Improved estimation of population variance in stratified successive sampling using calibrated weights under non-response

Abstract

This paper introduces a new method to estimate the population variance of a study variable in stratified successive sampling over two occasions, while accounting for random non-response. The method uses a logarithmic type estimator and leverages information from a highly positively correlated auxiliary variable. The paper also presents calibrated weights for the new estimator and examines its properties through numerical and simulation studies. The results indicate that the suggested estimator is more effective than the standard estimator for estimating the population variance.

1. Introduction

Stratified successive sampling is a statistical method employed to estimate population variance in situations where obtaining a random sample from the population proves to be challenging or cost-prohibitive. This technique involves dividing the population into distinct strata and selecting individuals randomly from within these strata. Successive sampling proves particularly advantageous in socio-economic research, where populations may be widely dispersed or access may be limited. The concept of stratified sampling for estimating population parameters, including population variance, was first introduced by [16]. Subsequently, [14] extended this method for estimating population variance in stratified successive sampling, while [19] and [25] explored population variance estimation in stratified successive sampling scenarios with unequal probabilities while [15] did the same with equal probability. [23] developed an effectual variance estimation strategy for two-occasion successive sampling, accounting for random non-response, whereas [27] worked on a family of robust-type estimators for population variance in simple and stratified random sampling. [1] improved the estimation of finite population variance using dual supplementary information under stratified random sampling, and [6] introduced efficient classes of estimators of population variance in two-phase successive sampling scenarios with random non-response. Moreover, [5] and [4] contributed to the field by developing memory-type ratio and product estimators under ranked-based sampling schemes, among others provide a comprehensive foundation for the utilization of stratified successive sampling and calibration approaches, enhancing the proposed method's accuracy and applicability in estimating population variance, especially under non-response.

[7] introduced a calibration approach using least squares adjustment, a method later adopted by statistical authorities in various organizations. The primary aim of the calibration approach is to formulate unbiased estimation procedures with minimal dispersion by leveraging information from auxiliary variables. [8] proposed a calibration estimation procedure that reduces the discrepancy between initial and final weights while still adhering to calibration equations and constraints. [10] explored model-assisted higher-order calibration of variance estimators, and [20] applied the calibration approach in survey theory and practice. [12] and [11] contributed to calibration approach estimators in stratified sampling. [26] introduced a calibration approach-based regression-type estimator to model the inverse relationship between study and auxiliary variables, and [13] applied calibration weighting in stratified random sampling. [17] introduced a new calibration estimator specifically designed for stratified sampling, and [9] developed calibration estimation for ratio estimators in stratified sampling with proportion allocation. [22] extended the calibration approach to estimate population variance in stratified successive sampling while accounting for random non-response. Recently, [2] conducted work on estimating the calibration of the mean by employing a double use of auxiliary information. Additionally, [3] focused on determining optimal calibrated weights while minimizing a variance function. In a related study, [21] explored L-moments and calibration-based variance estimators under a double stratified random sampling scheme, with a specific focus on their application during the COVID-19 pandemic. Moreover, [18] contributed to the field by developing a general class of improved population variance estimators under non-sampling errors using calibrated weights in stratified sampling.

Calibration finds its place as a pivotal technique in the estimation of population variance through stratified successive sampling. This approach enriches estimation accuracy by imbuing resulting estimates with greater population representativeness and reduced bias. This significance is particularly pronounced in stratified successive sampling, where the goal centers on ensuring representation of each stratum within the final estimate. Additionally, this proposed estimation strategy finds application in domains as diverse as the estimation of variance in gas turbine exhaust pressure, as evidenced by real-data illustration in later sections of this manuscript.

2. Motivation of the study

The dynamic nature of our world entails constant variation, with implications across various realms. In medicine, variations in body temperature, blood pressure, and pulse rate hold diagnostic significance. Similarly, diverse consumer responses to products drive pricing and quality decisions, while variations in climatic factors guide agricultural planning.

The estimation of population variance assumes a pivotal role in diverse fields, ranging from socio-economic studies to engineering applications. Stratified successive sampling offers a strategic approach, particularly when random sampling from the entire population is intricate. However, the presence of non-response poses a challenge, potentially introducing bias.

In this context, this research article proposed a novel solution. We have introduced a logarithmic-type estimator that incorporates information from a highly correlated auxiliary variable. Moreover, calibrated weights are integrated into the estimation process to counteract the impact of non-response bias.

The motivation for this study emanates from various factors, including:

Practical Relevance: Stratified successive sampling is commonly used in scenarios where populations are geographically dispersed or hard to access due to non-homogeneity. However, non-response can distort estimates, making it essential to account for this bias through calibrated weights.

Calibration Approach: We use the calibration technique to improve the accuracy of the estimates. By incorporating calibration into the estimation process, resulting estimates are more representative of the population and have reduced bias. Leveraging auxiliary variables to enhance estimation accuracy has shown effectiveness. Applying this calibration technique to stratified successive sampling aims to yield more representative and unbiased population variance estimates.

Engineering Applications: The proposed methodology holds practical significance in fields like engineering, particularly in quality control and process improvement. Precise population variance estimation is vital for assessing manufacturing process variability, enabling informed decisions for process optimization and quality control strategies.

Socio-Economic Research: In socio-economic research, understanding the relationship between variables like education and income is pivotal for informed decision-making. By enhancing population variance estimation, the proposed method provides deeper insights into these relationships, guiding policy and strategy decisions.

