Compromised-Imputation and EWMA-Based Memory-Type Mean Estimators Using Quantile Regression

Abstract

Survey sampling commonly faces the challenge of missing information, prompting the development of various imputation-based mean estimation methods to address this concern. Among these, ratio-type regression estimators have been devised to compute population parameters using only current sample data. However, recent pioneering research has revolutionized this approach by integrating both past and current sample information through the application of exponentially weighted moving averages (EWMA). This groundbreaking methodology has given rise to the creation of memory-type estimators tailored for surveys conducted over time. In this paper, we present novel imputation-based memory-type mean estimators that leverage EWMA and quantile regression to handle missing observations. For the performance assessment between traditional, adapted and proposed estimators, real-life time-scaled datasets related to the stock market and humidity are considered. Furthermore, we conduct a simulation study using an asymmetric dataset to further validate the effectiveness of the introduced estimators. The humidity data results show that the proposed estimators ($T_{p_{q_{\left(0.25\right)}}}$, $T_{p_{q_{\left(0.45\right)}}}$, $T_{p_{q_{\left(0.25\right)}}}^{{}^{\prime }}$, $T_{p_{q_{\left(0.45\right)}}}^{{}^{\prime }}$, $T_{p_{q_{\left(0.25\right)}}}^{{}^{″}}$, $T_{p_{q_{\left(0.45\right)}}}^{{}^{″}}$) have the minimum MSE. The stock market data results show that the proposed estimators ($T_{p_{q_{\left(0.85\right)}}}$, $T_{p_{q_{\left(0.85\right)}}}^{{}^{\prime }}$, $T_{p_{q_{\left(0.85\right)}}}^{{}^{″}}$) also have the minimum MSE. Additionally, the simulation results demonstrate that the proposed estimators ($T_{p_{q_{\left(0.45\right)}}}$, $T_{p_{q_{\left(0.45\right)}}}^{{}^{\prime }}$, $T_{p_{q_{\left(0.45\right)}}}^{{}^{″}}$) have the minimum MSE when compared to traditional and adapted estimators. Therefore, in conclusion, the use of the proposed estimators is recommended over traditional and adapted ones.

1. Introduction

The occurrence of missing data within a dataset can be attributed to various factors. These include human or machine errors, unanswered questions during surveys, equipment malfunctions, latent variables, and misjudgments. By employing the imputation approach, we can investigate the root causes behind the presence of missing data in the dataset. This entails identifying whether the missingness is inherent to the fundamental observations or originates from the broader population that serves as the source of the sample. Before Rubin’s [1] pioneering work, the consideration of whether missingness was connected to underlying values was largely disregarded. Rubin’s classifications, as outlined in reference [1], are therefore widely recognized as the cornerstone of missing data mechanism understanding and represent fundamental divisions in the realm of data incompleteness. Rubin’s work primarily categorized its concepts into three distinct mechanisms based on whether a particular missing value is linked to fundamental observations. These mechanisms are referred to as Missing Completely at Random (MCAR), Missing at Random (MAR), and Not Missing at Random (NMAR).

Prior to employing a suitable statistical methodology to analyze a dataset with missing values, it is crucial to ascertain which of the aforementioned conditions is at play within the dataset. A dataset is labeled as MCAR when the probability of data being missing is not connected to any observable or unobservable measurements present in the data. Conversely, a dataset is termed as MAR when the probability of missing values is tied to observed measurements but not to unobserved ones. Lastly, a dataset is assigned to the NMAR category when the probability of missing data is influenced by both observable and unobservable measurements. For a more comprehensive understanding, further details can be found in Gordon’s work [2].

The arithmetic mean stands as a significant measure of central tendency, renowned as one of the prevalent and widely recognized types of averages, with applications spanning across various scientific and artistic domains. Consequently, the estimation of means holds a pivotal position not solely within survey sampling but also across numerous other fields. Given these considerations, it becomes imperative to enhance mean estimation techniques to achieve more accurate estimates, particularly in scenarios involving missing observations. The occurrence of missing information is a prevalent challenge in sampling surveys. To address the predicament of missing observations, imputation methods offer a promising avenue for resolution. For instance, many authors developed imputation-based mean estimators. For example, Kadilar and Cingi [3] provided ratio-type mean estimators. Diana and Perri [4] developed regression-type mean estimators. Al-Omari et al. [5] extended the idea of Kadilar and Cingi [3] and suggested traditional ratio-type estimators using known correlation coefficients. Al-Omari and Bouza [6] attempted to improve the performance of imputed mean estimators by using ranked set sampling. Bhushan and Pandey [7,8] defined an optimal imputation strategy for mean estimation. Prasad [9] developed exponential-type estimators for missing observations. Taking motivation from Zaman Bulut [10,11], Shahzad et al. [12] constructed imputation-based mean estimators using robust covariance matrices. For more about imputation-based mean estimators, interested readers can refer to Bushan et al. [13,14].

It is worth noting that extreme observations may also mislead the mean parameter results. Recently, a novel approach using quantile regression was introduced by Shahzad et al. [15]. This approach is particularly effective when dealing with non-normally distributed data contaminated by outliers. Anas et al. [16] extended this concept by employing L-moments to develop central-tendency estimators for estimating the mean parameter in the presence of outliers within a non-normal dataset under simple random sampling. This integration of quantile regression proved especially advantageous for accurately estimating the population mean even when dealing with missing observations. Shahzad et al. [17,18] proposed a group of robust regression estimators under a stratified and systematic random sampling framework using quantile regression. Taking a step further, Shahzad et al. [19] harnessed the potential of quantile regression in conjunction with measures derived from the minimum covariance determinant to establish another group of estimators. By extending Shahzad et al.’s [17] work, Koc and Koc [20], and Yadav and Prasad [21] also provided regression- and exponential-type mean estimators using quantile regression coefficients.

However, no studies are available on mean estimation for time-scaled surveys using quantile regression in the presence of missing observations. Therefore, the aims and objectives of the present study are to develop mean estimators for time-based observations using EWMA for handling missing data. To achieve these aims and objectives, we employ both EWMA and quantile regression to estimate the population mean. To the best of our knowledge, no studies are available that utilize both EWMA and quantile regression to calculate population mean estimates. Thus, the present study is considered novel. It is noteworthy that this study proposes a class of compromised-imputation- and EWMA-based quantile-regression-type mean estimators under a simple random sampling scheme for handling missing observations. The major advantage of the proposed estimators is their efficiency compared to the estimators used in previous research, as illustrated in Section 5.

