Mean Estimation Using Memory-Type Estimators in Systematic Sampling for Time-Scaled Surveys

Abstract

In this paper, we propose memory-type ratio, product, exponential ratio, and exponential product estimators based on an exponentially weighted moving average (EWMA) statistic for mean estimation in time-scaled surveys using systematic sampling. The approximate expressions for bias and mean squared error (MSE) of the proposed memory-type estimators are derived using Taylor and exponential series expansions up to the second order. Mathematical conditions are also established under which the memory-type ratio, product, exponential ratio, and exponential product estimators outperform their corresponding conventional estimators in terms of MSE. A simulation study is carried out to evaluate the performance of the proposed estimators compared to the conventional estimators. Additionally, real data application is also used to support the simulation findings. The results indicate that the proposed memory-type estimators have smaller MSEs than conventional estimators for time-scaled surveys using systematic sampling.

1. Introduction

Systematic sampling is a widely used probability sampling design that offers a good balance between operational simplicity and statistical efficiency while ensuring that the sample adequately represents the natural populations, as discussed by [1,2,3] noted that systematic sampling is frequently used either independently or in combination with other sampling methods, and is considered one of the most extensively applied sampling techniques in practice. According to [4], systematic sampling may provide implicit stratification, which often leads to improved estimates when the sampling frame is ordered. Moreover, Buckland [5] agreed with Finney [1,2] that systematic sampling is efficient and practically convenient, especially in the sampling of natural populations like forests. Although, systematic sampling has been applied in various applications (agriculture, forestry, and meteorology), the theory of systematic sampling was first studied by [6,7,8] continued the development of the theory of systematic sampling, which was later reviewed by [5,9] presented a critical review of the recent developments in system systematic.

The use of auxiliary information in systematic sampling is beneficial to enhancing the accuracy and efficiency of mean estimation. It enables researchers to make better use of available knowledge about the population and reduces the potential for bias and uncertainty in the estimation process. By leveraging auxiliary data effectively, systematic sampling can produce reliable and informative estimates of the population mean. Many authors have used systematic sampling with or without auxiliary data. In the presence of auxiliary information, refs. [10,11] proposed ratio and product estimators for the estimation of population mean using systematic sampling and discussed the properties as well. Ref. [12] proposed modified ratio and product-type estimators for population mean, whereas ref. [13] proposed exponential-type estimators for the estimation of study variables in systematic sampling design. Refs. [14,15], respectively, proposed chain ratio- and exponential ratio-type estimators, whereas ref. [16] proposed a semi-exponential ratio estimator using two auxiliary variables under a systematic sampling design. Recently, refs. [17,18] suggested various estimators using two auxiliary variables for mean estimation in systematic sampling.

The EWMA statistic is a technique that integrates both the current and previous sample information to generate smoothed estimates, giving more weight to recent data while not entirely discarding the older observations under the setting of time-scaled surveys. The estimators based on EWMA statistics, also known as memory-type estimators, not only improve the efficiency but also produce smoother estimates by weighting recent observations more heavily, which is useful in situations where data may be subject to short-term fluctuations. These estimators are less sensitive to the sudden changes in data and outliers, making them robust in time-scaled surveys where data consistency is essential.

In recent years, refs. [19] proposed memory-type ratio and product estimators, whereas ref. [20] proposed a log-type estimator for time-scaled surveys. Ref. [21] proposed memory-type variance estimators using auxiliary information in the presence and absence of measurement errors under simple random sampling (SRS). Ref. [22] introduced memory-type estimators with dual auxiliary variables for mean estimation. Ref. [22] extended memory-type estimators to a stratified design for heterogeneous populations. Ref. [23] proposed memory-type estimators that incorporate EWMA and regression imputation techniques to handle non-response. Ref. [24] developed memory-type estimators for survey accuracy using EWMA and related extended statistics.

In time-scaled surveys, where the population values may change over time, classical estimators may not capture temporal dynamics effectively, which can lead to suboptimal performance in terms of bias and MSE. To address this limitation, we propose memory-type estimators based on the EWMA statistic, which incorporate past information and provide improved estimation accuracy for estimation of the mean in time-scaled surveys. After this brief introduction, the rest of the paper is structured as follows. Section 2 is based on the methodology, formulas, and basic notations of systematic sampling. Some classical estimators are mentioned in Section 3. The proposed memory-type estimators are presented in Section 4, along with the derivations of bias and mean MSE expressions. Mathematical comparisons of memory-type estimators with conventional estimators are presented in Section 5. To assess the performance of the memory-type estimators, a simulation study is conducted in Section 6, while Section 7 is based on application to real data. Concluding remarks and future directions are presented in Section 8.

2. Sampling Methodology, Formulas, and Notations

The process of systematic sampling is straightforward to implement. Once the sampling interval is determined, selecting the sample elements becomes a systematic process, making it convenient for researchers to employ in practice. Consider a finite population P = (P₁, P₂, …, P_j, …, P_N) made up of N different elements labelled from 1 to N (1, 2, …, j, …, N), and organized in a particular order. Let us simply refer to the jth element of P as P_j. Additionally, suppose that N may be written as the product of two non-negative numbers, n and k, with the result being N = nk. Let Y stand for the primary variable of concern and X represent a supporting variable. The labels for the study variable and the auxiliary variable, respectively denoted as y = (y₁, y₂, …, y_j, …, y_N) and x = (x₁, x₂, …, x_j, …, x_N), can be used to define the values of both variables. The values of the primary study variable and the auxiliary variable for the jth unit (j = 1, 2, 3, …, k) in the ith (i = 1, 2, 3, …, n) systematic sample are represented here by y_ij and x_ij, respectively. We start by creating a random number between 1 and k (let us pretend it is j) in order to choose a sample of size n. Then, starting from j, we choose every kth unit such that it contains the j, j + k, j + 2k, …, j + (n − 1)k successive elements. As a result, there are a total of k potential samples with a total of n elements.

The means and variances for the study and the auxiliary variables for systematic (sys) samples may be obtained as

$${\overline{y}}_{sys}={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{j=1}^n{y}_{ij}} \mathrm{and} {\overline{x}}_{sys}={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{j=1}^n{x}_{ij}}$$

$$s_y^2={\displaystyle \frac{1}{\left(n-1\right)}}{{\displaystyle {\sum }_{j=1}^n\left(y_{ij}-\overline{y}\right)}}^2\mathrm{and} s_x^2={\displaystyle \frac{1}{\left(n-1\right)}}{{\displaystyle {\sum }_{j=1}^n\left(x_{ij}-\overline{x}\right)}}^2.$$

The means and variances for the study and the auxiliary variables for population mean may be obtained as

$$\overline{Y}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{i=1}^N{y}_i }\mathrm{and} \overline{X}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{i=1}^N{x}_i }.$$

$$S_y^2={\displaystyle \frac{1}{\left(N-1\right)}}{\displaystyle {\sum }_{i=1}^k{{\displaystyle {\sum }_{j=1}^n\left(y_{ij}-\overline{Y}\right)}}^2}\mathrm{and} S_x^2={\displaystyle \frac{1}{\left(N-1\right)}}{\displaystyle {\sum }_{i=1}^k{{\displaystyle {\sum }_{j=1}^n\left(x_{ij}-\overline{X}\right)}}^2}.$$

Ref. [25] first gave rise to the notion of the EWMA statistic to examine the change in the process mean. The EWMA statistic for the non-sensitive study variable and auxiliary variable are, respectively, defined as

$$w_{t,sys}=\lambda {\overline{y}}_{sys}+\left(1-\lambda \right)w_{t-1,sys}.$$

and

$$u_{t,sys}=\lambda {\overline{x}}_{sys}+\left(1-\lambda \right)u_{t-1,sys},$$

where ${\overline{y}}_{sys}$ and ${\overline{x}}_{sys}$ are means of the study variable and the auxiliary variable of the current sample using systematic sampling. Here, λ is the smoothing constant (0 $\le$λ$\le$ 1), which is known as the weight given to the observations. The larger the value of λ, the larger the weight given to the current values and the smaller the weight given to the past values; whereas, the smaller the value of λ, the smaller the values given to the current values and larger the weight given to the past values. The mean of the current sample receives all the weight for λ = 1, and the EWMA statistic would be equal to the conventional sample mean estimator. Further, t determines the number of the sample and the term “t − 1” is used to indicate the prior observation of the given statistic. The expected mean or the average of the prior sample is taken as its initial value.