Unique Challenges Addressed: The use of logarithmic-type estimators underscores a focus on addressing rare challenges, such as estimating diseases or socio-economic issues requiring specialized techniques. This approach showcases our dedication to tackling complex issues with precision.

In conclusion, the motivation behind this research lies in the imperative to enhance population variance estimation accuracy in stratified successive sampling, particularly by addressing non-response challenges. Through the utilization of calibrated weights and auxiliary information, we aspire to provide more resilient and effective estimation techniques, applicable across diverse fields from engineering to socio-economic research. By obtaining more accurate estimates of population variance, one can better understand the relationship between education level and income in the studied population. This understanding may hold important implications for policy decisions related to education and income inequality, as well as for businesses and organizations interested in targeting college-educated individuals for employment or marketing purposes.

3. Sample structure

Consider a finite population of size N divided into L non-overlapping strata, each containing $N_k$ (k=1,2,..., L) units. Let us use X and Y to represent the study character on the first and second occasions. It is assumed that information regarding an auxiliary variable Z is accessible on both occasions, and the population variance of Z is known.

Let us consider the $k^{\text{th}}$ strata, where k ranges from 1 to L. To begin with, we use simple random sampling without replacement (SRSWOR) to draw a preliminary sample of size $n_k$ from the population for the first occasion, where $r_{1k}$ units do not respond. From the responding part of this sample, we draw a second stage SRSWOR sample of size $m_k=n_k{\lambda }_k^{″}$, where ${\lambda }_k^{″}$ is the fraction of matched samples, and $r_{2k}$ units do not respond. We use this sample for the second occasion and collect information on the study variable Y. Additionally, we draw a fresh sample of size $u_k=n_k-m_k=n_k{\mu }_k^{″}$ from the population using SRSWOR on Y again. Here, $r_{3k}$ units do not respond. The fractions of matched and fresh samples on the current (second) occasion are represented by ${\lambda }_k^{″}$ and ${\mu }_k^{″}$, respectively, where ${\lambda }_k^{″}+{\mu }_k^{″}$ = 1.

3.1. Notations

From now on, we will use the following notations:

${\sigma }_{\text{Y}}^2$: The population variance of Y, i.e., the characteristics under study.

${\overline{\text{Z}}}_{{\text{N}}_{\text{k}}}=\frac{1}{{\text{N}}_{\text{k}}}{\sum }_{l=1}^{{\text{N}}_{\text{k}}}{\text{Z}}_{\text{kl}}$: The population mean of Z in the $k^{\text{th}}$ strata.

$\overline{\text{Z}}=\frac{1}{\text{N}}{\sum }_{l=1}^{{\text{N}}_{\text{k}}}{\text{N}}_{\text{k}}{\overline{\text{Z}}}_{{\text{N}}_{\text{k}}}$: The population mean of Z.

${\overline{\text{x}}}_{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}=\frac{1}{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}{\sum }_{l=1}^{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}{\text{x}}_{\text{kl}},{\overline{\text{x}}}_{{\text{m}}_{\text{k}}-{\text{r}}_{\text{2k}}}=\frac{1}{{\text{m}}_{\text{k}-{\text{r}}_{\text{2k}}}}{\sum }_{l=1}^{{\text{m}}_{\text{k}}-{\text{r}}_{\text{2k}}}{\text{x}}_{\text{kl}}$: The sample means of the study variable X in the $k^{\text{th}}$ strata based on the responding part of samples of sizes ${\text{n}}_{\text{k}}$ and ${\text{m}}_{\text{k}}$, respectively.

${\overline{\text{z}}}_{{\text{n}}_{\text{k}}}=\frac{1}{{\text{n}}_{\text{k}}}{\sum }_{l=1}^{{\text{n}}_{\text{k}}}{\text{z}}_{\text{kl}},{\overline{\text{z}}}_{{\text{u}}_{\text{k}}}=\frac{1}{{\text{u}}_{\text{k}}}{\sum }_{l=1}^{{\text{u}}_{\text{k}}}{\text{z}}_{\text{kl}}$: The sample means of the auxiliary variable Z in the $k^{\text{th}}$ strata based on samples of sizes ${\text{n}}_{\text{k}}$ and ${\text{u}}_{\text{k}}$, respectively.

${\text{S}}_{{\text{Y}}_{{\text{N}}_{\text{k}}}}^2$=$\frac{1}{{\text{N}}_{\text{k}}-1}{\sum }_{l=1}^{{\text{N}}_{\text{k}}}{({\text{Y}}_{\text{kl}}-{\overline{\text{Y}}}_{{\text{N}}_{\text{k}}})}^2$, ${\text{S}}_{{\text{X}}_{{\text{N}}_{\text{k}}}}^2$=$\frac{1}{{\text{N}}_{\text{k}}-1}{\sum }_{l=1}^{{\text{N}}_{\text{k}}}{({\text{X}}_{\text{kl}}-{\overline{\text{X}}}_{{\text{N}}_{\text{k}}})}^2$: The population mean squares of the $k^{\text{th}}$ stratum of the study variables Y and X, respectively.

${\text{S}}_{{\text{Z}}_{{\text{N}}_{\text{k}}}}^2$=$\frac{1}{{\text{N}}_{\text{k}}-1}{\sum }_{l=1}^{{\text{N}}_{\text{k}}}{({\text{Z}}_{\text{kl}}-{\overline{\text{Z}}}_{{\text{N}}_{\text{k}}})}^2$: The population mean squares of the $k^{\text{th}}$ stratum for the auxiliary variable Z.

