimators under the stratified random sampling scheme to estimate the population mean of the study variable. Specifically, we modify the estimators proposed by Aslam et al.⁵ and introduce the following improved versions.

Calibrated memory type estimators using EWMA statistic

The calibrated memory-type ratio and product estimators in stratified random sampling, assuming that each stratum has a different population mean, are defined as follows:

$$\:{\widehat{y}}_{mr}=\frac{\sum\:_{h=1}^{k}{{W}^{*}}_{h}{Z}_{th}}{\sum\:_{h=1}^{k}{{W}^{*}}_{h}{Q}_{th}}\sum\:_{h=1}^{k}{{W}^{*}}_{h}{\mu\:}_{xh},\:\:\:\:h=1,\:2,\:\dots\:,\:k$$

(29)

and

$$\:{\widehat{y}}_{mp}=\frac{\sum\:_{h=1}^{k}{{W}^{*}}_{h}{Z}_{th}}{\sum\:_{h=1}^{k}{{W}^{*}}_{h}{\mu\:}_{xh}}\sum\:_{h=1}^{k}{{W}^{*}}_{h}{Q}_{th},\:\:\:\:h=1,\:2,\:\dots\:,\:k.$$

(30)

The calibration constraint, which reflects the association between the auxiliary variable and the study variable, is given as follows:

$$\:\sum\:_{h=1}^{k}{{W}^{*}}_{h}{x}_{h}=\sum\:_{h=1}^{k}{X}_{h}.$$

(31)

To minimize the calibrated weights, the following distance measure is employed:

$$\:{\theta\:}_{h}=\sum\:_{h=1}^{k}\frac{1}{{q}_{h}{W}_{h}}{\left({W}_{h}^{*}-{W}_{h}\right)}^{2}.$$

(32)

Using the distance measure and calibration constraint specified in Eqs. (32) and (31), respectively, the Lagrange multiplier can be derived as follows:

$$\:\varDelta\:=\frac{{\left({W}_{h}^{*}-{W}_{h}\right)}^{2}}{{q}_{h}{W}_{h}}-2\lambda\:({{W}^{*}}_{h}{x}_{h}-{X}_{h})$$

(33)

Differentiating Eq. (32) with respect to $\:{W}_{h}^{*}$ and equating it to zero, we obtain

$$\:{W}_{h}^{*}=\lambda\:{x}_{h}{q}_{h}{W}_{h}+{W}_{h}.$$

(34)

Substituting the expression of $\:{W}_{h}^{\text{*}}$ into the calibration constraint given in Eq. (31) yields:

$$\:\sum\:_{h=1}^{k}(\lambda\:{x}_{h}{q}_{h}{W}_{h}+{W}_{h}){x}_{h}=\sum\:_{h=1}^{k}{X}_{h}.$$

(35)

By simplifying the above expression, the resulting value of $\:\lambda\:$ is obtained as follows:

$$\:\lambda\:=\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}$$

(36)

.

Substituting the value of $\:\lambda\:$ into Eq. (34) yields the calibrated weight:

$$\:{W}_{h}^{*}=\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\left({x}_{h}{q}_{h}{W}_{h}\right)+{W}_{h}.$$

(37)

Substituting the value of $\:{W}_{h}^{*}$ from Eq. (34) into Eq. (29), the expression for the memory-type ratio estimator in stratified sampling is obtained as follows:

$$\:{\widehat{y}}_{mr}\left.\begin{array}{c}=\frac{\sum\:_{h=1}^{k}{Z}_{th}{W}_{h}+\frac{\left\{\sum\:_{h=1}^{k}{Z}_{th}{x}_{h}{q}_{h}{W}_{h}\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}}{\sum\:_{h=1}^{k}{Q}_{th}{W}_{h}+\frac{\left\{\sum\:_{h=1}^{k}{Q}_{th}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}}\\\:\times\:\sum\:_{h=1}^{k}\left\{\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\left({x}_{h}{q}_{h}{W}_{h}\right)+{W}_{h}\right\}{\mu\:}_{xh}\end{array}\right]$$