It is quite obvious that no survey is conducted unless the results obtained from the pilot survey are trustworthy. Usually, the initial values of w_t,sys and u_t,sys are assumed to be the expected means, which can be estimated from the pilot survey. In this study, we consider them to be zero.

Let ${\zeta }_y$ and ${\zeta }_x$ represent the sampling errors associated with the study variable and the auxiliary variable, respectively, defined as

$${\mathrm{\zeta }}_y={\displaystyle \frac{w_{t,sys}-\overline{Y}}{\overline{Y}}} \mathrm{and} {\mathrm{\zeta }}_y={\displaystyle \frac{u_{t,sys}-\overline{X}}{\overline{X}}}.$$

Some important notations, formulas, and statistical properties are presented in Equation (1), and are required for the derivation of the bias and MSE of the proposed estimator.

$$\left.\begin{array}{l} w_{t,sys}=\overline{Y}\left(1+{\mathrm{\zeta }}_y\right) \mathrm{and} u_{t,sys}=\overline{X}\left(1+{\mathrm{\zeta }}_x\right)\\ E\left({\mathrm{\zeta }}_y\right)=E\left({\mathrm{\zeta }}_x\right)=0\\ E\left({\mathrm{\zeta }}_y^2\right)=\mathrm{\theta }\left({\displaystyle \frac{\mathrm{\lambda }}{2-\mathrm{\lambda }}}\right){\displaystyle \frac{Var\left(w_{t,sys}\right)}{{\overline{Y}}^2}}=\mathrm{\theta }\left({\displaystyle \frac{\mathrm{\lambda }}{2-\mathrm{\lambda }}}\right){\mathrm{\rho }}_y^{\ast }{C}_y^2\\ E\left({\mathrm{\zeta }}_x^2\right)=\mathrm{\theta }\left({\displaystyle \frac{\mathrm{\lambda }}{2-\mathrm{\lambda }}}\right){\displaystyle \frac{Var\left(u_{t,sys}\right)}{{\overline{X}}^2}}=\mathrm{\theta }\left({\displaystyle \frac{\mathrm{\lambda }}{2-\mathrm{\lambda }}}\right){\mathrm{\rho }}_x^{\ast }{C}_x^2\\ E\left({\mathrm{\zeta }}_y{\mathrm{\zeta }}_{\mathrm{x}}\right)=\mathrm{\theta }{\displaystyle \frac{Cov\left(w_{t,sys},u_{t,sys}\right)}{\overline{Y} \overline{X}}}=\mathrm{\theta }\left({\displaystyle \frac{\mathrm{\lambda }}{2-\mathrm{\lambda }}}\right){\mathrm{\rho }}_{yx}\sqrt{{\mathrm{\rho }}_y^{\ast }{\mathrm{\rho }}_x^{\ast }}{C}_y{C}_x\\ {\mathrm{\rho }}_y^{\ast }=\left(1+\left(n-1\right){\mathrm{\rho }}_y\right), {\mathrm{\rho }}_y={\displaystyle \frac{\left[E\left(y_{ij}-\overline{Y}\right)\left(y_{i{j}^{\ast }}-\overline{Y}\right)\right]}{E{\left(y_{ij}-\overline{Y}\right)}^2}} \\ {\mathrm{\rho }}_x^{\ast }=\left(1+\left(n-1\right){\mathrm{\rho }}_x\right), {\mathrm{\rho }}_x={\displaystyle \frac{\left[E\left(x_{ij}-\overline{X}\right)\left(x_{i{j}^{\ast }}-\overline{X}\right)\right]}{E{\left(x_{ij}-\overline{X}\right)}^2}} \\ C_y={\displaystyle \frac{S_y}{\overline{Y}}} , C_x={\displaystyle \frac{S_x}{\overline{X}}}, {\mathrm{\rho }}_{yx}={\displaystyle \frac{S_{yx}}{S_y{S}_x}} \mathrm{and} \mathrm{\theta }= {\displaystyle \frac{\left(N-1\right)}{Nn}}.\end{array}\right|$$

(1)

where ρ_y and ρ_x be the corresponding intra-class correlations for the study and the auxiliary variables, which actually measure the rate of homogeneity within clusters in the context of systematic sampling. The intra-class correlation typically depends on the y_ij (the ith observation of the jth cluster) and y_ij* (the ith observation of the j* cluster, which is different from jth cluster), and ρ_yx is the correlation coefficient between the study variable and the auxiliary variable. The correction factor is used to adjust the variance in sampling when dealing with a finite population. Moreover, the quantities C_y and C_x are population coefficients of variations for the study and auxiliary variables, respectively.

3. Classical Estimators Under Systematic Sampling

In this section, we present the classical ratio, product, exponential ratio, and exponential product estimator, along with the expression of bias and MSE expressions in systematic sampling.

• Sample Mean Estimator

The sample mean estimator along with the expression of its variance is given by

$$t_{0,sys}^y={\overline{y}}_{sys}.$$

(1)

and

$$Var\left(t_{0,sys}^y\right)=\theta {\overline{Y}}^2{\rho }_y^{\ast }{C}_y^2.$$

(2)

• Classical Ratio and Product Estimators

Refs. [10,11], respectively, proposed classical ratio and product estimators under systematic sampling, defined as follows:

$$t_{\mathrm{r},sys}^y=\left({\displaystyle \frac{{\overline{y}}_{sys}}{{\overline{x}}_{sys}}}\right)\overline{X}.$$

(3)

and

$$t_{\mathrm{p},sys}^y=\left({\displaystyle \frac{{\overline{y}}_{sys}}{\overline{X}}}\right){\overline{x}}_{sys}.$$

(4)

The expressions of bias and MSE of classical ratio and product estimators are given by

$$B\left(t_{\mathrm{r},sys}^y\right)\approx \theta \overline{Y}\left({\rho }_x^{\ast }{C}_x^2-\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(5)

$$B\left(t_{\mathrm{p},sys}^y\right)\approx \theta \overline{Y}\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x.$$

(6)

and

$$MSE\left(t_{\mathrm{r},sys}^y\right)\approx \theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\rho }_x^{\ast }{C}_x^2-2\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(7)

$$MSE\left(t_{\mathrm{p},sys}^y\right)\approx \theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\rho }_x^{\ast }{C}_x^2-2\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(8)

• Exponential Ratio and Exponential Product Estimators

Ref. [12] proposed classical exponential ratio and exponential product estimators under systematic sampling, defined as follows:

$$t_{\mathrm{e}\mathrm{r},sys}^y={\overline{y}}_{sys}exp\left({\displaystyle \frac{\overline{X}-{\overline{x}}_{sys}}{\overline{X}+{\overline{x}}_{sys}}}\right).$$

(9)

and

$$t_{\mathrm{e}\mathrm{p},sys}^y={\overline{y}}_{sys}exp\left({\displaystyle \frac{{\overline{x}}_{sys}-\overline{X}}{\overline{X}+{\overline{x}}_{sys}}}\right).$$

(10)

The expressions of bias and MSE of classical exponential ratio and exponential product estimators are given by

$$B\left(t_{\mathrm{e}\mathrm{r},sys}^y\right)\approx \theta \overline{Y}\left({\displaystyle \frac{3}{8}}{\rho }_x^{\ast }{C}_x^2-{\displaystyle \frac{1}{2}}\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(11)

$$B\left(t_{\mathrm{e}\mathrm{p},sys}^y\right)\approx \theta \overline{Y}\left({\displaystyle \frac{1}{2}}\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x-{\displaystyle \frac{1}{8}}{\rho }_x^{\ast }{C}_x^2\right).$$

(12)

and

$$MSE\left(t_{\mathrm{e}\mathrm{r},sys}^y\right)\approx \theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}{\rho }_x^{\ast }{C}_x^2-\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(13)

$$MSE\left(t_{\mathrm{e}\mathrm{p},sys}^y\right)\approx \theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}{\rho }_x^{\ast }{C}_x^2+\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(14)

4. Memory-Type Estimators Based on EWMA Statistic Using Systematic Sampling

In this section, we define the memory-type classical estimators and modified forms of classical estimators based on the EWMA statistic under systematic sampling. The expressions of approximate bias and MSE of the aforementioned estimators are also derived up to the second order using Taylor and exponential expansions.