${\text{s}}_{{\text{x}}_{{\text{n}}_{\text{k}}}-{\text{r}}_{\text{1k}}}^{{⁎}^2}=\frac{1}{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}-1}{\sum }_{l=1}^{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}{({\text{x}}_{\text{kl}}-{\overline{\text{x}}}_{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}})}^2$: Depending on the responding part of sample of size ${\text{n}}_{\text{k}}$, the sample mean square of study variable X for the $k^{\text{th}}$ stratum.

${\text{s}}_{{\text{x}}_{{\text{m}}_{\text{k}}}-{\text{r}}_{\text{2k}}}^{{⁎}^2}=\frac{1}{{\text{m}}_{\text{k}}-{\text{r}}_{\text{2k}}-1}{\sum }_{l=1}^{{\text{m}}_{\text{k}}-{\text{r}}_{\text{2k}}}{({\text{x}}_{\text{kl}}-{\overline{\text{x}}}_{{\text{u}}_{\text{k}}-{\text{r}}_{\text{2k}}})}^2$: Depending on the responding part of sample of size ${\text{m}}_{\text{k}}$, the sample mean square of study variable X for the $k^{\text{th}}$ stratum.

${\text{s}}_{{\text{y}}_{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}}^{{⁎}^2}=\frac{1}{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}-1}{\sum }_{l=1}^{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}{({\text{y}}_{\text{kl}}-{\overline{\text{y}}}_{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}})}^2$: Depending on the responding part of sample of size ${\text{n}}_{\text{k}}$, the sample mean square of study variable Y for the $k^{\text{th}}$ stratum.

${\text{s}}_{{\text{y}}_{{\text{u}}_{\text{k}}-{\text{r}}_{\text{3k}}}}^{{⁎}^2}=\frac{1}{{\text{u}}_{\text{k}}-{\text{r}}_{\text{3k}}-1}{\sum }_{l=1}^{{\text{u}}_{\text{k}}-{\text{r}}_{\text{3k}}}{({\text{y}}_{\text{kl}}-{\overline{\text{y}}}_{{\text{u}}_{\text{k}}-{\text{r}}_{\text{3k}}})}^2$: Depending on the responding part of sample of size ${\text{u}}_{\text{k}}$, the sample mean square of study variable Y for the $k^{\text{th}}$ stratum.

${\text{s}}_{{\text{z}}_{{\text{n}}_{\text{k}}}}^{{⁎}^2}=\frac{1}{{\text{n}}_{\text{k}}-1}{\sum }_{l=1}^{{\text{n}}_{\text{k}}}{({\text{z}}_{\text{kl}}-{\overline{\text{z}}}_{{\text{n}}_{\text{k}}})}^2$: Depending on the sample of size ${\text{n}}_{\text{k}}$, the sample mean square of auxiliary variable Z for the $k^{\text{th}}$ stratum.

${\text{s}}_{{\text{z}}_{{\text{m}}_{\text{k}}}}^{{⁎}^2}=\frac{1}{{\text{m}}_{\text{k}-1}}{\sum }_{l=1}^{{\text{m}}_{\text{k}}}{({\text{z}}_{\text{kl}}-{\overline{\text{z}}}_{{\text{m}}_{\text{k}}})}^2$: Depending on the sample of size ${\text{m}}_{\text{k}}$, the sample mean square of auxiliary variable Z for the $k^{\text{th}}$ stratum.

${\text{s}}_{{\text{z}}_{{\text{u}}_{\text{k}}}}^{{⁎}^2}=\frac{1}{{\text{u}}_{\text{k}}-1}{\sum }_{l=1}^{{\text{u}}_{\text{k}}}{({\text{z}}_{\text{kl}}-{\overline{\text{z}}}_{{\text{u}}_{\text{k}}})}^2$: Depending on the sample of size ${\text{m}}_{\text{k}}$, the sample mean square of auxiliary variable Z for the $k^{\text{th}}$ stratum.

${\text{W}}_{\text{k}}$ =$\frac{{\text{N}}_{\text{k}}}{\text{N}}$: The original weight of the $k^{\text{th}}$ stratum, k= 1, 2,...,L

${\mathrm{\Omega }}_{\text{k}}^{⁎}$: The calibrated weight of the $k^{\text{th}}$ stratum, k= 1, 2,...,L based on the sample of size $m_k$

${\mathrm{\Omega }}_{\text{k}}^{⁎⁎}$: The calibrated weight of the $k^{\text{th}}$ stratum, k= 1, 2,...,L based on the sample of size $u_k$

${\text{Q}}_{\text{k}}$: The independent weight of the $k^{\text{th}}$ stratum, k= 1, 2,...,L.

4. Non-response probability model

The $k^{\text{th}}$ stratum is considered using [24] random non-response model. Consider a sample $S_{n_k}$ of size $n_k$ for which some data on X could not be collected due to random non-response. Let $r_{1k}$, where $r_{1k}$ ranges from 0, 1, 2,...,($n_k$ - 2), represent the number of such cases in $S_{n_k}$. Similarly, for a sample $S_{m_k}$ of size $m_k$, let $r_{2k}$ represent the number of units for which Y information on the second occasion could not be acquired due to random non-response, and $r_{3k}$ represent the same for a sample of size $u_k$. It is presumed that $r_{1k}$, $r_{2k}$, and $r_{3k}$ fall within their respective bounds. We assume that $0\le r_{1k}\le (n_k-2)$, $0\le r_{2k}\le (m_k-2)$ and $0\le r_{3k}\le (u_k-2)$. If $p_1$, $p_2$, and $p_3$ probabilities of non-response among the ($n_k$-2), ($m_k$-2), and ($u_k$-2) possible values of non-responses respectively, the discrete probability distributions for $r_{1k}$, $r_{2k}$, and $r_{3k}$ are represented by