After simplification, the memory-type calibrated ratio estimator in stratified sampling is obtained as:

$$\:{\widehat{y}}_{mr}=\left.\begin{array}{c}\frac{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}\:\sum\:_{h=1}^{k}{Z}_{th}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Z}_{th}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{Q}_{th}{W}_{h}\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Q}_{th}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}\\\:\times\:\left\{\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\sum\:_{h=1}^{k}{x}_{h}{q}_{h}{W}_{h}{\mu\:}_{xh}+\sum\:_{h=1}^{k}{W}_{h}{\mu\:}_{xh}\right\}\end{array}\right].$$

(38)

Similarly, the calibrated memory-type product estimator in stratified sampling is given by:

$$\:{\widehat{y}}_{mp}=\left.\begin{array}{c}\frac{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}\:\sum\:_{h=1}^{k}{Z}_{th}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Z}_{th}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{\mu\:}_{xh}{W}_{h}\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}+\left\{\sum\:_{h=1}^{k}{\mu\:}_{xh}{x}_{h}{q}_{h}{W}_{h}\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}\:\\\:\times\:\left\{\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\sum\:_{h=1}^{k}{x}_{h}{q}_{h}{W}_{h}{Q}_{th}+\sum\:_{h=1}^{k}{W}_{h}{Q}_{th}\right\}\end{array}\right].$$

(39)

Calibrated memory type estimators using EEWMA statistic

Assuming that each stratum has a different population mean, the suggested calibrated memory-type ratio and product estimators based on the Exponentially Weighted Moving Average (EEWMA) statistic are defined as follows:

• Calibrated memory-type ratio estimator using EEWMA:

$$\:{\widehat{y}}_{mre}=\left.\begin{array}{c}\frac{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}\sum\:_{h=1}^{k}{Z}_{teh}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Z}_{teh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{Q}_{teh}{W}_{h}\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Q}_{teh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}\\\:\times\:\left\{\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\sum\:_{h=1}^{k}{x}_{h}{q}_{h}{W}_{h}{\mu\:}_{xh}+\sum\:_{h=1}^{k}{W}_{h}{\mu\:}_{xh}\right\}\end{array}\right].$$

(40)

• Calibrated memory-type product estimator using EEWMA:

$$\:{\widehat{y}}_{mpe}=\left.\begin{array}{c}\frac{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}\sum\:_{h=1}^{k}{Z}_{teh}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Z}_{teh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{\mu\:}_{xh}{W}_{h}\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}+\left\{\sum\:_{h=1}^{k}{\mu\:}_{xh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}\\\:\times\:\left\{\frac{\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\sum\:_{h=1}^{k}{x}_{h}{q}_{h}{W}_{h}{Q}_{teh}+\sum\:_{h=1}^{k}{W}_{h}{Q}_{teh}\right\}\end{array}\right],$$

(41)

where $\:{Z}_{teh}={\lambda\:}_{1}{\stackrel{-}{y}}_{st}{-\lambda\:}_{2}{\stackrel{-}{y}}_{st-1}+\left(1-{\lambda\:}_{1}+{\lambda\:}_{2}\right){Z}_{teh-1}$ and $\:{Q}_{teh}={\lambda\:}_{1}{\stackrel{-}{x}}_{st}{-\lambda\:}_{2}{\stackrel{-}{x}}_{t-1}+(1-{\lambda\:}_{1}+{\lambda\:}_{2}){Q}_{teh-1}$ are EEWMA statistic for study as well as assisting variables respectively.

Calibrated memory type estimators using HEWMA statistic

Using the HEWMA statistic, the calibrated memory-type ratio and product estimators in stratified random sampling—assuming each stratum has a different population mean—are defined as follows:

• Calibrated memory-type ratio estimator (HEWMA-based):

$$\:{\widehat{y}}_{mrh}=\left.\begin{array}{c}\frac{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}\:\sum\:_{h=1}^{k}{Z}_{thh}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Z}_{thh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{Q}_{thh}{W}_{h}\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Q}_{thh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}\\\:\times\:\left\{\frac{\sum\:_{h=1}^{k}{x}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\sum\:_{h=1}^{k}{x}_{h}{q}_{h}{W}_{h}{\mu\:}_{xh}+\sum\:_{h=1}^{k}{W}_{h}{\mu\:}_{xh}\right\}\end{array}\right]$$