• Memory-Type Mean Estimator

The memory-type sample mean estimator based on the EWMA statistic [25] under systematic sampling is defined as

$$t_{mt,sys}^0=w_{t,sys}=\lambda {\overline{y}}_{sys}+\left(1-\lambda \right)w_{t-1,sys}.$$

(15)

The variance of the memory-type mean estimator is given by

$$Var\left(t_{mt,sys}^0\right)=\theta {\overline{Y}}^2\left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_y^{\ast }{C}_y^2.$$

(16)

• Memory-Type Ratio Estimator

The memory-type ratio estimator, based on the classical ratio estimator of [10] under systematic sampling, replaces the sample means of y and x with their EWMA statistics, and is defined as

$$t_{mt,sys}^r=\left({\displaystyle \frac{w_{t,sys}}{u_{t,sys}}}\right)\overline{X}.$$

(17)

To derive the expression of bias, we re-write Equation (17) in terms of sampling error using the notations given in Equation (1) as

$$t_{mt,sys}^r={\displaystyle \frac{\overline{Y}\left(1+{\zeta }_{yt,sys}\right)}{\overline{X}\left(1+{\zeta }_{xt,sys}\right)}}\overline{X}.$$

(18)

Applying the Taylor series up to the second order in Equation (18), we have

$$t_{mt,sys}^r\approx \overline{Y}\left(1+{\zeta }_{yt,sys}\right)\left(1-{\zeta }_{xt,sys}+{\zeta }_{xt,sys}^2\right).$$

(19)

Simplifying Equation (19) and retaining the terms up to the second order, we get

$$t_{mt,sys}^r-\overline{Y}\approx \overline{Y}\left({\zeta }_{xt,sys}^2-{\zeta }_{yt,sys}{\zeta }_{xt,sys}\right).$$

(20)

Applying expectation to both sides of Equation (20), we find that the expression for the bias of the memory-type ratio estimator based on the EWMA statistic is given by

$$B\left(t_{mt,sys}^r\right)\approx \overline{Y}\left[\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2-\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right].$$

(21)

To derive the expression of MSE, ignoring the term beyond the first order, squaring and taking expectations on both sides of Equation (19), we get

$$E{\left(t_{mt,sys}^r-\overline{Y}\right)}^2\approx {\overline{Y}}^{2}E\left({\zeta }_{yt,sys}^2+{\zeta }_{xt,sys}^2-2{\zeta }_{xt,sys}{\zeta }_{xt,sys}\right).$$

(22)

The final expression for the MSE of the memory-type ratio estimator based on EWMA statistics is given by

$$MSE\left(t_{mt,sys}^r\right)\approx {\overline{Y}}^2\left(\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_y^{\ast }{C}_y^2+\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2-2\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(23)

• Memory-Type Product Estimator

The memory-type product estimator, based on the classical product estimator of [11] under systematic sampling, replaces the sample means of y and x with their EWMA statistics, and is defined as

$$t_{mt,sys}^p=\left({\displaystyle \frac{w_{t,sys}}{\overline{X}}}\right)u_{t,sys}.$$

(24)

To derive the expression of bias of the memory-type product estimator, we re-write Equation (24) in terms of sampling error using the notations given in Equation (1) as

$$t_{mt,sys}^p={\displaystyle \frac{\overline{Y}\left(1+{\zeta }_{yt,sys}\right)}{\overline{X}}}\overline{X}\left(1+{\zeta }_{xt,sys}\right).$$

(25)

On simplification of Equation (25), we have

$$t_{mt,sys}^p-\overline{Y}\approx \overline{Y}\left({\zeta }_{yt,sys}+{\zeta }_{xt,sys}+{\zeta }_{yt,sys}{\zeta }_{xt,sys}\right).$$

(26)

Applying expectation to both sides of Equation (26), we find that the final expression for the bias of the memory-type product estimator based on EWMA statistics is given by

$$B\left(t_{mt,sys}^p\right)\approx \theta \overline{Y}\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x.$$

(27)

To derive the expression of MSE, ignoring the term beyond the first order, squaring and taking expectations on both sides of Equation (26), we get

$$E{\left(t_{mt,sys}^p-\overline{Y}\right)}^2\approx {\overline{Y}}^{2}E\left({\zeta }_{yt,sys}^2+{\zeta }_{xt,sys}^2+2{\zeta }_{yt,sys}{\zeta }_{xt,sys}\right).$$

(28)

The final expression of MSE of the memory-type product estimator based on EWMA statistics is given by

$$MSE\left(t_{mt,sys}^p\right)\approx {\overline{Y}}^2\left(\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_y^{\ast }{C}_y^2+\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2+2\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(29)

• Memory-Type Exponential Ratio Estimator

The memory-type exponential ratio estimator is an adaptation of the traditional exponential ratio estimator proposed by [12] under systematic sampling with sample means replaced by the EWMA statistics of y and x, and is defined as

$$t_{mt,sys}^{er}=w_{t,sys}exp\left({\displaystyle \frac{\overline{X}-u_{t,sys}}{\overline{X}+u_{t,sys}}}\right).$$

(30)

To derive the expression of bias of the memory-type exponential ratio estimator, we re-write Equation (30) in terms of sampling error using the notations given in Equation (1) as

$$t_{mt,sys}^{er}=\overline{Y}\left(1+{\zeta }_{yt,sys}\right)exp\left({\displaystyle \frac{\overline{X}-\overline{X}\left(1+{\zeta }_{xt,sys}\right)}{\overline{X}+\overline{X}\left(1+{\zeta }_{xt,sys}\right)}}\right).$$

(31)

Simplifying, and applying Taylor and exponential series up to the second order in Equation (31), we have

$$t_{mt,sys}^{er}\approx \overline{Y}\left(1+{\zeta }_{yt,sys}\right)\left(1-{\displaystyle \frac{{\zeta }_{xt,sys}}{2}}+{\displaystyle \frac{{\zeta }_{xt,sys}^2}{4}}+{\displaystyle \frac{{\zeta }_{xt,sys}^2}{8}}\right).$$

(32)

On simplification of Equation (32), we have

$$t_{mt,sys}^{er}-\overline{Y}\approx \overline{Y}\left({\displaystyle \frac{{\zeta }_{xt,sys}}{2}}+{\displaystyle \frac{3}{8}}{\zeta }_{xt,sys}^2+{\zeta }_{yt,sys}-{\displaystyle \frac{{\zeta }_{yt,sys}{\zeta }_{xt,sys}}{2}}\right).$$

(33)

Applying expectation to both sides of Equation (33), we find that the final expression for the bias of the memory-type exponential ratio estimator based on the EWMA statistic is given by

$$B\left(t_{mt,sys}^{er}\right)\approx \overline{Y}\left({\displaystyle \frac{3}{8}}\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2-{\displaystyle \frac{1}{2}}\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(34)

To derive the expression of MSE of the memory-type exponential ratio estimator, simplifying Equation (32) and ignoring the terms beyond the first order, we get

$$t_{mt,sys}^{er}\approx \overline{Y}\left(1+{\zeta }_{yt,sys}-{\displaystyle \frac{{\zeta }_{xt,sys}}{2}}\right).$$

(35)

Simplifying, squaring, and taking expectation on both sides of Equation (35), we obtain

$$E{\left(t_{mt,sys}^{er}-\overline{Y}\right)}^2\approx {\overline{Y}}^{2}E\left({\zeta }_{yt,sys}^2+{\displaystyle \frac{1}{4}}{\zeta }_{xt,sys}^2-{\zeta }_{yt,sys}{\zeta }_{xt,sys}\right).$$

(36)

The final expression for the MSE of the memory-type exponential ratio estimator based on EWMA statistics is given by

$$MSE\left(t_{mt,sys}^{er}\right)\approx {\overline{Y}}^2\left(\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2-\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(37)