$$P(r_{1k})=\frac{n_k-r_{1k}}{n_k{q}_1+2p_1}\left(\begin{array}{c}{n}_k-2\\ r_{1k}\end{array}\right)p_1^{r_{1k}}{q}_1^{n_k-r_{1k}-2};r_{1k}=0,1,2,...,n_k-2,P(r_{2k})=\frac{m_k-r_{2k}}{n_k{q}_2+2p_2}\left(\begin{array}{c}{m}_k-2\\ r_{2k}\end{array}\right)p_2^{r_{2k}}{q}_2^{m_k-r_{2k}-2};r_{2k}=0,1,2,...,m_k-2$$

and

$$P(r_{3k})=\frac{u_k-r_{3k}}{n_k{q}_3+2p_3}\left(\begin{array}{c}{u}_k-2\\ r_{3k}\end{array}\right)p_3^{r_{3k}}{q}_2^{u_k-r_{3k}-2};r_{3k}=0,1,2,...,u_k-2,\text{respectively}.$$

The number of ways to obtain $r_{lk}$ (l = 1, 2, 3) non-responses from all potential non-response values for the three samples are represented by $\left(\begin{array}{c}{n}_k-2\\ r_{1k}\end{array}\right)$, $\left(\begin{array}{c}{m}_k-2\\ r_{2k}\end{array}\right)$, and $\left(\begin{array}{c}{u}_k-2\\ r_{3k}\end{array}\right)$.

5. Proposed estimator

We have developed a set of estimators to estimate the population mean square $S_{Y_{N_k}}^2$ of the $k^{\text{th}}$ stratum, where k may range from 1 to L. These estimators are based on two samples: $S_{m_k}$, which is a common sample of size $m_k$ collected on previous occasions, and $S_{u_k}$, which is a fresh sample of size $u_k$ collected on the current occasion. We refer to the estimators based on these samples as $T_{m_k}$ and $T_{u_k}$, respectively.

$$T_{m_k}=s_{y_{n_k}-r_{1k}}^2+a_k\log [1+|s_{x_{n_k}}^{⁎2}-s_{x_{m_k}}^{⁎2}|]$$ (1)

$$T_{u_k}=s_{u_k-r_{3k}}^2+b_k\log [1+|1-\frac{s_{z_{u_k}}^{⁎2}}{S_{Z_{N_k}}^2}|]$$ (2)

where

$$s_{x_{n_k}}^{⁎2}=s_{x_{n_k}-r_{1k}}^2+\log [1+|1-\frac{s_{z_{n_k}}^{⁎2}}{S_{Z_{N_k}}^2}|]$$

$$s_{x_{m_k}}^{⁎2}=s_{x_{n_k}-r_{2k}}^2+\log [1+|1-\frac{s_{z_{m_k}}^{⁎2}}{S_{Z_{N_k}}^2}|]$$

The constants $a_k$ and $b_k$ may be established by minimizing the mean square errors of the estimators.

In order to estimate the population variance ${\sigma }_Y^2$, we use the estimators $T_{m_k}$ and $T_{u_k}$, which are defined in Equations (1) and (2), and we suggest the matched sample estimator $T_m$ and fresh sample estimator $T_u$ as follows:

$$T_m=\sum _{k=1}^L{\mathrm{\Omega }}^{⁎2}{T}_{m_k}$$ (3)

and

$$T_u=\sum _{k=1}^L{\mathrm{\Omega }}^{⁎⁎2}{T}_{u_k}$$ (4)

Remark 1

The argument of the logarithmic function must be non-negative; otherwise, the function is undefined. When assessing the practical applicability of an estimator, it is crucial to establish its domain of valid values. The presented estimators fall under the category of the log(x) function, where x must be greater than zero, limiting their use to situations with positive values of x. In contrast, for the $\log [1+|x|]$ function, there are no such restrictions on the domain of x, allowing for a broader range of values and increased versatility in real-world applications.

Remark 2

We have observed that the structure remains consistent in large sample approximations. Specifically, if ${\text{m}}_{\text{k}}$, ${\text{u}}_{\text{k}}$, and ${\text{n}}_{\text{k}}$ → ${\text{N}}_{\text{k}}$, then ${\text{s}}_{{\text{z}}_{{\text{n}}_{\text{k}}}}^2$ → ${\text{S}}_{{\text{Z}}_{{\text{N}}_{\text{k}}}}^2$, ${\text{s}}_{{\text{z}}_{{\text{m}}_{\text{k}}}}^2$ → ${\text{S}}_{{\text{Z}}_{{\text{N}}_{\text{k}}}}^2$, ${\text{s}}_{{\text{z}}_{{\text{u}}_{\text{k}}}}^2$ → ${\text{S}}_{{\text{Z}}_{{\text{N}}_{\text{k}}}}^2$, ${\text{s}}_{{\text{y}}_{{\text{n}}_{\text{k}}-{\text{r}}_{\text{1k}}}}^2$ → ${\text{S}}_{{\text{Y}}_{{\text{N}}_{\text{k}}}}^2$, and ${\text{s}}_{{\text{y}}_{{\text{u}}_{\text{k}}-{\text{r}}_{\text{3k}}}}^2$ → ${\text{S}}_{{\text{Y}}_{{\text{N}}_{\text{k}}}}^2$. Using the fact that $\log (1)=0$, we may conclude that ${\text{s}}_{{\text{x}}_{{\text{n}}_{\text{k}}}}^2$ → ${\text{S}}_{{\text{X}}_{{\text{N}}_{\text{k}}}}^2$, and also ${\text{s}}_{{\text{x}}_{{\text{m}}_{\text{k}}}}^2$ → ${\text{S}}_{{\text{X}}_{{\text{N}}_{\text{k}}}}^2$. Therefore, the estimator is consistent in large-sample approximations, as it may be inferred that ${\text{T}}_{{\text{m}}_{\text{k}}}$ → ${\text{S}}_{{\text{Y}}_{{\text{N}}_{\text{k}}}}^2$ and ${\text{T}}_{{\text{u}}_{\text{k}}}$ → ${\text{S}}_{{\text{Y}}_{{\text{N}}_{\text{k}}}}^2$.