(42)

• Calibrated memory-type product estimator (HEWMA-based):

$$\:{\widehat{y}}_{mph}=\left.\begin{array}{c}\frac{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}\:\sum\:_{h=1}^{k}{Z}_{thh}{W}_{h}+\left\{\sum\:_{h=1}^{k}{Z}_{thh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}{\sum\:_{h=1}^{k}{\mu\:}_{xh}{W}_{h}\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}+\left\{\sum\:_{h=1}^{k}{\mu\:}_{xh}{x}_{h}{q}_{h}{W}_{h}\:\:\left(\sum\:_{h=1}^{k}{X}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}\right)\right\}}\\\:\times\:\left\{\frac{\sum\:_{h=1}^{k}{x}_{h}-\sum\:_{h=1}^{k}{W}_{h}{x}_{h}}{\sum\:_{h=1}^{k}{x}_{h}^{2}{q}_{h}{W}_{h}}\sum\:_{h=1}^{k}{x}_{h}{q}_{h}{W}_{h}{Q}_{thh}+\sum\:_{h=1}^{k}{W}_{h}{Q}_{thh}\right\}\end{array}\right],$$

(43)

where $\:{Z}_{thh}$​ and $\:{Q}_{thh}$​ are the HEWMA statistics for the study and auxiliary variables in the hth stratum, respectively. These are recursively defined as:

• $$\:{E}_{tyh}={\lambda\:}_{2}{\stackrel{-}{y}}_{st}+\left(1-{\lambda\:}_{2}\right){E}_{tyh-1},$$
• $$\:{Z}_{thh}={\lambda\:}_{1}{E}_{tyh}+\left(1-{\lambda\:}_{1}\right){Z}_{thh-1},$$
• $$\:{E}_{txh}={\lambda\:}_{2}{\stackrel{-}{x}}_{st}+\left(1-{\lambda\:}_{2}\right){E}_{txh-1},$$
• $$\:{Q}_{thh}={\lambda\:}_{1}{E}_{txh}+\left(1-{\lambda\:}_{1}\right){Q}_{thh-1},$$

where $\:{\lambda\:}_{1}$ and $\:{\lambda\:}_{2}$ are smoothing parameters such that $\:0<{\lambda\:}_{1},{\lambda\:}_{2}\le\:1,$ $\:{\stackrel{-}{y}}_{st}$ and $\:{\stackrel{-}{x}}_{st}$​ are the stratified sample means for the study and auxiliary variables, respectively, and $\:{x}_{h}$, $\:{q}_{h}$, $\:{W}_{h}$, and $\:{\mu\:}_{xh}$ are defined as per earlier notation for the hth stratum.

Simulation study

The simulation study is carried out to evaluate the performance of the proposed calibrated memory-type ratio and product estimators. Specifically, we compare the proposed calibrated estimators—$\:{\widehat{y}}_{mr,}{\widehat{y}}_{mp},{\widehat{y}}_{mre},{\widehat{y}}_{mpe},{\widehat{y}}_{mrh},{\widehat{y}}_{mph}$​—with the memory-type estimator developed by Noor-ul-Amin⁴. Additionally, a detailed comparative analysis is performed between the proposed calibrated estimators and the simple stratified memory-type estimators introduced by Aslam et al.⁵, under the assumption that the auxiliary variable has different means across strata.

The MSE and RE of the proposed estimators are computed based on 50,000 replications. The values of the correlation coefficient (ρ) for each stratum are set at 0.05,0.25,0.50,0.750.05, 0.25, 0.50, 0.750.05,0.25,0.50,0.75, and 0.950.950.95, while the smoothing parameter $\:\lambda\:$ takes values 0.1, 0.15, 0.25, 0.50, 0.75 and 1.00, following the simulation settings of Noor-ul-Amin⁴.