• Memory-Type Exponential Product Estimator

The memory-type exponential product estimator is an adaptation of the traditional exponential product estimator proposed by [12] under systematic sampling, where EWMA statistics are used in place of the sample means of y and x, and it is defined as

$$t_{mt,sys}^{ep}=w_{t,sys}exp\left({\displaystyle \frac{u_{t,sys}-\overline{X}}{u_{t,sys}+\overline{X}}}\right).$$

(38)

To derive the expression of bias of the memory-type exponential product estimator, we re-write Equation (38) in terms of sampling error using the notations given in Equation (1) as

$$t_{mt,sys}^{ep}=\overline{Y}\left(1+{\zeta }_{yt,sys}\right)exp\left({\displaystyle \frac{\overline{X}\left(1+{\zeta }_{xt,sys}\right)-\overline{X}}{\overline{X}\left(1+{\zeta }_{xt,sys}\right)+\overline{X}}}\right).$$

(39)

Simplifying and applying Taylor and exponential series up to second order on Equation (39), we have

$$t_{mt,sys}^{ep}\approx \overline{Y}\left(1+{\zeta }_{yt,sys}\right)\left(1+{\displaystyle \frac{1}{2}}{\zeta }_{xt,sys}-{\displaystyle \frac{1}{8}}{\zeta }_{xt,sys}^2\right).$$

(40)

On simplification of Equation (40), we have

$$t_{mt,sys}^{ep}-\overline{Y}\approx \overline{Y}\left({\zeta }_{yt,sys}+{\displaystyle \frac{1}{2}}{\zeta }_{xt,sys}+{\displaystyle \frac{1}{2}}{\zeta }_{yt,sys}{\zeta }_{xt,sys}-{\displaystyle \frac{1}{8}}{\zeta }_{xt,sys}^2\right).$$

(41)

Applying expectation to both sides of Equation (41), we find that the final expression for the bias of the memory-type exponential product estimator based on EWMA statistics, is given by

$$B\left(t_{mt,sys}^{ep}\right)\approx \overline{Y}\left({\displaystyle \frac{1}{2}}\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x-{\displaystyle \frac{1}{8}}\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2\right).$$

(42)

In order to derive the expression of MSE of the memory-type exponential product estimator, simplifying Equation (40) up to the first order, and expanding the exponential series, we have

$$t_{mt,sys}^{ep}\approx \overline{Y}\left(1+{\zeta }_{yt,sys}+{\displaystyle \frac{{\zeta }_{xt,sys}}{2}}\right).$$

(43)

Simplifying, squaring, and taking expectation on both sides of Equation (43), we obtain

$$E{\left(t_{mt,sys}^{er}-\overline{Y}\right)}^2\approx {\overline{Y}}^{2}E\left({\zeta }_{yt,sys}^2+{\displaystyle \frac{1}{4}}{\zeta }_{xt,sys}^2+{\zeta }_{yt,sys}{\zeta }_{xt,sys}\right).$$

(44)

The final expression for the MSE of the memory-type exponential product estimator based on EWMA statistics is given by

$$MSE\left(t_{mt,sys}^{ep}\right)\approx {\overline{Y}}^2\left(\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_x^{\ast }{C}_x^2+\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right).$$

(45)

5. Mathematical Comparisons

In this section, we have compared the conventional estimators with their corresponding memory-type estimators under systematic sampling.

• The proposed memory-type mean estimator based on the EWMA statistic performs better than the conventional mean estimator under systematic sampling if

$$var\left(t_{mt,sys}^0\right)<var\left(t_{0,sys}^y\right).$$

which implies

$$\theta {\overline{Y}}^2\left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\rho }_y^{\ast }{C}_y^2<\theta {\overline{Y}}^2{\rho }_y^{\ast }{C}_y^2$$

or

$$\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)<1$$

• The proposed memory-type ratio estimator based on the EWMA statistic performs better than the conventional ratio estimator under systematic sampling if

$$MSE\left(t_{mt,sys}^r\right)<MSE\left(t_{r,sys}^y\right).$$

which implies

$$\theta {\overline{Y}}^2\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\left({\rho }_y^{\ast }{C}_y^2+{\rho }_x^{\ast }{C}_x^2-2\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)<\theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\rho }_x^{\ast }{C}_x^2-2\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)$$

or

$$\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)<1$$

• The proposed memory-type product estimator based on the EWMA statistic performs better than the conventional product estimator under systematic sampling if

$$MSE\left(t_{mt,sys}^p\right)<MSE\left(t_{p,sys}^y\right).$$

which implies

$$\theta {\overline{Y}}^2\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)\left({\rho }_y^{\ast }{C}_y^2+{\rho }_x^{\ast }{C}_x^2+2\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)<\theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\rho }_x^{\ast }{C}_x^2+2\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)$$

or

$$\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)<1$$

• The memory-type exponential ratio estimator based on the EWMA statistic performs better than the conventional exponential ratio estimator under systematic sampling if

$$MSE\left(t_{mt,sys}^{er}\right)<MSE\left(t_{er,sys}^y\right).$$

which implies

$$\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}{\rho }_x^{\ast }{C}_x^2-\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)<\theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}{\rho }_x^{\ast }{C}_x^2-\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)$$

or

$$\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)<1$$

• The memory-type exponential product estimator based on the EWMA statistic performs better than the conventional exponential product estimator under systematic sampling if

$$MSE\left(t_{mt,sys}^{ep}\right)<MSE\left(t_{ep,sys}^y\right).$$

which implies

$$\theta \left({\displaystyle \frac{\lambda }{2-\lambda }}\right){\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}{\rho }_x^{\ast }{C}_x^2+\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)<\theta {\overline{Y}}^2\left({\rho }_y^{\ast }{C}_y^2+{\displaystyle \frac{1}{4}}{\rho }_x^{\ast }{C}_x^2+\sqrt{{\rho }_y^{\ast }{\rho }_x^{\ast }}{\rho }_{yx}{C}_y{C}_x\right)$$

or

$$\left({\displaystyle \frac{\lambda }{2-\lambda }}\right)<1$$

For practical applications, the choice of estimator depends primarily on the correlation between the study variable and the auxiliary variable. If the correlation is positive, ratio-type estimators are generally preferred, whereas product-type estimators are more suiable when the correlation is negative. Note that, as the EWMA weight constant $\lambda$ ranges from 0 to 1, the proposed memory-type ratio and product estimators will perform better than conventional ratio and product type estimators for $\lambda <1$. The performance of both memory-type and conventional estimators will perform equally for $\lambda = 1.$

6. Simulation Study

We have conducted an extensive simulation study to judge the performance of the proposed estimators with their conventional counterparts. A bivariate normal population is generated using [26] with the below parameters.

$${\mu }_{yx}=\left[\begin{array}{cc}5& 6\end{array}\right], S_y^2=125, S_x^2=100.$$

The intra-class correlations for the simulated populations at various levels of correlation are summarized in

Table 1. Intra-class Correlation of Study Variable and Auxiliary Variable.

ρyx	ρy	ρx	ρyx	ρy	ρx
0.75	0.01691	0.01390	−0.75	0.01179	0.00123
0.80	0.01879	0.01831	−0.80	0.01543	0.01408
0.85	0.02766	0.02635	−0.85	0.02167	0.02181
0.90	0.03571	0.02658	−0.90	0.02599	0.02209
0.95	0.03880	0.03280	−0.95	0.04097	0.04077

.