6. Suggested calibration technique

The new calibrated weights for the estimator of the population variance ($T_m={\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎2}{T}_{m_k}$) under stratified sampling are obtained by minimizing the chi-square distance function ${\sum }_{k=1}^L\frac{{({\mathrm{\Omega }}_k^{⁎}-W_k)}^2}{Q_k{W}_k}$ while adhering to specific calibration constraints. These constraints are as follows:

• 1.${\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎}=1$
• 2.${\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎}\log ({\overline{z}}_{n_k})={\sum }_{k=1}^L{W}_k\log ({\overline{Z}}_k)$
• 3.${\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎}{c}_{x_{m_k-r_{2k}}}={\sum }_{k=1}^L{W}_k{c}_{x_{n_k-r_{1k}}}$

Where ${\mathrm{\Omega }}_k^{⁎}$ represents the calibrated weights, $W_k$ are the initial weights, and $c_{x_{m_k-r_{2k}}}$=$\frac{s_{x_{m_k-r_{2k}}}}{{\overline{x}}_{m_k-r_{2k}}}$ and $c_{x_{n_k-r_{1k}}}$=$\frac{s_{x_{n_k-r_{1k}}}}{{\overline{x}}_{n_k-r_{1k}}}$.

Next, we consider the estimation of the population variance on the current occasion ($T_u={\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎2}{T}_{u_k}$) using another set of calibrated weights. These calibrated weights are derived through the minimization of the chi-square distance function ${\sum }_{k=1}^L\frac{{({\mathrm{\Omega }}_k^{⁎⁎}-W_k)}^2}{Q_k{W}_k}$ while adhering to specific calibration constraints, which are as follows:

• 1.${\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎}=1$
• 2.${\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎}{c}_{{\overline{z}}_{u_k}}=C_Z$

Where ${\mathrm{\Omega }}_k^{⁎⁎}$ represents the calibrated weights, $c_{z_{u_k}}$=$\frac{s_{z_{u_k}}}{{\overline{z}}_{u_k}}$ and $C_Z$=$\frac{S_Z}{\overline{Z}}$.

It is important to note that $Q_k>0$ are appropriately determined weights that will determine the estimator form.

In Appendix A, detailed derivations have been given.

We propose a way to estimate the population variance of a study variable on the current (second) occasion of a two-occasion successive sampling. Our proposed estimator is a combination of two estimators, $T_u$ and $T_m$, using a scalar ϕ $(0\le \varphi \le 1)$ that minimizes the mean square error of the estimator T.

i.e.

$$T=\varphi T_u+(1-\varphi )T_m$$

If we want to estimate the population variance on each occasion, we choose ϕ=1 and use $T_u$. If we want to estimate the change from one occasion to the next, we choose ϕ=0 and use $T_m$. To deal with both problems at the same time, we need to determine an optimum value for ϕ.

7. Properties of the estimator that has been proposed

The first-order bias and mean squared errors of proposed estimators $T_u$ and $T_m$ are derived, assuming large sample size and utilizing the stated transformations.

$s_{x_{n_k-r_{1k}}}^2=S_{X_{N_k}}^2(1+{ϵ}_{0k})$	$s_{z_{n_k}}^2=S_{Z_{N_k}}^2(1+{ϵ}_{1k})$	$s_{x_{m_k-r_{2k}}}^2=S_{X_{N_k}}^2(1+{ϵ}_{2k})$
$s_{z_{m_k}}^2=S_{Z_{N_k}}^2(1+{ϵ}_{3k})$	$s_{y_{n_k-r_{1k}}}^2=S_{Y_{N_k}}^2(1+{ϵ}_{4k})$	$s_{y_{u_k-r_{3k}}}^2=S_{Y_{N_k}}^2(1+{ϵ}_{5k})$
$s_{z_{u_k}}^2=S_{Z_k}^2(1+{ϵ}_{6k})$	$s_{y_{m_k-r_{2k}}}^2=S_{Y_{N_k}}^2(1+{ϵ}_{7k})$

Based on the calculations, the bias of the suggested estimators $T_m$ and $T_u$, as well as the mean squared error (MSE) of $T_m$ and $T_u$, have been computed accurately up to the first order of approximation.