The MSEs for each estimator are computed using the following formula:

$$\:MSE\left({y}_{i}\right)=\frac{\sum\:_{k=1}^{50000}{\left({y}_{i}-\stackrel{-}{y}\right)}^{2}}{50000}.$$

(44)

The MSE and RE values for both the simple and stratified memory-type estimators, as well as the proposed calibrated memory-type ratio and product estimators based on EWMA, EEWMA, and HEWMA statistics, are reported in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17, respectively. These tables illustrate the comparative performance of all considered estimators under various correlation levels and smoothing parameter settings.

The RE of an estimator is computed using the following formula:

$$\:RE=\frac{MSE\left({y}^{h}\right)}{MSE\left(\widehat{y}\right)},$$

(45)

where $\:{y}^{h}={y}^{*},{y}^{**},\:{{\stackrel{-}{y}}^{M}}_{rt},\:{{\stackrel{-}{y}}^{M}}_{pt}\&\:\widehat{y}={\widehat{y}}_{mr},{\widehat{y}}_{mp},{\widehat{y}}_{mpe},{\widehat{y}}_{mre},{\widehat{y}}_{mph},{\widehat{y}}_{mrh}$

Algorithm for computing the MSE and RE of the proposed estimators.

Following the approach of Noor-ul-Amin⁴:

• 1. Generate a population of size 5000 from a bivariate normal distribution.

• 2. Select appropriate values for the parameter $\:\lambda\:$.

• 3. Draw samples of sizes ni = 10, 20, 30, 50, 200, and 500 units from each stratum separately.

• 4. Based on the samples obtained in Step 3, generate 50,000 values for each estimator.

• 5. For each sample size, compute the MSE of the estimators using Eq. (44).

• 6. Compute the RE of the proposed estimators using Eq. (45).

Table 1 MSE of stratified and calibrated memory-type ratio estimators based on EWMA statistic at various values of ρ, smoothing parameters $\:\lambda\:=0.10,\:0.15,\:0.25,$ and sample sizes $\:n$.

Table 2 MSE of stratified and calibrated memory-type ratio estimators based on EWMA statistic at various values of ρ, smoothing parameters $\:\lambda\:=0.50,\:0.75,\:1.00,$ and sample sizes $\:n$.

Table 3 RE of stratified and proposed calibrated memory-type ratio estimators at various values of ρ, smoothing parameters $\:\lambda\:=0.10,\:0.15,\:0.25,$ and sample sizes $\:n$.

Table 4 MSE of stratified and calibrated memory-type product estimators based on EWMA statistic at various values of ρ, smoothing parameters $\:\lambda\:=0.50,\:0.75,\:1.00,$ and sample sizes $\:n$.

Table 5 RE of stratified and proposed calibrated memory-type ratio estimators at various values of ρ, smoothing parameters $\:\lambda\:=0.25,\:0.50,\:0.75,$ and sample sizes $\:n$.

Table 6 MSE of memory-type and proposed calibrated ratio estimators based on EWMA statistic at different values of correlation coefficient ρ, smoothing parameters $\:\lambda\:=0.10,\:0.15,\:0.25,$ and sample sizes $\:n$.

Table 7 MSE of memory-type and proposed calibrated ratio estimators based on EWMA statistic at different values of correlation coefficient ρ, smoothing parameters $\:\lambda\:=0.50,\:0.75,\:1.00,$ and sample sizes $\:n$.

Table 8 RE of memory-type and proposed calibrated ratio estimators at different values of correlation coefficient ρ, smoothing parameters $\:\lambda\:=0.10,\:0.15,\:0.25,$ and sample sizes $\:n$.

Table 9 MSE of memory-type and proposed calibrated ratio estimators based on EWMA statistic at different values of correlation coefficient ρ, smoothing parameters $\:\lambda\:=0.10,\:0.15,\:0.25,$ and sample sizes $\:n$.

Table 10 MSE of memory-type and proposed calibrated ratio estimators based on EWMA statistic at different values of correlation coefficient ρ, smoothing parameters $\:\lambda\:=0.50,\:0.75,\:1.00,$ and sample sizes $\:n$.

Table 11 RE of memory-type and proposed calibrated product estimators at different values of correlation coefficient ρ, smoothing parameters $\:\lambda\:=0.10,\:0.15,\:0.25,\:0.50,\:0.75,\:1.00$ and sample sizes $\:n$.