A well-known mvnorm function is used to generate the populations of x and y under the library of mvtnorm. The following formulas are used to compute the relative efficiency of the considered estimators as follows:

$$RE\left(Est,t_{mt,sys}^0\right)={\displaystyle \frac{\mathit{var}\left(t_{mt,sys}^0\right)}{MSE\left(Es{t}_j\right)}},$$

and

$$MSE\left(Es{t}_j\right)={\displaystyle \frac{1}{\mathrm{50,000}}}{\displaystyle {\sum }_{j=1}^{\mathrm{50,000}}{\left(Est-\overline{Y}\right)}^2,}$$

where $Est=t_{0,sys}^y,t_{r,sys}^y,t_{p,sys}^y,t_{er,sys}^y,t_{ep,sys}^y, t_{mt,sys}^y,t_{mt,sys}^r{t}_{mt,sys}^{er},t_{mt,sys}^p and t_{mt,sys}^{ep}.$

The output of MSEs and relative efficiencies are computed at different levels of correlation along with the different weights of constant using the algorithm of the following simulation:

• Bivariate populations of size N (=5000) are generated with different positive/negative levels of correlation (=0.75, 0.80, 0.85, 0.90, and 0.95) and the true mean of the auxiliary variable is computed. Some robust parameters associated with the auxiliary variable are also computed.
• Different choices of constant λ (=0.1, 0.2, 0.3, 0.5, and 0.75) are selected to assign the weights to the current and previous sample means.
• Different samples of size n (=50, 100, 200, 300, and 500) are selected using systematic sampling from the populations simulated in Step-1.
• Step-3 is repeated 50,000 times and the estimators considered in this study are computed.
• The MSE and RE of each estimator are computed for different sample sizes, different levels of correlation, and different weight constants, and are summarized in

  Table 2. MSE of Conventional and Memory-Type Ratio Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.75$
  		${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$
  0.75	50	386.18	15.77	389.66	34.52	391.99	50.29	394.35	81.82	398.29	129.13
  	100	176.85	6.80	178.44	16.14	179.51	22.94	180.59	36.53	182.40	56.93
  	200	86.28	3.69	87.06	9.23	87.58	12.93	88.10	20.31	88.99	31.39
  	300	52.03	2.37	52.50	3.80	52.81	6.16	53.13	10.89	53.66	17.99
  	500	29.06	1.37	29.32	1.90	29.50	3.26	29.67	5.99	29.97	10.09
  0.80	50	313.85	12.39	316.67	31.14	318.57	43.54	320.49	68.33	323.69	105.51
  	100	139.98	5.10	141.24	14.44	142.09	19.54	142.94	29.75	144.37	45.05
  	200	66.48	2.60	67.08	8.14	67.48	10.73	67.89	15.92	68.56	23.71
  	300	45.30	2.06	45.71	3.49	45.98	5.55	46.26	9.67	46.72	15.84
  	500	26.91	1.26	27.15	1.79	27.32	3.05	27.48	5.57	27.75	9.360
  0.85	50	229.51	8.21	231.58	26.96	232.97	35.16	234.36	51.57	236.71	76.19
  	100	111.75	3.81	112.76	13.15	113.43	16.96	114.11	24.58	115.25	36.01
  	200	47.68	1.66	48.11	7.20	48.40	8.87	48.69	12.19	49.18	17.18
  	300	36.07	1.74	36.39	3.17	36.61	4.91	36.83	8.39	37.20	13.61
  	500	18.71	0.85	18.88	1.38	18.99	2.22	19.11	3.91	19.30	6.450
  0.90	50	159.71	5.20	161.15	23.95	162.11	29.14	163.09	39.53	164.72	55.12
  	100	77.87	2.96	78.57	12.30	79.04	15.26	79.52	21.17	80.31	30.05
  	200	32.34	1.35	32.63	6.89	32.83	8.24	33.02	10.94	33.35	14.98
  	300	20.71	0.80	20.90	2.23	21.02	3.03	21.15	4.62	21.36	7.02
  	500	11.54	0.46	11.64	0.99	11.71	1.45	11.78	2.37	11.90	3.76
  0.95	50	79.04	1.96	79.75	20.71	80.23	22.68	80.71	26.61	81.52	32.50
  	100	37.39	0.85	37.73	10.19	37.95	11.04	38.18	12.74	38.56	15.30
  	200	16.16	0.68	16.31	6.22	16.40	6.90	16.50	8.26	16.67	10.30
  	300	10.19	0.45	10.28	1.88	10.34	2.32	10.41	3.21	10.51	4.55
  	500	5.78	0.27	5.83	0.80	5.87	1.06	5.90	1.59	5.96	2.39

  ,

  Table 3. RE of Conventional and Memory-Type Ratio Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.75$
  		${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$
  0.75	50	1.05	7.76	1.06	6.40	1.07	4.30	1.08	3.10	1.11	1.00
  	100	1.11	8.26	1.12	7.16	1.13	4.94	1.14	3.49	1.17	1.27
  	200	1.26	7.09	1.27	8.95	1.28	6.43	1.29	4.41	1.33	1.90
  	300	1.26	7.15	1.27	11.03	1.28	8.31	1.29	5.88	1.33	3.16
  	500	1.34	8.53	1.35	11.84	1.36	8.93	1.37	6.10	1.41	3.18
  0.80	50	1.27	11.96	1.28	10.60	1.29	8.07	1.30	6.01	1.34	3.47
  	100	1.30	11.63	1.31	11.53	1.32	8.93	1.33	6.72	1.37	4.12
  	200	1.47	11.43	1.48	13.29	1.49	10.34	1.51	7.45	1.55	4.50
  	300	1.46	10.43	1.47	13.79	1.48	10.77	1.50	7.73	1.54	4.71
  	500	1.46	10.70	1.47	14.97	1.48	11.83	1.50	8.55	1.54	5.41
  0.85	50	1.64	19.64	1.65	18.28	1.66	15.01	1.68	11.46	1.73	8.18
  	100	1.84	21.53	1.86	20.43	1.87	16.75	1.88	12.37	1.94	8.69
  	200	2.02	22.14	2.04	23.00	2.05	18.97	2.07	13.91	2.13	9.88
  	300	1.81	18.92	1.83	25.18	1.84	20.72	1.85	14.80	1.91	10.34
  	500	2.06	21.85	2.08	27.37	2.09	22.67	2.11	16.27	2.17	11.57
  0.90	50	2.45	33.77	2.47	32.41	2.49	27.52	2.51	20.74	2.58	15.85
  	100	2.59	38.34	2.61	37.24	2.63	32.06	2.65	24.71	2.73	19.54
  	200	2.74	36.37	2.76	40.08	2.78	34.22	2.81	25.50	2.89	19.64
  	300	2.73	35.57	2.75	39.63	2.77	34.11	2.80	26.07	2.88	20.55
  	500	3.27	45.24	3.30	49.15	3.32	41.50	3.35	29.20	3.45	21.54
  0.95	50	4.54	78.32	4.58	46.96	4.61	37.89	4.65	22.75	4.79	13.68
  	100	4.75	78.29	4.79	50.94	4.82	40.94	4.86	23.94	5.01	13.94
  	200	5.17	86.24	5.22	56.18	5.25	45.24	5.30	26.36	5.45	15.42
  	300	5.41	87.41	5.46	60.35	5.49	48.52	5.54	27.86	5.71	16.03
  	500	5.57	87.94	5.62	60.54	5.65	49.19	5.70	29.50	5.88	18.15

  ,

  Table 4. MSE of Conventional and Memory-Type Exponential Ratio Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.78$
  		${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$
  0.75	50	387.75	17.34	391.24	36.09	394.37	51.86	395.95	83.39	398.32	130.70
  	100	178.12	8.07	179.72	17.41	181.16	24.21	181.89	37.80	182.98	58.20
  	200	87.40	4.81	88.19	10.35	88.89	14.05	89.25	21.43	89.78	32.51
  	300	53.04	3.38	53.52	4.81	53.95	7.17	54.16	11.90	54.49	19.00
  	500	29.66	1.97	29.93	2.50	30.17	3.86	30.29	6.59	30.47	10.69
  0.80	50	315.42	13.96	318.26	32.71	320.80	45.11	322.09	69.90	324.02	107.08
  	100	141.25	6.37	142.52	15.71	143.66	20.81	144.24	31.02	145.10	46.32
  	200	67.60	3.72	68.21	9.26	68.75	11.85	69.03	17.04	69.44	24.83
  	300	46.31	3.07	46.73	4.50	47.10	6.56	47.29	10.68	47.57	16.85
  	500	27.51	1.86	27.76	2.39	27.98	3.65	28.09	6.17	28.26	9.96
  0.85	50	231.08	9.78	233.16	28.53	235.02	36.73	235.97	53.14	237.38	77.76
  	100	113.02	5.08	114.04	14.42	114.95	18.23	115.41	25.85	116.10	37.28
  	200	48.80	2.78	49.24	8.32	49.63	9.99	49.83	13.31	50.13	18.30
  	300	37.08	2.75	37.41	4.18	37.71	5.92	37.86	9.40	38.09	14.62
  	500	19.31	1.45	19.48	1.98	19.64	2.82	19.72	4.51	19.84	7.05
  0.90	50	161.28	6.77	162.73	25.52	164.03	30.71	164.69	41.10	165.68	56.69
  	100	79.14	4.23	79.85	13.57	80.49	16.53	80.81	22.44	81.30	31.32
  	200	33.46	2.47	33.76	8.01	34.03	9.36	34.17	12.06	34.37	16.11
  	300	21.72	1.81	21.92	3.24	22.09	4.04	22.18	5.63	22.31	8.03
  	500	12.14	1.06	12.25	1.59	12.35	2.05	12.40	2.97	12.47	4.36
  0.95	50	80.61	3.53	81.34	22.28	81.99	24.25	82.31	28.18	82.81	34.07
  	100	38.66	2.12	39.01	11.46	39.32	12.31	39.48	14.01	39.71	16.57
  	200	17.28	1.80	17.44	7.34	17.58	8.02	17.65	9.38	17.75	11.42
  	300	11.20	1.46	11.30	2.89	11.39	3.33	11.44	4.22	11.51	5.56
  	500	6.38	0.87	6.44	1.40	6.49	1.66	6.51	2.19	6.55	2.99