The bias of $T_m$:

$$Bias(T_m)=\sum _{k=1}^L{a}_k{\mathrm{\Omega }}_k^{⁎2}\left[\frac{S_{X_{N_k}}^2}{2}{f}_{4k}\right({\lambda }_{202k-1})-\frac{S_{X_{N_k}}^4}{2}{f}_{5k}({\lambda }_{400k-1}\left)\right]$$ (5)

Mean Squared Error (MSE) of $T_m$:

$$MSE(T_m)=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎4}\left[\begin{array}{c}\hfill f_{3k}{S}_{Y_{N_k}}^4({\lambda }_{040k}-1)+a_k^2{f}_{5k}{S}_{X_{N_k}}^4({\lambda }_{400k}-1)+a_k^2{f}_{2k}({\lambda }_{004k}-1)\hfill \\ \hfill -2a_k^2{f}_{4k}{S}_{X_{N_k}}^2({\lambda }_{202k}-1)-2a_k{f}_{1k}{S}_{Y_{N_k}}^2({\lambda }_{022k}-1)\hfill \end{array}\right]$$ (6)

and bias of $T_u$:

$$Bias(T_u)=\sum _{k=1}^L{b}_k{\mathrm{\Omega }}_k^{⁎⁎2}{f}_{7k}({\lambda }_{004k}-1)$$ (7)

Mean Squared Error (MSE) of $T_u$:

$$MSE(T_u)=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎4}\left[\begin{array}{c}\hfill f_{6k}{S}_{Y_{N_k}}^4({\lambda }_{040k}-1)+b_k^2{f}_{7k}({\lambda }_{004k}-1)-2b_k{f}_{7k}{S}_{Y_{N_k}}^2({\lambda }_{022k}-1)\hfill \end{array}\right]$$ (8)

where

$$f_{1k}=(\frac{1}{n_k{q}_1+2p_1}-\frac{1}{n_k}),f_{2k}=(\frac{1}{m_k}-\frac{1}{n_k}),f_{3k}=(\frac{1}{n_k{q}_1+2p_1}-\frac{1}{N_k})f_{4k}=(\frac{1}{n_k{q}_1+2p_1}-\frac{1}{m_k}),f_{5k}=(\frac{1}{m_k{q}_2+2p_2}-\frac{1}{n_k{q}_1+2p_1}),f_{6k}=(\frac{1}{u_k{q}_3+2p_3}-\frac{1}{N_k})f_{7k}=(\frac{1}{u_k}-\frac{1}{N_k}),f_{8k}=(\frac{1}{n_k}-\frac{1}{N_k}),f_{9k}=(\frac{1}{m_k}-\frac{1}{N_k})f_{10k}=(\frac{1}{m_k{q}_2+2p_2}-\frac{1}{N_k})$$

and

$${\lambda }_{\alpha \beta \gamma k}=\frac{{\mu }_{\alpha \beta \gamma k}}{\sqrt{{\mu }_{200k}^{\alpha }{\mu }_{020k}^{\beta }{\mu }_{002k}^{\gamma }}},{\mu }_{\alpha \beta \gamma k}=\frac{1}{N_k-1}\sum _{j=1}^{N_k}{(Y_{kj}-{\overline{Y}}_k)}^{\alpha }{(X_{kj}-{\overline{X}}_k)}^{\beta }{(Z_{kj}-{\overline{Z}}_k)}^{\gamma }$$

In Appendix B, detailed derivations have been given.

Thus, we may obtain the bias and MSE of estimator T by combining the biases and MSEs of two non-overlapping samples $T_u$ and $T_m$, as shown below:

$$Bias(T)=\varphi Bias(T_u)+(1-\varphi )Bias(T_m)$$ (9)

and

$$MSE(T)={\varphi }^{2}MSE(T_u)+{(1-\varphi )}^{2}MSE(T_m)$$ (10)

Where equations (5), (7), (6), and (8) provide the expressions for Bias($T_m$), Bias($T_u$), MSE($T_m$), and MSE($T_u$), respectively. As $T_u$ and $T_m$ are based on non-overlapping samples of sizes u and m, respectively, the covariance term may be ignored as it is of order $N^{-1}$, and c$(T_u,T_m)$=0.

8. The estimator's minimum mean squared error (MSE)

MSEs of $T_m$ and $T_u$ depend on $a_k$ and $b_k$. To find the best $a_k$ and $b_k$, we minimize the MSEs in Equations (6) and (8). We get optimal $a_k$ and $b_k$ values.

$$a_{k_{opt}}=\frac{f_{1k}{S}_{Y_{N_k}}^2({\lambda }_{022k}-1)}{f_{5k}{S}_{X_{N_k}}^4({\lambda }_{400k}-1)+f_{2k}({\lambda }_{004k}-1)-2f_{4k}{S}_{X_{N_k}}^2({\lambda }_{202k}-1)}$$ (11)

and

$$b_{k_{opt}}=\frac{S_{Y_{N_k}}^2({\lambda }_{022k}-1)}{({\lambda }_{004k}-1)}$$ (12)

Using $a_{k_{opt}}$ from Eq. (11) in Eq. (6) and $b_{k_{opt}}$ from Eq. (12) in Eq. (8) gives the minimum MSE of $T_m$ and $T_u$.

$$MinMSE(T_m)=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎4}{S}_{Y_{N_k}}^4\left[\begin{array}{c}\hfill f_{3k({\lambda }_{040k}-1)}-\frac{f_{1k}^2{({\lambda }_{022k}-1)}^2}{f_{5k}{S}_{Y_{N_k}}^4({\lambda }_{400k}-1)+f_{2k}({\lambda }_{004k}-1)-2f_{4k}{S}_{X_{N_k}}^2({\lambda }_{202k}-1)}\hfill \end{array}\right]$$ (13)

and

$$MinMSE(T_u)=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎4}{S}_{Y_{N_k}}^4\left[\begin{array}{c}\hfill f_{6k}({\lambda }_{040k}-1)-\frac{f_{7k}{({\lambda }_{022k}-1)}^2}{{\lambda }_{004k}-1}\hfill \end{array}\right]$$ (14)