The rest of the article is organized into five distinct sections, with each section contributing to the exploration of imputation-based memory-type mean estimators utilizing the EWMA statistic and quantile regression. Section 2 provides foundational concepts about the EWMA statistic, subsequently constructing a class of traditional imputation-based memory-type mean estimators to tackle the issue of missing observations. In Section 3, adapted estimators are introduced, employing both the EWMA statistic and quantile regression. These adapted estimators are built on the compromised-imputation- and ratio-type regression techniques introduced in Section 2. Section 4 presents the proposed estimators by unveiling a streamlined approach, which involves discarding the ratio component found in both the adapted and traditional estimators. This results in a more efficient solution. The numerical study is undertaken in Section 5, wherein real-world data, including humidity levels in Karachi and HBL share prices, along with simulated asymmetric data, are considered as diverse populations. Lastly, Section 6 provides a conclusion and a forward-looking perspective, expanding the work and suggesting potential avenues for future research in this domain.

2. EWMA Statistics and Traditional Imputation-Based Estimators

Roberts [22] was the first to introduce the EWMA statistic as a means of monitoring changes in the process mean. The EWMA statistic can be represented as follows:

$$Z_t=\lambda \overline{y}+(1-\lambda )Z_{t-1}$$

(1)

In Equation (1), $\overline{y}$ represents the mean of the current sample of the study variable, while $\lambda$ denotes the smoothing constant—a weight assigned to the observations. A higher $\lambda$ value indicates greater emphasis on recent data, while less importance is given to past values. Conversely, a smaller $\lambda$ value gives more weight to historical data and less to the latest observations. The range of $\lambda$ lies between 0 and 1. When $\lambda$ is equal to 1, the EWMA statistic becomes equivalent to the conventional sample mean, as it entirely focuses on the most recent values. In Equation (1), t indicates the sample number and the term $Z_{t-1}$ refers to past observations. The starting value, $Z_0$, is determined by taking the expected mean or average of the prior sample.

A majority of the estimators available in the existing literature rely solely on the present sample. However, by incorporating data from both the current and previous samples (such as ${\overline{y}}_t,{\overline{y}}_{t-1},{\overline{y}}_{t-2}$ for respective times $t,t-1,t-2$, and so forth), the efficiency of the estimator can be significantly improved. This is particularly crucial in scenarios where surveys are carried out at regular intervals, like monthly or annual surveys. Noor-ul-Amin [23,24] introduced memory-based ratio and product estimators for the population mean using a Hybrid Exponentially Weighted Moving Average (HEWMA) in time-scaled surveys conducted under a simple Random Sampling (SRS) approach. Building upon this, Aslam et al. [25,26] defined similar product and ratio estimators within the context of ranked-based and stratified sampling methods, respectively. Further exploration by Singh et al. [27] led to the development of an EWMA-based shrinkage estimator for the population mean in quality control processes. In the realm of time-based surveys, readers interested in this topic can refer to Chhaparwal and Kumar [28] and Qureshi et al. [29]. Recently, Bhushan et al. [30] introduced a novel approach to crafting EWMA-type estimators using logarithms. Logarithms hold wide-ranging significance in practical applications, serving as a means of measuring various phenomena. They are used, for instance, to assess earthquake magnitudes on the Richter scale, acidity levels on pH scales, and sound intensity in decibels. Furthermore, logarithms play a pivotal role in quantifying exponential decline and growth, as evident in areas such as interest rates, bacterial proliferation in a Petri dish, and the radioactive decay process in carbon dating. Given the extensive applicability of logarithms, Bhushan et al. [30,31] underscore the importance of investigating estimators based on logarithmic principles. Drawing inspiration from these studies, we will now present EWMA-based imputed traditional mean estimators in the following section.

Consider a variable denoted as Y, which serves as the focal study variable originating from a confined population denoted as U. This variable is defined across the entire population U and encompasses a range of N distinguishable units. Additionally, let X be an auxiliary variable employed to estimate $\overline{Y}$. For missing observations in study and auxiliary variables $(X,Y)$, the respective imputed EWMA-based sample mean estimators are:

$$Q_{t_r}=\lambda {\overline{x}}_r+(1-\lambda )Q_{t_r-1}$$

(2)

$$Z_{t_r}=\lambda {\overline{y}}_r+(1-\lambda )Z_{t_r-1}$$

(3)

Furthermore, the EWMA-based non-imputed sample mean of X is

$$Q_t=\lambda {\overline{x}}_r+(1-\lambda )Q_{t-1}$$

(4)

Let r represent the units that have responded within a sample, i.e., $(r<n)$. It is pertinent to highlight the three scenarios of EWMA-based traditional ratio-type regression estimators in the context of missing data:

• Case I: missing values exclusively exist within the target variable.
• Case II: both $(X,Y)$ contain missing values and ${\overline{X}}_a$ is known.
• Case III: both $(X,Y)$ contain missing values and ${\overline{X}}_a$ is unknown.

For the objective of estimating the mean, three distinct estimators employing ratio and mean imputation approaches are formulated for the aforementioned cases as follows:

$$T_1=\frac{\left[Z_{t_r}+b({\overline{X}}_a-Q_t)\right]}{Q_t}{\overline{X}}_a.$$

(5)

In Equation (5), $Q_t$ signifies an EWMA sample mean that remains non-imputed. The coefficient b represents the OLS regression coefficient, while ${\overline{X}}_a$ denotes the population mean of the auxiliary variable. The Mean Squared Error (MSE) of $T_1$ can be expressed as follows:

$$MSE\left(T_1\right)={\psi }_2^{*}{S}_y^2+{\psi }_1^{*}\left(R^2{S}_x^2-{\rho }_a^2{S}_y^2\right)$$

(6)

where ${\psi }_1^{*}={\displaystyle \frac{\lambda }{2-\lambda }}\left(\frac{1}{n}-\frac{1}{N}\right)$ and ${\psi }_2^{*}={\displaystyle \frac{\lambda }{2-\lambda }}\left(\frac{1}{r}-\frac{1}{N}\right)$.