  ,

  Table 5. RE of Conventional and Memory-Type Exponential Ratio Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.78$
  		${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$
  0.75	50	1.01	7.72	1.02	6.36	1.03	4.26	1.04	3.06	1.01	1.05
  	100	1.06	8.21	1.07	7.11	1.08	4.89	1.09	3.44	1.06	1.09
  	200	1.18	7.01	1.19	8.87	1.20	6.35	1.21	4.33	1.18	1.22
  	300	1.08	6.97	1.09	10.85	1.10	8.13	1.11	5.70	1.08	1.11
  	500	1.14	8.33	1.15	11.64	1.16	8.73	1.17	5.90	1.14	1.18
  0.80	50	1.03	11.72	1.04	10.36	1.05	7.83	1.06	5.77	1.03	1.06
  	100	1.01	11.34	1.02	11.24	1.03	8.64	1.04	6.43	1.01	1.04
  	200	1.14	11.10	1.15	12.96	1.16	10.01	1.17	7.12	1.14	1.18
  	300	1.08	10.05	1.09	13.41	1.10	10.39	1.11	7.35	1.08	1.11
  	500	1.04	10.28	1.05	14.55	1.06	11.41	1.07	8.13	1.04	1.07
  0.85	50	1.17	19.17	1.18	17.81	1.19	14.54	1.20	10.99	1.17	1.21
  	100	1.33	21.02	1.34	19.92	1.35	16.24	1.36	11.86	1.33	1.37
  	200	1.47	21.59	1.48	22.45	1.50	18.42	1.51	13.36	1.47	1.52
  	300	1.21	18.32	1.22	24.58	1.23	20.12	1.24	14.20	1.21	1.25
  	500	1.42	21.21	1.43	26.73	1.44	22.03	1.46	15.63	1.42	1.46
  0.90	50	1.76	33.08	1.78	31.72	1.79	26.83	1.80	20.05	1.76	1.82
  	100	1.86	37.61	1.88	36.51	1.89	31.33	1.91	23.98	1.86	1.92
  	200	1.96	35.59	1.98	39.30	1.99	33.44	2.01	24.72	1.96	2.02
  	300	1.91	34.75	1.93	38.81	1.94	33.29	1.96	25.25	1.91	1.97
  	500	2.40	44.37	2.42	48.28	2.44	40.63	2.46	28.33	2.40	2.48
  0.95	50	3.63	77.41	3.66	46.05	3.69	36.98	3.72	21.84	3.63	3.74
  	100	3.80	77.34	3.83	49.99	3.86	39.99	3.90	22.99	3.80	3.92
  	200	4.17	85.24	4.21	55.18	4.24	44.24	4.28	25.36	4.17	4.30
  	300	4.37	86.37	4.41	59.31	0.30	47.48	0.30	26.82	4.37	0.75
  	500	4.48	86.85	4.52	59.45	4.56	48.10	4.59	28.41	4.48	4.62

  ,

  Table 6. MSE of Conventional and Memory-Type Product Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.75$
  		${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$
  −0.75	50	411.83	23.93	413.48	26.80	413.89	32.54	415.55	44.03	418.04	61.27
  	100	174.83	10.90	175.53	12.67	175.70	16.20	176.41	23.27	177.47	33.87
  	200	84.68	5.85	85.02	7.11	85.10	9.64	85.44	14.71	85.96	22.31
  	300	52.98	3.52	53.19	4.29	53.25	5.84	53.46	8.93	53.78	13.56
  	500	31.53	1.77	31.66	1.93	31.69	2.24	31.81	2.87	32.01	3.81
  −0.80	50	346.97	19.83	348.36	22.70	348.71	28.45	350.10	39.94	352.20	57.17
  	100	149.17	9.49	149.77	11.26	149.92	14.79	150.52	21.86	151.42	32.45
  	200	59.41	4.46	59.65	5.73	59.71	8.26	59.95	13.33	60.31	20.92
  	300	43.83	3.16	44.01	3.93	44.05	5.47	44.23	8.56	44.49	13.19
  	500	24.57	1.47	24.67	1.63	24.69	1.94	24.79	2.57	24.94	3.51
  −0.85	50	231.29	14.42	232.22	17.29	232.45	23.03	233.38	34.52	234.78	51.76
  	100	113.80	7.67	114.26	9.44	114.37	12.97	114.83	20.03	115.52	30.63
  	200	48.50	3.88	48.69	5.15	48.74	7.68	48.94	12.75	49.23	20.35
  	300	33.11	2.57	33.24	3.35	33.28	4.89	33.41	7.98	33.61	12.61
  	500	19.96	1.21	20.04	1.37	20.06	1.68	20.14	2.31	20.26	3.25
  −0.90	50	163.93	11.09	164.59	13.96	164.75	19.70	165.41	31.19	166.40	48.42
  	100	74.58	5.77	74.88	7.54	74.95	11.07	75.25	18.14	75.70	28.73
  	200	34.63	3.19	34.77	4.45	34.80	6.98	34.94	12.05	35.15	19.65
  	300	22.38	2.02	22.47	2.79	22.49	4.34	22.58	7.42	22.72	12.06
  	500	13.73	0.90	13.78	1.06	13.80	1.37	13.85	2.00	13.94	2.94
  −0.95	50	78.99	7.14	79.31	10.01	79.39	15.75	79.70	27.24	80.18	44.48
  	100	38.22	3.90	38.37	5.66	38.41	9.19	38.56	16.26	38.80	26.86
  	200	15.85	2.23	15.91	3.49	15.93	6.02	15.99	11.09	16.09	18.69
  	300	10.77	1.44	10.81	2.21	10.82	3.75	10.87	6.84	10.93	11.47
  	500	6.27	0.66	6.30	0.97	6.30	1.59	6.33	2.83	6.36	4.69