MSE of T depends on ϕ. To find the best ϕ, we minimize the MSE. The optimal ϕ is

$${\varphi }_{opt}=\frac{MinMSE(T_m)}{MinMSE(T_m)+MinMSE(T_u)}$$ (15)

We may use this ${\varphi }_{opt}$ value to get the best MSE of T, which is

$$MSE{(T)}_{opt}=\frac{MinMSE(T_m)MinMSE(T_u)}{MinMSE(T_m)+MinMSE(T_u)}$$ (16)

Equations (13) and (14) give the expressions for $MinMSE(T_m)$ and $MinMSE(T_u)$, respectively.

9. Empirical study

Before employing an estimator in practical situations, it is crucial to assess its performance based on its inherent properties. In light of this, an empirical analysis has been carried out in this section utilizing both real and simulated data to evaluate the suggested estimator.

To accomplish this, we will conduct a comparison between the suggested estimator T and an alternative estimator τ, which is also designed to handle random non-response and is defined in the same manner. The purpose of this comparison is to evaluate how well the suggested estimator T performs under conditions of random non-response. It is important to note that this evaluation will also include a comparison with the standard estimator because no estimator is proposed in stratified successive sampling under non-response.

$$\tau =\psi {\tau }_u+(1-\psi ){\tau }_m$$

The values of ${\tau }_u={\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎2}{s}_{y_{u_k-r_{3k}}}^2$, ${\tau }_m={\sum }_{k=1}^L{\mathrm{\Omega }}_k^{⁎2}{s}_{y_{m_k-r_{2k}}}^2$ and $\psi (0\le \psi \le 1)$ are unknown. The constant ψ needs to be determined by minimizing the Mean Squared Error (MSE) of estimator τ.

$$MSE({\tau }_m)=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎4}{S}_{Y_{N_k}}^4(\frac{1}{m_k{q}_2+2p_2}-\frac{1}{N_k})({\lambda }_{040k}-1)$$ (17)

and

$$MSE({\tau }_u)=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎4}{S}_{Y_{N_k}}^4(\frac{1}{u_k{q}_3+2p_3}-\frac{1}{N_k})({\lambda }_{040k}-1)$$ (18)

The minimum Mean Squared Error (MSE) of estimator τ by the combination of (17) and (18), up to the first order of approximations, may be expressed as:

$$MSE(\tau )=\frac{MSE({\tau }_u)MSE({\tau }_m)}{MSE({\tau }_u)+MSE({\tau }_m)}$$ (19)

The proposed estimator T may be evaluated in terms of its Percentage Relative Efficiency (PRE) i.e. (20) with respect to the estimator τ. This may be calculated using the following formula:

$$PRE=\frac{MSE(\tau )}{MSE{(T)}_{opt}}\times 100,$$ (20)

where $MSE{(T)}_{opt}$ and $MSE(\tau )$ are defined in Equations (16) and (19), respectively.

The following $Q_k$ values have been considered:

• Case A:

  $Q_k=1.0$

• Case B:

  $Q_k=\frac{1}{W_k}$

• Case C:

  $Q_k={\overline{Z}}_k$

• Case D:

  $Q_k=S_{Z_{N_k}}^2$

The resulting calibrated stratum weights are shown in Tables (2), (9) and (16), respectively.

9.1. Simulation study

We generated data based on our theoretical findings using the statistical computing software R. To create data from a normal distribution with specific parameters and correlation coefficients for both study and auxiliary variables, we utilized the mvrnorm function from the MASS package and the genCorgen function from the Simstudy package. The population parameters for the generated data following a Poisson distribution are outlined in Table 1, while the generated data following a Normal distribution are outlined in Table 8. Simulations were conducted to analyze the effects of the controlling parameter $Q_k$, both with and without random non-response. The results of these simulations are presented in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14.

Table 1.

The statistical parameters correspond to the simulated data from a Poisson distribution.

Stratum	Nk	nk	r1k	mk	r2k	uk	ρxy	ρyz	ρzx
Strata 1	10000	3000	500	2000	400	1000	0.9	0.9	0.9
Strata 2	15000	3000	500	2000	400	1000	0.8	0.8	0.8
Strata 3	5000	2000	400	1200	300	800	0.8	0.8	0.8
Strata 4	8000	800	50	500	50	300	0.6	0.6	0.6

Table 8.

The statistical parameters correspond to the simulated data from a Normal distribution.

Stratum	Nk	nk	r1k	mk	r2k	uk	ρxy	ρyz	ρzx
Strata 1	12000	3600	600	2400	480	1200	0.95	0.95	0.95
Strata 2	17000	3600	600	2400	480	1200	0.85	0.85	0.85
Strata 3	6000	2400	480	1440	360	960	0.85	0.85	0.85
Strata 4	9000	900	57	563	57	337	0.65	0.65	0.65

Table 2.

Calibrated strata weights for simulated data following a Poisson distribution.