Table 12 MSE of stratified and calibrated memory-type ratio estimators based on EEWMA at different values of ρ, $\:{\lambda\:}_{1}=0.25,\:0.50,\:0.75$, $\:{\lambda\:}_{2}=\:0.05,\:0.15,\:0.20$, and sample sizes $\:n$.

Table 13 MSE of stratified and calibrated memory-type product estimators based on EEWMA at different values of ρ, $\:{\lambda\:}_{1}=0.25,\:0.50,\:0.75$, $\:{\lambda\:}_{2}=\:0.05,\:0.15,\:0.20$, and sample sizes $\:n$.

Table 14 RE of stratified and calibrated memory-type ratio and product estimators based on EEWMA at different values of ρ, $\:{\lambda\:}_{1}=0.25,\:0.50,\:0.75$, $\:{\lambda\:}_{2}=\:0.05,\:0.15,\:0.20$, and sample sizes $\:n$.

Table 15 MSE of stratified and calibrated memory-type ratio estimators based on EEWMA at different values of ρ, $\:{\lambda\:}_{1}=0.25,\:0.50,\:0.75$, $\:{\lambda\:}_{2}=\:0.05,\:0.15,\:0.20$, and sample sizes $\:n$.

Table 16 MSE of stratified and calibrated memory-type product estimators based on HEWMA at different values of ρ, $\:{\lambda\:}_{1}=0.25,\:0.50,\:0.75$, $\:{\lambda\:}_{2}=\:0.05,\:0.15,\:0.20$, and sample sizes $\:n$.

Table 17 RE of stratified and calibrated memory-type ratio and product estimators based on HEWMA at different values of ρ, $\:{\lambda\:}_{1}=0.25,\:0.50,\:0.75$, $\:{\lambda\:}_{2}=\:0.05,\:0.15,\:0.20$, and sample sizes $\:n$.

Fig. 1

MSE of stratified and proposed calibrated ratio estimators at λ = 0.1 and ρ = 0.05.

Fig. 2

MSE of stratified and proposed calibrated ratio estimators at λ = 0.15 and ρ = 0.50.

Fig. 3

MSE of stratified and proposed calibrated ratio estimators at λ = 0.25 and ρ = 0.95.

Fig. 4

MSE of memory type and proposed calibration estimators at λ = 0.50 and ρ = 0.25.

Fig. 5

MSE of memory type and proposed calibration estimators at λ = 0.75 and ρ = 0.75.

Fig. 6

MSE of memory type and proposed calibration estimators at λ = 1.00 and ρ = 0.95.

Fig. 7

RE of stratified and proposed calibrated ratio estimators at ρ = 0.05.

Fig. 8

RE of stratified and proposed calibrated ratio estimators at ρ = 0.25.

Fig. 9

RE of stratified and proposed calibrated ratio estimators at ρ = 0.75.

Fig. 10

RE of memory type classical and calibrated ratio estimators at ρ = 0.05.

Fig. 11

RE of memory type classical and calibrated ratio estimators at ρ = 0.50.

Fig. 12

RE of memory type classical and calibrated ratio estimators at ρ = 0.95.

The results of the simulation study, in terms of MSEs and REs, are summarized in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17. Tables 1 and 2 present the MSEs, while Table 3 reports the REs of both the proposed and existing estimators. Tables 4 and 5 display the MSEs and REs for the stratified and suggested calibrated memory-type product estimators. Furthermore, Tables 6, 7, 8, 9, 10 and 11 illustrate the MSEs and REs of the simple memory-type estimators and the proposed calibrated memory-type estimators based on the EWMA statistic. Additionally, Tables 12, 13, 14, 15, 16 and 17 provide the corresponding results using the EEWMA and HEWMA statistics, respectively.

Overall, it is evident from the tables that the proposed calibrated memory-type ratio and product estimators consistently yield lower MSE values compared to the other estimators considered. These results demonstrate that the calibrated memory-type estimators outperform their counterparts in terms of estimation efficiency.