  ,

  Table 7. RE of Conventional and Memory-Type Product Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.75$
  		${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$	${\mathit{t}}_{\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{p}}$
  −0.75	50	1.03	17.56	1.03	10.22	1.04	6.67	1.04	3.14	1.05	1.14
  	100	1.23	20.73	1.23	13.39	1.24	9.84	1.24	6.31	1.25	3.31
  	200	1.23	21.17	1.23	13.83	1.24	10.28	1.24	6.75	1.25	3.75
  	300	1.24	21.30	1.24	13.95	1.25	10.41	1.25	6.88	1.26	3.88
  	500	1.24	22.13	1.24	14.79	1.25	11.24	1.25	7.71	1.26	4.71
  −0.80	50	1.47	26.31	1.48	18.97	1.48	15.42	1.48	11.89	1.49	8.89
  	100	1.35	23.01	1.36	15.66	1.36	12.12	1.36	8.59	1.37	5.59
  	200	1.48	24.77	1.49	17.43	1.49	13.88	1.49	10.35	1.50	7.35
  	300	1.42	26.22	1.43	18.87	1.43	15.32	1.43	11.80	1.44	8.80
  	500	1.44	26.43	1.45	19.08	1.45	15.54	1.45	12.01	1.46	9.01
  −0.85	50	1.83	32.48	1.84	25.14	1.84	21.59	1.85	18.06	1.86	15.06
  	100	1.82	31.23	1.83	23.88	1.83	20.34	1.84	16.81	1.85	13.81
  	200	2.14	36.08	2.15	28.74	2.15	25.19	2.16	21.66	2.17	18.66
  	300	2.18	36.21	2.19	28.86	2.19	25.32	2.20	21.79	2.21	18.79
  	500	2.05	36.86	2.06	29.52	2.06	25.97	2.07	22.44	2.08	19.44
  −0.90	50	2.67	47.55	2.68	40.21	2.68	36.66	2.69	33.13	2.71	30.13
  	100	2.69	45.33	2.70	37.99	2.70	34.44	2.71	30.91	2.73	27.91
  	200	3.00	50.22	3.01	42.87	3.02	39.33	3.03	35.80	3.05	32.80
  	300	2.89	50.68	2.90	43.33	2.90	39.79	2.92	36.26	2.93	33.26
  	500	2.66	52.25	2.67	44.91	2.67	41.36	2.68	37.83	2.70	34.83
  −0.95	50	5.16	88.54	5.18	81.20	5.19	77.65	5.21	74.12	5.24	71.12
  	100	6.05	92.93	6.07	85.59	6.08	82.04	6.10	78.51	6.14	75.51
  	200	5.26	93.77	5.28	86.43	5.29	82.88	5.31	79.35	5.34	76.35
  	300	5.88	94.37	5.90	87.03	5.91	83.48	5.93	79.95	5.97	76.95
  	500	5.95	95.99	5.97	88.65	5.98	85.10	6.00	81.57	6.04	78.57

  ,

  Table 8. MSE of Conventional and Memory-Type Exponential Product Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.78$
  		${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$
  −0.75	50	410.26	22.36	412.31	25.23	414.37	30.97	417.69	42.46	410.26	421.45
  	100	173.56	9.63	174.43	11.40	175.30	14.93	176.70	22.00	173.56	178.29
  	200	83.56	4.73	83.98	5.99	84.40	8.52	85.07	13.59	83.56	85.84
  	300	51.97	2.51	52.23	3.28	52.49	4.83	52.91	7.92	51.97	53.39
  	500	30.93	1.17	31.08	1.33	31.24	1.64	31.49	2.27	30.93	31.77
  −0.80	50	345.40	18.26	347.13	21.13	348.86	26.88	351.65	38.37	345.40	354.82
  	100	147.90	8.22	148.64	9.99	149.38	13.52	150.58	20.59	147.90	151.93
  	200	58.29	3.34	58.58	4.61	58.87	7.14	59.35	12.21	58.29	59.88
  	300	42.82	2.15	43.03	2.92	43.25	4.46	43.60	7.55	42.82	43.99
  	500	23.97	0.87	24.09	1.03	24.21	1.34	24.40	1.97	23.97	24.62
  −0.85	50	229.72	12.85	230.87	15.72	232.02	21.46	233.88	32.95	229.72	235.98
  	100	112.53	6.40	113.09	8.17	113.66	11.70	114.57	18.76	112.53	115.60
  	200	47.38	2.76	47.62	4.03	47.85	6.56	48.24	11.63	47.38	48.67
  	300	32.10	1.56	32.26	2.34	32.42	3.88	32.68	6.97	32.10	32.98
  	500	19.36	0.61	19.46	0.77	19.55	1.08	19.71	1.71	19.36	19.89
  −0.90	50	162.36	9.52	163.17	12.39	163.99	18.13	165.30	29.62	162.36	166.79
  	100	73.31	4.50	73.68	6.27	74.04	9.80	74.64	16.87	73.31	75.31
  	200	33.51	2.07	33.68	3.33	33.85	5.86	34.12	10.93	33.51	34.42
  	300	21.37	1.01	21.48	1.78	21.58	3.33	21.76	6.41	21.37	21.95
  	500	13.13	0.30	13.20	0.46	13.26	0.77	13.37	1.40	13.13	13.49
  −0.95	50	77.42	5.57	77.81	8.44	78.20	14.18	78.82	25.67	77.42	79.53
  	100	36.95	2.63	37.13	4.39	37.32	7.92	37.62	14.99	36.95	37.96
  	200	14.73	1.11	14.80	2.37	14.88	4.90	15.00	9.97	14.73	15.13
  	300	9.76	0.43	9.81	1.20	9.86	2.74	9.94	5.83	9.76	10.03
  	500	5.67	0.06	5.70	0.37	5.73	0.99	5.77	2.23	5.67	5.82

  and

  Table 9. RE of Conventional and Memory-Type Exponential Product Estimators using EWMA Statistic.

  ${\mathit{\rho }}_{\mathit{y}\mathit{x}}$	n	$\mathit{\lambda }$ = 0.1		$\mathit{\lambda }=0.2$		$\mathit{\lambda }=0.3$		$\mathit{\lambda }=0.5$		$\mathit{\lambda }=0.78$
  		${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$	${\mathit{t}}_{\mathit{e}\mathit{p},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{p}}$
  −0.75	50	1.27	17.80	1.28	10.46	1.28	6.91	1.29	3.38	1.30	1.38
  	100	1.38	20.88	1.39	13.54	1.39	9.99	1.40	6.46	1.42	3.46
  	200	1.33	21.27	1.34	13.93	1.34	10.38	1.35	6.85	1.37	3.85
  	300	2.18	22.24	2.19	14.89	2.20	11.35	2.22	7.82	2.24	4.82
  	500	1.80	22.69	1.81	15.35	1.82	11.80	1.83	8.27	1.85	5.27
  −0.80	50	2.30	27.14	2.31	19.80	2.32	16.25	2.34	12.72	2.36	9.72
  	100	2.33	23.99	2.34	16.64	2.35	13.10	2.37	9.57	2.39	6.57
  	200	2.60	25.89	2.61	18.55	2.63	15.00	2.65	11.47	2.67	8.47
  	300	2.69	27.49	2.70	20.14	2.72	16.59	2.74	13.07	2.76	10.07
  	500	2.85	27.84	2.86	20.49	2.88	16.95	2.90	13.42	2.93	10.42
  −0.85	50	3.39	34.04	3.41	26.70	3.42	23.15	3.45	19.62	3.48	16.62
  	100	3.53	32.94	3.55	25.59	3.57	22.05	3.59	18.52	3.63	15.52
  	200	3.99	37.93	4.01	30.59	4.03	27.04	4.06	23.51	4.10	20.51
  	300	4.18	38.21	4.20	30.86	4.22	27.32	4.26	23.79	4.29	20.79
  	500	4.19	39.00	4.21	31.66	4.23	28.11	4.27	24.58	4.30	21.58
  −0.90	50	4.96	49.84	4.98	42.50	5.01	38.95	5.05	35.42	5.10	32.42
  	100	5.12	47.76	5.15	40.42	5.17	36.87	5.21	33.34	5.26	30.34
  	200	5.58	52.80	5.61	45.45	5.64	41.91	5.68	38.38	5.73	35.38
  	300	5.61	53.40	5.64	46.05	5.67	42.51	5.71	38.98	5.76	35.98
  	500	5.53	55.12	5.56	47.78	5.59	44.23	5.63	40.70	5.68	37.70
  −0.95	50	8.17	91.55	8.21	84.21	8.25	80.66	8.32	77.13	8.39	74.13
  	100	9.21	96.09	9.26	88.75	9.30	85.20	9.38	81.67	9.46	78.67
  	200	8.56	97.07	8.60	89.73	8.65	86.18	8.71	82.65	8.79	79.65
  	300	9.33	97.82	9.38	90.48	9.42	86.93	9.50	83.40	9.58	80.40
  	500	9.54	99.58	9.59	92.24	9.64	88.69	9.71	85.16	9.80	82.16

  , respectively.