In cases where both variables $(X,Y)$ turn out to be imputed, then the traditional mean estimator is

$$T_2=\frac{\left[Z_{t_r}+b({\overline{X}}_a-Q_{t_r})\right]}{Q_{t_r}}{\overline{X}}_a$$

(7)

In Equation (7), $Q_{t_r}$ is an imputation-based EWMA sample mean of ${\overline{X}}_a$. The MSE of $T_2$ can be expressed as

$$MSE\left(T_2\right)={\psi }_2^{*}\left[S_y^2(1-{\rho }_a^2)+R^2{S}_x^2\right].$$

(8)

For the MCAR scenario, the traditional estimator $T_3$ is

$$T_3=\frac{\left[Z_{t_r}+b(\overline{x}-Q_{t_r})\right]}{Q_{t_r}}\overline{x}.$$

(9)

In Equation (9), $\overline{x}$ represents the first-stage sample mean because ${\overline{X}}_a$ is unknown in this case. The MSE of $T_3$ can be expressed as:

$$MSE\left(T_3\right)={\psi }_2^{*}{S}_y^2+{\psi }_3^{*}\left(R^2{S}_x^2-{\rho }_a^2{S}_y^2\right)$$

(10)

Note that $R={\displaystyle \frac{\overline{Y}}{\overline{X}}}$ is a ratio, $(S_x^2,S_y^2)$ are variances, and ${\rho }_a$ is the correlation between $(X,Y)$. Furthermore, ${\psi }_3^{*}={\displaystyle \frac{\lambda }{2-\lambda }}\left(\frac{1}{r}-\frac{1}{n}\right)$.

3. Adapted Estimators Using Quantile Regression

It is worth highlighting that all the studies regarding estimators discussed in the preceding section rely on specific functional forms, including chains, logarithmic functions, exponential functions, and linear combinations. However, to the best of our knowledge, no study is available about imputation-type EWMA mean estimators using quantile regression under SRS. So, in the next sub-section we will provide EWMA-based adapted estimators and in Section 4, we will provide the proposed estimators.

Adapted Estimators

In the realm of statistics, the Ordinary Least Squares (OLS) technique stands as a renowned parametric approach. Its widespread usage can be attributed to its computational simplicity, often leveraged to estimate function parameters. Particularly for the determination of coefficients in linear regression, OLS methodology provides the optimal estimation, as long as the appropriate conditions, as outlined by the OLS assumptions, are met. OLS estimators boast the least variance among unbiased estimators, earning them the designation of “best linear unbiased estimator” (BLUE), as cited by Al-Noor and Mohammad [32]. Nevertheless, the OLS estimators’ accuracy can be significantly undermined when outliers are present within the dataset, resulting in less reliable outcomes. The concept of “breakdown value”, which defines the tiny proportion of values capable of rendering an estimator meaningless, is applicable here (as discussed in references such as Hampel et al. [33] and Rousseeuw and Leroy [34]). For OLS fitting, this breakdown value is either $1/n$ or $0\%$, indicating the susceptibility of results to outliers. The sensitivity toward outliers in OLS is elaborated by Seber and Lee [35], who highlight two key factors:

• When employing squared residuals to gauge the size of residuals, those with substantial magnitudes will disproportionately contribute to the overall size;
• Utilizing measures of central tendency like the arithmetic mean, which lack robustness against outliers, exerts a disproportionately strong influence on measures when dealing with larger squared values. This inconsistency significantly impacts regression estimates.

Hence, in this article, we will construct mean estimators based on the EWMA technique. For these rationales, we employ a quantile regression coefficient.

In contrast to Ordinary Least Squares (OLS), quantile regression introduces a distinct regression approach that diverges from the assumptions of a normally distributed error term and constant variance. This flexible approach does not demand certain assumptions due to its unique nature [36]. Within the framework of quantile regression, coefficients are established based on quantiles, as outlined by Chen and Wei [37]. The quantile regression coefficient is

$$b_{q_{\left(j\right)}}={\mathrm{argmin}}_{\mathrm{\Omega }ϵR^{\mathrm{p}}}{\rho }_{\mathrm{q}}\left(v\right)\sum _{i=1}^n(y_{\mathrm{i}}-\langle x_{\mathrm{i}},\mathrm{\Omega }\rangle ).$$

(11)

In Equation (11), ${\rho }_{\mathrm{q}}\left(v\right)$ is a continuous piecewise linear function (or asymmetric absolute loss function), for quantile $qϵ(0,1),$ but nondifferentiable at $v=0$. However, $b_{q_{\left(j\right)}}$ is the quantile regression coefficient for $p=2$ variables. To study more regarding quantile regression, see Koc and Koc [20] and Yadav and Prasad [21].

Survey sampling is one of the most important pillars of statistics. The use of auxiliary information is very common for better mean estimation. For example, Abbasi et al. [38] uses auxiliary information in calibration constraints for efficient estimation of the empirical CDF. Zaman [39] utilized auxiliary information with robust regression methods. Koc et al. [40] merged auxiliary information with Poisson regression coefficients. So, taking motivation from relveant studies [15,16,17,18,19,20,21], the EWMA-based adapted quantile regression type mean estimators, i.e., ($T_{z{b}_{q_{\left(0.25\right)}}}$, $T_{z{b}_{q_{\left(0.45\right)}}}$, $T_{z{b}_{q_{\left(0.65\right)}}}$, $T_{z{b}_{q_{\left(0.85\right)}}}$) for Case-I, $(T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{\prime }}$, $T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{\prime }}$, $T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{\prime }}$, $T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{\prime }})$ for Case-II, and $(T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{″}}$, $T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{″}}$, $T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{″}}$, $T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{″}})$ for Case-III are given by:

$$T_{z{b}_{q_{\left(0.25\right)}}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.25\right)}}({\overline{X}}_a-Q_t)\right]}{Q_t}}{\overline{X}}_a$$

(12)

$$T_{z{b}_{q_{\left(0.45\right)}}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.45\right)}}({\overline{X}}_a-Q_t)\right]}{Q_t}}{\overline{X}}_a$$

(13)

$$T_{z{b}_{q_{\left(0.65\right)}}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.65\right)}}({\overline{X}}_a-Q_t)\right]}{Q_t}}{\overline{X}}_a$$

(14)

$$T_{z{b}_{q_{\left(0.85\right)}}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.85\right)}}({\overline{X}}_a-Q_t)\right]}{Q_t}}{\overline{X}}_a$$

(15)

$$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{\prime }}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.25\right)}}({\overline{X}}_a-Q_{t_r})\right]}{Q_{t_r}}}{\overline{X}}_a$$

(16)

$$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{\prime }}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.45\right)}}({\overline{X}}_a-Q_{t_r})\right]}{Q_{t_r}}}{\overline{X}}_a$$