Similarly, the MSE results for the previously developed memory-type estimators by Noor-ul-Amin⁴ and the proposed calibrated memory-type estimators based on different statistics are presented in the corresponding tables alongside their REs. It is evident that the MSE values for the calibrated memory-type ratio and product estimators constructed using EWMA, EEWMA, and HEWMA statistics are consistently lower compared to the existing memory-type estimators.

Moreover, it is observed that increasing the correlation coefficient (ρ) from 0.00 to 0.95 leads to a noticeable reduction in MSEs, thereby enhancing the efficiency of the proposed calibrated estimators. This trend is further supported by Figs. 1, 2, 3, 4, 5 and 6, which depict the MSEs at various smoothing constants and correlation coefficient values, highlighting the superior performance of the proposed estimators over the existing ones. Additionally, Figs. 7, 8, 9, 10, 11 and 12 illustrate the REs of the estimators, reaffirming the improved performance of the suggested calibrated memory-type estimators.

Real life application

In this section, we apply the proposed estimators to a real-time dataset involving apple production, originally considered by Kadilar and Cingi¹⁴. The dataset includes information on apple production ($\:Y$) and the number of apple trees ($\:X$) across 854 communities in Turkey for the year 1999. The data were collected by the Turkish Ministry of Education to explore practical applications. The dataset is stratified into six regions—Marmara, Aegean, Mediterranean, Central Anatolia, Black Sea, and East and Southeast Anatolia—each treated as a separate stratum. The performance of the proposed calibrated memory-type ratio estimators is evaluated by assessing the strength of association between the study and auxiliary variables in this real-world application.

We consider a sample size of $\:n=140$, with smoothing parameters λ = 0.25, λ₁ = 0.25, λ₂ = 0.05. The observed statistics for the population, strata, and sample sizes are presented in Table 18.

Table 18 Population size, sample size, means, and weights for each stratum in the Apple production dataset.

Table 19 presents the estimated values obtained from the proposed calibrated memory type estimators, while Fig. 13 illustrates the corresponding MSEs of both existing stratified memory type estimators and the proposed calibrated estimators. From the results in Table 19 and the trends depicted in Fig. 13, it is evident that the proposed calibrated memory type ratio estimators yield more accurate and reliable estimates compared to the existing stratified memory type estimators when using the EWMA, EEWMA, and HEWMA statistics. Notably, the MSE values associated with the proposed estimators are consistently lower than those of the existing ones, highlighting the improved efficiency of the calibration approach. These findings demonstrate that incorporating memory type estimators into the calibration framework enhances estimation accuracy over time.

Table 19 Memory type stratified and calibrated ratio estimators with their mses.

Fig. 13

MSE of stratified and proposed memory type calibrated ratio estimators.

Conclusion

The estimation of parameters plays a crucial role in sampling techniques. The use of auxiliary information significantly enhances the accuracy and reliability of the estimates. Moreover, calibration techniques, particularly those incorporating prior information, are vital in the estimation process. These techniques utilize distance measures to minimize the discrepancy between the calibrated and original sampling weights under various calibration constraints. In this study, we proposed calibrated ratio and product-type estimators that incorporate both past sample data and current information using EWMA, EEWMA, and HEWMA statistics. A simulation study was carried out to evaluate the performance of the proposed estimators. Based on the simulation results (Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17), it was observed that the proposed memory-type calibrated estimators are more efficient and accurate in estimating the population mean compared to existing estimators, as reflected in their lower mean squared error (MSE) and higher relative efficiency (RE) values. Additionally, an empirical study was conducted to validate the findings of the simulation. The results, presented in Table 19, further confirmed that the proposed ratio and product-type memory estimators outperform previous estimators, exhibiting the minimum MSE values among the compared methods.

Future studies may extend the proposed memory-type calibrated estimators to more complex sampling designs such as stratified or cluster sampling. Incorporating multiple auxiliary variables could further enhance estimator efficiency and flexibility. It is also recommended to explore the robustness of these estimators in the presence of outliers or missing data. Additionally, integrating alternative memory-based statistics or adaptive calibration techniques could improve performance under dynamic conditions. Finally, applying the proposed methods to real-world datasets across various domains would help validate their practical applicability.