Table 2, Table 4, Table 6 and Table 8, together with Table 3, Table 5, Table 7 and Table 9, represent the MSE and RE of the conventional and memory-type estimators based on EWMA statistics for different combinations of sample size, correlation, and smoothing constant λ. The results indicate that the MSE of memory-type estimators decreases as the sample size increases for fixed values of λ and correlations. It is clear that, regardless of the degree of positive correlation and weight constants, the memory-type ratio and exponential ratio estimators beat the conventional ratio and exponential ratio estimators. For varying degrees of negative correlation and weight constants, the memory-type product and exponential product estimators performed better than the traditional product and exponential product estimators. The values of RE further confirm that the proposed memory-type estimators are generally more efficient than the conventional estimators across various n, λ, and correlations. The efficiency gain is more pronounced when the correlation between the variables is strong. In addition, the results suggest that smaller values of λ, which assign higher weight to the past observations in EWMA statistics, tend to improve the estimator performance in time-scaled survey settings. However, when λ = 1, the EWMA statistic reduces to the current sample mean, and therefore, the proposed estimators perform similar to the conventional estimators.

Overall, the simulation results demonstrate that the proposed memory-type estimators, which employ the EWMA statistic, are effective and useful for estimating the population means using systematic sampling, particularly for time-scaled surveys where integrating past information can improve estimation accuracy.

7. Real Data Application

We considered a real dataset from [27] to assess the performance of the proposed memory-type estimators over the conventional estimators based on the EWMA statistic in systematic sampling. Density is considered to be the main study variable Y, while stiffness is taken to be the auxiliary variable X with the parameters given as

$$\overline{Y}=15.47 \mathrm{a}\mathrm{n}\mathrm{d} \overline{X}=\mathrm{34,666.83},$$

$${ S}_y^2=34.023, { S}_x^2=\mathrm{643,900,000} \mathrm{a}\mathrm{n}\mathrm{d} {\rho }_{YX}=0.890.$$

We only considered the ratio- and exponential ratio-type estimators as the correlation coefficients between the study variable and the auxiliary data are positive. We considered regular time intervals of 25 samples, each of size 5, selected using systematic sampling. The chosen samples, MSEs and REs of conventional ratio, exponential ratio and memory-type ratio, and exponential estimators are presented in

Table 10. MSEs of Conventional and Memory-Type Ratio and Exponential Ratio Estimators for Real Data.

Sr. No.	${\mathit{t}}_{0,\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$
1	68.7185	30.5384	28.9313	24.7317	30.3233	25.7541
2	37.7675	36.7046	67.5823	34.1794	68.9742	35.2027
3	51.6425	44.7978	22.8203	25.3045	24.2122	26.3271
4	46.3615	42.0463	45.7473	27.3056	47.1393	28.3285
5	47.4835	40.2644	24.7093	25.6263	26.1014	26.6492
6	44.9885	38.3038	57.6073	28.6454	58.9993	29.6689
7	60.8835	48.3072	37.7293	30.7022	39.1214	31.7258
8	50.9575	49.0786	45.6536	33.0132	47.0455	34.0364
9	53.0875	49.4399	39.4553	33.5643	40.8472	34.5872
10	91.2885	64.2592	23.6438	25.3122	25.0355	26.3351
11	91.3995	75.4035	45.8681	29.9074	47.2674	30.9307
12	64.1825	61.9562	21.5702	28.9584	22.9629	29.9814
13	55.7315	52.2584	17.4337	26.9393	18.8253	27.9625
14	49.2895	45.8747	68.0292	28.8458	69.4215	29.8686
15	39.4500	38.8434	79.0484	29.8827	80.4443	30.9054
16	42.9715	34.5585	20.1364	27.4625	21.5288	28.4857
17	44.7125	41.4556	40.3901	30.6509	41.7822	31.6732
18	49.2285	40.2576	27.4444	28.6968	28.8368	29.7198
19	62.2985	50.1261	45.7974	33.3659	47.1892	34.3882
20	56.0295	50.966	48.0564	35.6674	49.4485	36.6904
21	48.2185	47.1334	29.3413	33.3743	30.7333	34.3977
22	46.0785	45.7283	86.5196	38.5845	87.9117	39.6072
23	55.5175	47.5342	44.1709	38.9742	45.5626	39.9784
24	48.5755	47.1536	55.0264	41.3335	56.4185	42.3561
25	44.2025	43.2739	23.3809	35.3115	24.7724	36.3341

and

Table 11. REs of Conventional and Memory-Type Ratio and Exponential Ratio Estimators for Real Data.

Sr. No.	${\mathit{t}}_{0,\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{r}}$	${\mathit{t}}_{\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{y}}$	${\mathit{t}}_{\mathit{m}\mathit{t},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{r}}$
1	0.4444	1.0000	1.0555	1.2348	1.0071	1.1858
2	0.9719	1.0000	0.5431	1.0739	0.5321	1.0427
3	0.8675	1.0000	1.9631	1.7703	1.8502	1.7016
4	0.9069	1.0000	0.9191	1.5398	0.8920	1.4842
5	0.8480	1.0000	1.6295	1.5712	1.5426	1.5109
6	0.8514	1.0000	0.6649	1.3372	0.6492	1.2910
7	0.7934	1.0000	1.2804	1.5734	1.2348	1.5226
8	0.9631	1.0000	1.0750	1.4866	1.0432	1.4419
9	0.9313	1.0000	1.2531	1.4730	1.2104	1.4294
10	0.7039	1.0000	2.7178	2.5387	2.5667	2.4401
11	0.8250	1.0000	1.6439	2.5212	1.5953	2.4378
12	0.9653	1.0000	2.8723	2.1395	2.6981	2.0665
13	0.9377	1.0000	2.9976	1.9399	2.7760	1.8689
14	0.9307	1.0000	0.6743	1.5903	0.6608	1.5359
15	0.9846	1.0000	0.4914	1.2999	0.4829	1.2568
16	0.8042	1.0000	1.7162	1.2584	1.6052	1.2132
17	0.9272	1.0000	1.0264	1.3525	0.9922	1.3089
18	0.8178	1.0000	1.4669	1.4029	1.3960	1.3546
19	0.8046	1.0000	1.0945	1.5023	1.0622	1.4577
20	0.9096	1.0000	1.0605	1.4289	1.0307	1.3891
21	0.9775	1.0000	1.6064	1.4123	1.5336	1.3702
22	0.9924	1.0000	0.5285	1.1851	0.5202	1.1545
23	0.8562	1.0000	1.0761	1.2196	1.0433	1.1890
24	0.9707	1.0000	0.8569	1.1408	0.8358	1.1133
25	0.9790	1.0000	1.8508	1.2255	1.7469	1.1910

.

The results presented in Table 10 and Table 11 show that the proposed memory-type estimators achieve lower MSE and higher RE than the conventional estimators for systematic sampling. The estimators without the auxiliary information have higher variability as compared to the other estimators utilizing auxiliary information. Moreover, the memory-type ratio and exponential ratio estimators exhibit lower variance than the conventional ratio and exponential ratio estimators, except in a few cases. Based on the 25 samples, the average MSE values for the conventional ratio, exponential ratio, memory-type ratio, and memory-type exponential ratio estimators are 384.54, 397.98, 27.54, and 32.65, respectively. These results show that the memory-type estimators performed much better than the conventional estimators. In particular, the proposed ratio estimator provides the best performance among all the estimators for estimating the population mean using systematic sampling for time-scaled surveys.

8. Conclusions

In this study, we have emphasized the significance of mean estimation in survey sampling, as it serves as a crucial indicator for understanding population variability. To address this, we have suggested memory-type estimators based on EWMA statistic for time-scaled surveys using systematic sampling. By deriving the expressions of bias and mean square errors for these estimators, we explored their performance under different scenarios. We identified mathematical conditions in which our proposed estimators outperformed conventional estimators. Simulation and real data applications validated the superiority of our proposed memory-type estimators over conventional estimators at different levels of correlation and weight constants. Our findings underscore the importance of efficient estimators in drawing accurate and reliable conclusions from diverse data populations.

In this study, we considered only one auxiliary variable for mean estimation using memory-type estimators. In future, more than one auxiliary variable can be used in multi-phase sampling. Additionally, the proposed methodology can be extended to other sampling designs, such as ranked set sampling, and adapted to handle non-response, which are common challenges in time-scaled surveys.