(17)

$$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{\prime }}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.65\right)}}({\overline{X}}_a-Q_{t_r})\right]}{Q_{t_r}}}{\overline{X}}_a$$

(18)

$$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{\prime }}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.85\right)}}({\overline{X}}_a-Q_{t_r})\right]}{Q_{t_r}}}{\overline{X}}_a$$

(19)

$$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{″}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.25\right)}}(\overline{x}-Q_{t_r})\right]}{Q_{t_r}}}\overline{x}$$

(20)

$$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{″}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.45\right)}}(\overline{x}-Q_{t_r})\right]}{Q_{t_r}}}\overline{x}$$

(21)

$$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{″}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.65\right)}}(\overline{x}-Q_{t_r})\right]}{Q_{t_r}}}\overline{x}$$

(22)

$$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{″}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(0.85\right)}}(\overline{x}-Q_{t_r})\right]}{Q_{t_r}}}\overline{x}.$$

(23)

All the estimators given in Equations (12)–(23) can be written in a generalized form as given below

$$T_{z{b}_{q_{\left(j\right)}}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(j\right)}}({\overline{X}}_a-Q_t)\right]}{Q_t}}{\overline{X}}_a$$

(24)

$$T_{z{b}_{q_{\left(j\right)}}}^{{}^{\prime }}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(j\right)}}({\overline{X}}_a-Q_{t_r})\right]}{Q_{t_r}}}{\overline{X}}_a$$

(25)

$$T_{z{b}_{q_{\left(j\right)}}}^{{}^{″}}={\displaystyle \frac{\left[Z_{t_r}+b_{q_{\left(j\right)}}(\overline{x}-Q_{t_r})\right]}{Q_{t_r}}}\overline{x}.$$

(26)

In Equations (24)–(26), j represents different quantiles. In this article, we consider four quantile points, i.e., $j=0.25,0.45,0.65,0.85$. The MSE of $T_{z{b}_{q_{\left(j\right)}}},$$T_{z{b}_{q_{\left(j\right)}}}^{{}^{\prime }}$, and $T_{z{b}_{q_{\left(j\right)}}}^{{}^{″}}$ are as follows

$$MSE\left(T_{z{b}_{q_{\left(j\right)}}}\right)={\psi }_2^{*}{S}_y^2+{\psi }_1^{*}[R^2-B_{q\left(j\right)}^2]S_x^2$$

(27)

$$MSE\left(T_{z{b}_{q_{\left(j\right)}}}^{{}^{\prime }}\right)={\psi }_2^{*}[S_y^2+R^2{S}_x^2-B_{q\left(j\right)}{S}_{xy}]$$

(28)

$$MSE\left(T_{z{b}_{q_{\left(j\right)}}}^{{}^{″}}\right)={\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[{(R+B_{q\left(j\right)})}^2{S}_x^2-2(R+B_{q\left(j\right)})S_{xy}].$$

(29)

4. Proposed Estimators

The EWMA approach is a statistical tool employed to model and characterize time-series data. It has broad applications in the field of finance, particularly in activities like technical analysis and volatility modeling. When constructing the moving average, earlier data points are assigned decreasing weights as additional data points are incorporated. The term “exponentially weighted” indicates that these weights decline exponentially as more data points become part of the calculation. For users of EWMA, the selection of the appropriate parameter holds significant importance. This parameter plays a pivotal role in determining the influence of the current observation in the EWMA calculation. Opting for a higher value of this parameter enhances the accuracy of EWMA in monitoring the original time series. It is important to note that within the EWMA framework, estimators related to measures of central tendency and dispersion can also experience enhancements by utilizing supplementary information.

Based on the refined estimators, we introduce three novel classes of EWMA-based estimators that exhibit unquestionable and consistent efficacy. These suggested classes are characterized by their inherent simplicity, as they focus solely on the regression component while omitting the ratio-based imputation factor present in the adapted versions. These novel classes are delineated in the subsequent sections.

4.1. Case-I

The group of estimators suggested for scenario-I is

$$T_{p_{q_{\left(j\right)}}}=\left[Z_{t_r}+b_{q_{\left(j\right)}}({\overline{X}}_a-Q_t)\right].$$

(30)

Theorem 1.

The MSE of the proposed class is

$$MSE\left(T_{p_{q_{\left(j\right)}}}\right)={\psi }_2^{*}{S}_y^2+{\psi }_1^{*}{B}_{q_{\left(j\right)}}[B_{q_{\left(j\right)}}{S}_x^2-2S_{xy}].$$

Proof. 

To find the MSE for $T_{p_{q_{\left(j\right)}}}$, let ${\eta }_{y_{r1}}={\displaystyle \frac{Z_{t_r}-\overline{Y}}{\overline{Y}}}$, ${\eta }_x={\displaystyle \frac{Q_t-{\overline{X}}_a}{{\overline{X}}_a}}$, and ${\eta }_{x_{r1}}={\displaystyle \frac{Q_{t_r}-{\overline{X}}_a}{{\overline{X}}_a}}$. Furthermore, $E\left({\eta }_{y_{r1}}\right)=E\left({\eta }_x\right)=E\left({\eta }_{x_{r1}}\right)=0$, $E\left({\eta }_{y_{r1}}^2\right)={\psi }_2^{*}{C}_y^2$, $E\left({\eta }_x^2\right)={\psi }_1^{*}{C}_x^2$, $E\left({\eta }_{x_{r1}}^2\right)={\psi }_2^{*}{C}_x^2$, $E\left({\eta }_{y_{r1}}{\eta }_x\right)={\psi }_1^{*}{C}_{yx},$$E\left({\eta }_{y_{r1}}{\eta }_{x_{r1}}\right)={\psi }_2^{*}{C}_{yx},$ and $E\left({\eta }_{x_{r1}}{\eta }_x\right)={\psi }_1^{*}{C}_x^2$.

Expanding $T_{p_{q_{\left(j\right)}}}$ in terms of ${\eta }^{\prime }s$

$$T_{p_{q_{\left(j\right)}}}=\overline{Y}(1+{\eta }_{y_{r1}})-B_{q_{\left(j\right)}}{\overline{X}}_a{\eta }_x$$

(31)

$$T_{p_{q_{\left(j\right)}}}-\overline{Y}=\overline{Y}(1+{\eta }_{y_{r1}})-B_{q_{\left(j\right)}}{\overline{X}}_a{\eta }_x-\overline{Y}.$$

(32)

By taking the square of Equation (32), applying expectation, and performing simplification, the MSE of $T_{p_{q_{\left(j\right)}}}$ is

$$MSE\left({\widehat{\overline{y}}}_{p_{q_{\left(j\right)}}}\right)={\psi }_2^{*}{S}_y^2+{\psi }_1^{*}{B}_{q_{\left(j\right)}}[B_{q_{\left(j\right)}}{S}_x^2-2S_{xy}]$$

which proves the theorem. The expressions of MSE for different family members of $T_{p_{q_{\left(j\right)}}}$ at the considered quantile points are

$$MSE\left(T_{p_{q_{\left(j\right)}}}\right)=\left\{\begin{array}{c}{\psi }_2^{*}{S}_y^2+{\psi }_1^{*}{B}_{q_{\left(0.25\right)}}[B_{q_{\left(0.25\right)}}{S}_x^2-2S_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.25\\ {\psi }_2^{*}{S}_y^2+{\psi }_1^{*}{B}_{q_{\left(0.45\right)}}[B_{q_{\left(0.45\right)}}{S}_x^2-2S_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.45\\ {\psi }_2^{*}{S}_y^2+{\psi }_1^{*}{B}_{q_{\left(0.65\right)}}[B_{q_{\left(0.65\right)}}{S}_x^2-2S_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.65\\ {\psi }_2^{*}{S}_y^2+{\psi }_1^{*}{B}_{q_{\left(0.85\right)}}[B_{q_{\left(0.85\right)}}{S}_x^2-2S_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.85\end{array}\right.$$

(33)

It is worth highlighting that the class of estimators $T_{p_{q_{\left(j\right)}}}$ come into use when missing values solely occur in the targeted variable Y. Hence, the estimate (result) of estimator $T_{p_{q_{\left(j\right)}}}$ is solely reliant on $Z_{t_r}$, coupled with ${\overline{X}}_a$. □

4.2. Case-II

When missing values occur in both $(X,Y)$, the class of estimators for such circumstances is

$$T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}=\left[Z_{t_r}+b_{q_{\left(j\right)}}({\overline{X}}_a-Q_{t_r})\right].$$

(34)

Theorem 2.

The MSE of $T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}$ is

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}\right)={\psi }_2^{*}[S_y^2+B_{q_{\left(j\right)}}^2{S}_x^2-2B_{q_{\left(j\right)}}{S}_{xy}].$$

Proof. 

Expanding $T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}$ w.r.t. ${\eta }^{\prime }s$

$$T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}=\overline{Y}(1+{\eta }_{y_{r1}})-b_{q_{\left(j\right)}}{\overline{X}}_a{\eta }_{x_{r1}}$$

(35)

$$T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}-\overline{Y}=\overline{Y}(1+{\eta }_{y_{r1}})-b_{q_{\left(j\right)}}{\overline{X}}_a{\eta }_{x_{r1}}-\overline{Y}.$$

(36)

By taking square of Equation (36), applying expectation and performing simplification, the MSE of $T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}$ is

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}\right)={\psi }_2^{*}[S_y^2+B_{q_{\left(j\right)}}^2{S}_x^2-2B_{q_{\left(j\right)}}{S}_{xy}].$$

which proves the theorem. The expressions of MSE for different family members of $T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}$ at considered quantile points are

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}\right)=\left\{\begin{array}{c}{\psi }_2^{*}[S_y^2+B_{q_{\left(0.25\right)}}^2{S}_x^2-2B_{q_{\left(0.25\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.25\\ {\psi }_2^{*}[S_y^2+B_{q_{\left(0.45\right)}}^2{S}_x^2-2B_{q_{\left(0.45\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.45\\ {\psi }_2^{*}[S_y^2+B_{q_{\left(0.65\right)}}^2{S}_x^2-2B_{q_{\left(0.65\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.65\\ {\psi }_2^{*}[S_y^2+B_{q_{\left(0.85\right)}}^2{S}_x^2-2B_{q_{\left(0.85\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.85\end{array}\right.$$

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It is worth highlighting that the class of estimators $T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}$ come into use when missing values occur in both variables $(X,Y)$. Hence, the estimate (result) of estimator $T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}$ is reliant on both $(Z_{t_r},Q_{t_r})$. □

4.3. Case-III

In situations where both variables $(X,Y)$ experience the problem of missing observations, the group of estimators $T_{p_{q_{\left(j\right)}}}^{{}^{″}}$ is

$$T_{p_{q_{\left(j\right)}}}^{{}^{″}}=\left[Z_{t_r}+b_{q_{\left(j\right)}}(\overline{x}-Q_{t_r})\right],$$

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Theorem 3.

The MSE of the proposed class is

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{″}}\right)={\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[B_{q_{\left(j\right)}}^2{S}_x^2-2B_{q_{\left(j\right)}}{S}_{xy}].$$

Proof. 

By expanding $T_{p_{q_{\left(j\right)}}}^{{}^{″}}$ w.r.t. ${\eta }^{\prime }s$ we have

$$T_{p_{q_{\left(j\right)}}}^{{}^{″}}=\overline{Y}(1+{\eta }_{y_{r1}})-b_{q_{\left(j\right)}}{\overline{X}}_a({\eta }_x-{\eta }_{x_{r1}})$$

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$$T_{p_{q_{\left(j\right)}}}^{{}^{″}}-\overline{Y}=\overline{Y}(1+{\eta }_{y_{r1}})-b_{q_{\left(j\right)}}{\overline{X}}_a({\eta }_x-{\eta }_{x_{r1}})-\overline{Y}.$$

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By taking the square of Equation (40), applying expectation, and performing simplification, the MSE of $T_{p_{q_{\left(j\right)}}}^{{}^{″}}$ is

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{″}}\right)={\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[B_{q_{\left(j\right)}}^2{S}_x^2-2B_{q_{\left(j\right)}}{S}_{xy}].$$

which proves the theorem. The expressions of MSE for different family members of $T_{p_{q_{\left(j\right)}}}^{{}^{″}}$ at the considered quantile points are

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{″}}\right)=\left\{\begin{array}{c}{\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[B_{q_{\left(0.25\right)}}^2{S}_x^2-2B_{q_{\left(0.25\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.25\\ {\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[B_{q_{\left(0.45\right)}}^2{S}_x^2-2B_{q_{\left(0.45\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.45\\ {\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[B_{q_{\left(0.65\right)}}^2{S}_x^2-2B_{q_{\left(0.65\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.65\\ {\psi }_2^{*}{S}_y^2+{\psi }_3^{*}[B_{q_{\left(0.85\right)}}^2{S}_x^2-2B_{q_{\left(0.85\right)}}{S}_{xy}]\hspace{1em}\mathrm{for}\hspace{1em}j=0.85\end{array}\right.$$

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The proposed group of estimators $T_{p_{q_{\left(j\right)}}}^{{}^{″}}$ is applicable for the MCAR scenario, parallel to $T_3$.

The idea behind $(T_{p_{q_{\left(j\right)}}},T_{p_{q_{\left(j\right)}}}^{{}^{\prime }},T_{p_{q_{\left(j\right)}}}^{{}^{″}})$ is to remove the ratio component while retaining the regression aspect of the traditional $(T_1,T_2,T_3)$ and adapted $(T_{z{b}_{q_{\left(j\right)}}},T_{z{b}_{q_{\left(j\right)}}}^{{}^{\prime }},T_{z{b}_{q_{\left(j\right)}}}^{{}^{″}})$ estimators. □

5. Numerical Study

In order to facilitate the implementation of the reviewed, adapted, and suggested estimators, we commonly draw upon a combination of empirical and theoretical analyses. To achieve this, we consider three distinct populations, chosen with care. The evaluation and ultimate conclusions are based on Mean Squared Error (MSE). Consequently, two of these populations mirror real-world datasets. The important features of these populations are provided in

Table 1. Features of the humidity and HBL data.

	$\overline{Y}=83.02466$	${\overline{X}}_a=76.4547$	$b_{q_{\left(0.25\right)}}=0.0416$
	$S_y=10.5662$	$S_x=13.7950$	$b_{q_{\left(0.45\right)}}=0.0344$
(Population-1)	${\rho }_a=0.0606$	$S_{xy}=8.8404$	$b_{q_{\left(0.65\right)}}=0.0212$
	$S_x^2=190.3036$	$S_y^2=111.6450$	$b_{q_{\left(0.85\right)}}=0.0002$
	$N=365$	$n=30$	$r=21$
	$\overline{Y}=175.9964$	${\overline{X}}_a=146.7845$	$b_{q_{\left(0.25\right)}}=-0.0765$
	$S_y=5.3498$	$S_x=8.3574$	$b_{q_{\left(0.45\right)}}=-0.3229$
(Population-2)	${\rho }_a=-0.3266$	$S_{xy}=-14.6033$	$b_{q_{\left(0.65\right)}}=-0.4174$
	$S_x^2=69.8466$	$S_y^2=28.6210$	$b_{q_{\left(0.85\right)}}=-0.2904$
	$N=42$	$n=10$	$r=8$

. The third population is generated from a gamma distribution and used for simulation purposes.

5.1. Karachi Humidity Data (Population-1)

The humidity prevalent in Karachi stems from its geographical placement adjacent to the Arabian Sea. Due to its close proximity to this expansive water body, the city becomes susceptible to the influence of moisture-laden sea breezes, transporting humidity into its atmospheric layers. Over the course of each year, the inhabitants of Karachi encounter diverse levels of humidity, experiencing fluctuations between arid and moist environments. Notably, the monsoon period, spanning from June to September, heralds a substantial surge in humidity levels as damp air is pulled inland from the Arabian Sea, enveloping the city in a dense and humid curtain. For the purposes of the article, we consider the daily maximum percentage of humidity. The dataset was sourced from a publicly accessible website (https://www.wunderground.com, accessed on 29 June 2023), and formal consent was not needed for its use. Here

X: Humidity (%) in 2021;

Y: Humidity (%) in 2022.

5.2. HBL Data (Population-2)

In this study, we performed a numerical examination utilizing data sourced from the Islamabad Stock Market, specifically those of Habib Bank Limited (HBL). The data consist of daily time-based highs of the share price of HBL. The dataset was sourced from a publicly accessible website (https://pkfinance.info, accessed on 14 September 2021), and formal consent was not needed for its use. Here,

X: Highs of the share price January–February 2019;

Y: Highs of the share price January–February 2020.

5.3. Simulation Study: Asymmetric Data (Population-3)

In order to assess the effectiveness of the suggested estimators, a simulation study was conducted using asymmetric data. We generated a population of size $N=1000$ using the Shahzad et al. [18] model

$$Y=\delta +KX+\epsilon X^p$$

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where $X\sim Gamma(2.6,3.8)$, $\epsilon$ follows standard Normal distribution, and $\delta =5$, $p=1.6,K=2$.

Steps involved in the simulation study:

• Draw 5000 samples in total using SRSWOR;
• Randomly remove $rr$ units from each sample, resulting in $r=n-rr$;
• Calculate mean of r observations from each sample;
• Calculate MSE as follows

  $$MSE\left({\widehat{y}}_j\right)={\displaystyle \frac{{\sum }_{j=1}^{5000}{\left({\widehat{y}}_j-\widehat{\overline{y}}\right)}^2}{5000}},$$

  where ${\widehat{y}}_j$ denotes estimators and $\widehat{\overline{y}}=\frac{{\sum }_{j=1}^{5000}{\widehat{y}}_j}{5000}.$

5.4. Discussion

The results for each of the numerical populations are detailed in

Table 2. MSE for case-1 using the Humidity, HBL, and Asymmetric datasets.

	$\mathit{\lambda }$	0.05	0.10	0.25	0.50	0.75	1.00
	$T_1$	0.3551	0.7354	2.0264	4.6791	7.9582	11.8636
	$T_{z{b}_{q_{\left(0.25\right)}}}$	0.5933	1.1866	2.9665	5.9330	8.8995	11.8661
	$T_{z{b}_{q_{\left(0.45\right)}}}$	0.5934	1.1869	2.9673	5.9346	8.9019	11.8692
	$T_{z{b}_{q_{\left(0.65\right)}}}$	0.5936	1.1873	2.9683	5.9367	8.9051	11.8735
Population-1	$T_{z{b}_{q_{\left(0.85\right)}}}$	0.5938	1.1876	2.9690	5.9381	8.9071	11.8762
	$T_{p_{q_{\left(0.25\right)}}}$	0.2499	0.4998	1.2495	2.4990	3.7485	4.9981
	$T_{p_{q_{\left(0.45\right)}}}$	0.2499	0.4998	1.2497	2.4994	3.7491	4.9988
	$T_{p_{q_{\left(0.65\right)}}}$	0.2501	0.5001	1.2504	2.5008	3.7512	5.0016
	$T_{p_{q_{\left(0.85\right)}}}$	0.2505	0.5010	1.2526	2.5052	3.7579	5.0105
	$T_1$	0.3781	0.7707	2.0354	4.4330	7.1925	10.3141
	$T_{z{b}_{q_{\left(0.25\right)}}}$	0.5257	1.0515	2.6288	5.2577	7.8866	10.5155
	$T_{z{b}_{q_{\left(0.45\right)}}}$	0.4995	0.9991	2.4979	4.9959	7.4938	9.9918
	$T_{z{b}_{q_{\left(0.65\right)}}}$	0.4809	0.9619	2.4048	4.8097	7.2146	9.6195
Population-2	$T_{z{b}_{q_{\left(0.85\right)}}}$	0.5048	1.0097	2.5244	5.0489	7.5734	10.0978
	$T_{p_{q_{\left(0.25\right)}}}$	0.1378	0.2757	0.6892	1.3785	2.0678	2.7570
	$T_{p_{q_{\left(0.45\right)}}}$	0.13662	0.2732	0.6831	1.3662	2.0493	2.7325
	$T_{p_{q_{\left(0.65\right)}}}$	0.1447	0.2894	0.7236	1.4472	2.1709	2.8945
	$T_{p_{q_{\left(0.85\right)}}}$	0.1349	0.2698	0.6746	1.3493	2.0240	2.6987
	$T_1$	0.01115	0.02247	0.05743	0.11902	0.18475	0.25463
	$T_{z{b}_{q_{\left(0.25\right)}}}$	0.01296	0.02593	0.06484	0.12969	0.19454	0.25939
	$T_{z{b}_{q_{\left(0.45\right)}}}$	0.01274	0.02548	0.06370	0.12741	0.19112	0.25483
	$T_{z{b}_{q_{\left(0.65\right)}}}$	0.01246	0.02492	0.06232	0.12464	0.18696	0.24928
Population-3	$T_{z{b}_{q_{\left(0.85\right)}}}$	0.01186	0.02373	0.05934	0.11868	0.17802	0.23737
	$T_{p_{q_{\left(0.25\right)}}}$	0.00128	0.00257	0.00643	0.01286	0.01929	0.02573
	$T_{p_{q_{\left(0.45\right)}}}$	0.00123	0.00247	0.00618	0.01237	0.01856	0.02475
	$T_{p_{q_{\left(0.65\right)}}}$	0.00127	0.00254	0.00635	0.01270	0.01905	0.02541
	$T_{p_{q_{\left(0.85\right)}}}$	0.00147	0.00294	0.00736	0.01472	0.02208	0.02944

,

Table 3. MSE for case-2 using the Humidity, HBL, and Asymmetric datasets.

	$\mathit{\lambda }$	0.05	0.10	0.25	0.50	0.75	1.00
	$T_2$	0.7531	1.5063	3.7659	7.5318	11.2977	15.0637
	$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{\prime }}$	0.7532	1.5065	3.7664	7.5328	11.2992	15.0656
	$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{\prime }}$	0.7534	1.5068	3.7671	7.5342	11.3013	15.0684
	$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{\prime }}$	0.7536	1.5073	3.7684	7.5368	11.3052	15.0736
Population-1	$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{\prime }}$	0.7541	1.5082	3.7705	7.5410	11.3116	15.0821
	$T_{p_{q_{\left(0.25\right)}}}^{{}^{\prime }}$	0.2496	0.4992	1.2480	2.4961	3.7442	4.9923
	$T_{p_{q_{\left(0.45\right)}}}^{{}^{\prime }}$	0.2496	0.4993	1.2483	2.4966	3.7450	4.9933
	$T_{p_{q_{\left(0.65\right)}}}^{{}^{\prime }}$	0.2498	0.4997	1.2493	2.4987	3.7481	4.9975
	$T_{p_{q_{\left(0.85\right)}}}^{{}^{\prime }}$	0.2505	0.5010	1.2526	2.5052	3.7579	5.0105
	$T_2$	0.6374	1.2748	3.1870	6.3740	9.5610	12.7481
	$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{\prime }}$	0.6472	1.2944	3.2360	6.4720	9.7080	12.9440
	$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{\prime }}$	0.6289	1.2579	3.1449	6.2899	9.4349	12.5798
	$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{\prime }}$	0.6220	1.2440	3.1100	6.2201	9.3301	12.4402
Population-2	$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{\prime }}$	0.6313	1.2627	3.1569	6.3139	9.4709	12.6279
	$T_{p_{q_{\left(0.25\right)}}}^{{}^{\prime }}$	0.1355	0.2711	0.6778	1.3557	2.0335	2.7114
	$T_{p_{q_{\left(0.45\right)}}}^{{}^{\prime }}$	0.1339	0.2678	0.6697	1.3394	2.0091	2.6788
	$T_{p_{q_{\left(0.65\right)}}}^{{}^{\prime }}$	0.1447	0.2894	0.7235	1.4470	2.1705	2.8940
	$T_{p_{q_{\left(0.85\right)}}}^{{}^{\prime }}$	0.1316	0.2633	0.6584	1.3169	1.9754	2.6339
	$T_2$	0.01955	0.03910	0.09775	0.19551	0.29326	0.39102
	$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{\prime }}$	0.01978	0.03956	0.09891	0.19782	0.29673	0.39565
	$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{\prime }}$	0.01955	0.03911	0.09779	0.19559	0.29338	0.39118
	$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{\prime }}$	0.01936	0.03872	0.09680	0.19361	0.29042	0.38723
Population-3	$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{\prime }}$	0.01904	0.03808	0.09521	0.19043	0.28564	0.38086
	$T_{p_{q_{\left(0.25\right)}}}^{{}^{\prime }}$	0.00105	0.00211	0.00528	0.01057	0.01585	0.02114
	$T_{p_{q_{\left(0.45\right)}}}^{{}^{\prime }}$	0.00097	0.00195	0.00489	0.00978	0.01467	0.01956
	$T_{p_{q_{\left(0.65\right)}}}^{{}^{\prime }}$	0.00103	0.00206	0.00515	0.01031	0.01546	0.02062
	$T_{p_{q_{\left(0.85\right)}}}^{{}^{\prime }}$	0.00135	0.00271	0.00678	0.01357	0.02035	0.02714

and

Table 4. MSE for case-3 using the Humidity, HBL, and Asymmetric datasets.

	$\mathit{\lambda }$	0.05	0.10	0.25	0.50	0.75	1.00
	$T_3$	0.4105	0.8210	2.0526	4.1053	6.1579	8.2106
	$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{″}}$	0.4091	0.8182	2.0456	4.0912	6.1368	8.1824
	$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{″}}$	0.4070	0.8140	2.0350	4.0701	6.1052	8.1403
	$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{″}}$	0.4031	0.8063	2.0159	4.0318	6.0477	8.0636
Population-1	$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{″}}$	0.3971	0.7942	1.9855	3.9710	5.9566	7.9421
	$T_{p_{q_{\left(0.25\right)}}}^{{}^{″}}$	0.2502	0.5004	1.2511	2.5023	3.7535	5.0047
	$T_{p_{q_{\left(0.45\right)}}}^{{}^{″}}$	0.2502	0.5005	1.2512	2.5025	3.7538	5.0050
	$T_{p_{q_{\left(0.65\right)}}}^{{}^{″}}$	0.2503	0.5006	1.2516	2.5032	3.7548	5.0064
	$T_{p_{q_{\left(0.85\right)}}}^{{}^{″}}$	0.2505	0.5010	1.2526	2.5052	3.7579	5.0105
	$T_3$	0.2665	0.5330	1.3325	2.6650	3.9976	5.3301
	$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{″}}$	0.2957	0.5915	1.4789	2.9579	4.4369	5.9159
	$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{″}}$	0.2438	0.4876	1.2190	2.4380	3.6570	4.8761
	$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{″}}$	0.2266	0.4533	1.1333	2.2667	3.4001	4.5335
Population-2	$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{″}}$	0.2500	0.5001	1.2502	2.5005	3.7508	5.0011
	$T_{p_{q_{\left(0.25\right)}}}^{{}^{″}}$	0.1425	0.2850	0.7126	1.4252	2.1379	2.8505
	$T_{p_{q_{\left(0.45\right)}}}^{{}^{″}}$	0.1421	0.2842	0.7106	1.4212	2.1318	2.8424
	$T_{p_{q_{\left(0.65\right)}}}^{{}^{″}}$	0.1447	0.2895	0.7239	1.4478	2.1717	2.8956
	$T_{p_{q_{\left(0.85\right)}}}^{{}^{″}}$	0.1415	0.2831	0.7078	1.4157	2.1235	2.8314
	$T_3$	0.00847	0.01695	0.04239	0.08478	0.12718	0.16957
	$T_{z{b}_{q_{\left(0.25\right)}}}^{{}^{″}}$	0.00758	0.01517	0.03793	0.07587	0.11380	0.15174
	$T_{z{b}_{q_{\left(0.45\right)}}}^{{}^{″}}$	0.00844	0.01689	0.04223	0.08446	0.12669	0.16892
	$T_{z{b}_{q_{\left(0.65\right)}}}^{{}^{″}}$	0.00925	0.01850	0.04626	0.09253	0.13880	0.18507
Population-3	$T_{z{b}_{q_{\left(0.85\right)}}}^{{}^{″}}$	0.01064	0.02129	0.05322	0.10645	0.15968	0.21290
	$T_{p_{q_{\left(0.25\right)}}}^{{}^{″}}$	0.00142	0.00285	0.00714	0.01429	0.02144	0.02859
	$T_{p_{q_{\left(0.45\right)}}}^{{}^{″}}$	0.00139	0.00279	0.00699	0.01399	0.02099	0.02799
	$T_{p_{q_{\left(0.65\right)}}}^{{}^{″}}$	0.00141	0.00283	0.00709	0.01419	0.02129	0.02839
	$T_{p_{q_{\left(0.85\right)}}}^{{}^{″}}$	0.00154	0.00308	0.00772	0.01544	0.02316	0.03088

. These tables provide insights into the following observations:

For case-I

$$MSE\left(T_{p_{q_{\left(j\right)}}}\right)<MSE\left(T_{z{b}_{q_{\left(j\right)}}}\right)<MSE\left(T_1\right)$$

For case-II

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{\prime }}\right)<MSE\left(T_{z{b}_{q_{\left(j\right)}}}^{{}^{\prime }}\right)<MSE\left(T_2\right)$$

For case-III

$$MSE\left(T_{p_{q_{\left(j\right)}}}^{{}^{″}}\right)<MSE\left(T_{z{b}_{q_{\left(j\right)}}}^{{}^{″}}\right)<MSE\left(T_3\right).$$

Some key findings that can be derived from the results presented in Table 2, Table 3 and Table 4 are as given below:

• In both the existing and proposed estimators, increasing the value of $\lambda$ results in an increase in the mean squared error (MSE);
• Among all the existing and proposed estimators, the traditional ratio-type estimators $(T_1,T_2,T_3)$ exhibit the highest MSE;
• The adapted estimators $(T_{z{b}_{q_{\left(j\right)}}},T_{z{b}_{q_{\left(j\right)}}}^{{}^{\prime }},T_{z{b}_{q_{\left(j\right)}}}^{{}^{″}})$ have lower MSE than $(T_1,T_2,T_3)$;
• The proposed estimators $(T_{p_{q_{\left(j\right)}}},T_{p_{q_{\left(j\right)}}}^{{}^{\prime }},T_{p_{q_{\left(j\right)}}}^{{}^{″}})$ have lower MSE than the traditional and adapted estimators;
• Consequently, the proposed classes outperform their counterparts in all three instances of missing data.

6. Conclusions

Time-scaled surveys require simultaneous consideration of past and present data from the sample to enhance the performance of estimators. Missing information can also mislead the results of such surveys. In order to address this challenge, we utilized the imputation technique with the EWMA statistic and introduced quantile-regression-based memory-type mean estimators under SRS. This paper centers on the adaptation of estimators devised in references [15,16,17,18,19,20,21], tailored specifically for the SRS design—a widely recognized sampling method across various disciplines. The findings of the numerical study demonstrate that the proposed quantile-regression-based memory-type mean estimators outperform the reviewed and adapted estimators. The numerical study culminates in a detailed interpretation of the results, shedding light on the effectiveness of the proposed estimators. Based on the values of MSE in Table 2, Table 3 and Table 4, it is evident that the proposed estimators significantly improve the estimation of the population mean. Future research endeavors could delve into the exploration of all three scenarios in systematic sampling by employing the approach introduced by Shahzad et al. [18].