Case	Stratum	Wk	${\mathrm{\Omega }}_k^{⁎}$	${\mathrm{\Omega }}_k^{⁎⁎}$
I	1	0.2631579	0.30121385	0.3129923
	2	0.3947368	0.55203331	0.2801177
	3	0.1315789	0.08592402	0.1564961
	4	0.2105263	0.06082881	0.2503938
II	1	0.2631579	0.33327331	0.3013643
	2	0.3947368	0.52933590	0.2801177
	3	0.1315789	0.07974788	0.1697853
	4	0.2105263	0.05764291	0.2487327
III	1	0.2631579	0.30140728	0.3132156
	2	0.3947368	0.55189637	0.2801177
	3	0.1315789	0.08588676	0.1564564
	4	0.2105263	0.06080959	0.2502102
IV	1	0.2631579	0.30107846	0.3124016
	2	0.3947368	0.55212917	0.2801177
	3	0.1315789	0.08595011	0.1565251
	4	0.2105263	0.06084227	0.2509556

Table 3.

For Case I:Q₁ = 1.0, Q₂ = 1.0, Q₃ = 1.0, and Q₄ = 1.0. Subsequently, Bias and PRE were obtained from simulated data following a Poisson distribution, where p₁, p₂, and p₃ represent the probabilities of non-response among the (n_k - 2), (m_k - 2), and (u_k - 2) possible values of non-responses, respectively.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	142.9	141.9	141.1	140.4	0.0001505	0.0001462	0.0001416	0.0001367
0.05	0.10	146.1	145.3	144.6	144.0	0.0001505	0.0001462	0.0001416	0.0001367
0.05	0.15	149.5	148.7	148.1	147.7	0.0001505	0.0001462	0.0001416	0.0001367
0.05	0.20	152.9	152.3	151.8	151.6	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.05	139.3	138.2	137.3	136.5	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.10	142.5	141.5	140.6	140.0	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.15	145.8	144.9	144.1	143.6	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.20	149.1	148.3	147.7	147.3	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.05	136.1	134.9	133.8	132.9	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.10	139.2	138.1	137.1	136.3	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.15	142.4	141.4	140.5	139.8	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.20	145.7	144.8	144.0	143.4	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.05	133.2	131.9	130.7	129.7	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.10	136.3	135.0	133.9	133.0	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.15	139.4	138.2	137.3	136.4	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.20	142.6	141.6	140.7	140.0	0.0001505	0.0001462	0.0001416	0.0001367

Table 4.

For Case II, the values are Q₁ = 3.80, Q₂ = 2.53, Q₃ = 7.60, and Q₄ = 4.75. Subsequently, Bias and PRE were obtained from simulated data following a Poisson distribution.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	143.0	142.1	141.4	140.8	0.000144	0.0001399	0.0001353	0.0001305
0.05	0.10	146.4	145.6	145.0	144.6	0.000144	0.0001399	0.0001353	0.0001305
0.05	0.15	149.9	149.3	148.8	148.5	0.000144	0.0001399	0.0001353	0.0001305
0.05	0.20	153.5	153.0	152.7	152.5	0.000144	0.0001399	0.0001353	0.0001305
0.10	0.05	139.2	138.3	137.4	136.7	0.000144	0.0001399	0.0001353	0.0001305
0.10	0.10	142.6	141.7	140.9	140.3	0.000144	0.0001399	0.0001353	0.0001305
0.10	0.15	146.0	145.2	144.6	144.1	0.000144	0.0001399	0.0001353	0.0001305
0.10	0.20	149.5	148.9	148.4	148.0	0.000144	0.0001399	0.0001353	0.0001305
0.15	0.05	135.9	134.8	133.8	133.0	0.000144	0.0001399	0.0001353	0.0001305
0.15	0.10	139.1	138.1	137.2	136.5	0.000144	0.0001399	0.0001353	0.0001305
0.15	0.15	142.5	141.6	140.8	140.2	0.000144	0.0001399	0.0001353	0.0001305
0.15	0.20	145.9	145.1	144.5	144.0	0.000144	0.0001399	0.0001353	0.0001305
0.20	0.05	132.9	131.7	130.6	129.6	0.000144	0.0001399	0.0001353	0.0001305
0.20	0.10	136.1	134.9	133.9	133.1	0.000144	0.0001399	0.0001353	0.0001305
0.20	0.15	139.3	138.3	137.4	136.6	0.000144	0.0001399	0.0001353	0.0001305
0.20	0.20	142.7	141.8	141.0	140.4	0.000144	0.0001399	0.0001353	0.0001305

Table 5.

For Case III:Q₁=10.141, Q₂=10.07, Q₃= 10.08, and Q₄= 10.05. Subsequently, Bias and PRE were obtained from simulated data following a Poisson distribution.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	142.8901	141.9	141.1	140.5	0.0001505	0.0001462	0.0001416	0.0001367
0.05	0.10	146.1507	145.3	144.6	144.1	0.0001505	0.0001462	0.0001416	0.0001367
0.05	0.15	149.5087	148.8	148.2	147.8	0.0001505	0.0001462	0.0001416	0.0001367
0.05	0.20	152.9686	152.3	151.9	151.6	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.05	139.3401	138.3	137.3	136.5	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.10	142.5191	141.5	140.7	140.0	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.15	145.7932	144.9	144.2	143.6	0.0001505	0.0001462	0.0001416	0.0001367
0.10	0.20	149.1664	148.4	147.8	147.3	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.05	136.1238	134.9	133.8	132.9	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.10	139.2287	138.1	137.1	136.3	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.15	142.4261	141.4	140.5	139.8	0.0001505	0.0001462	0.0001416	0.0001367
0.15	0.20	145.7203	144.8	144.0	143.5	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.05	133.2532	131.9	130.8	129.7	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.10	136.2912	135.1	134.0	133.0	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.15	139.4197	138.3	137.3	136.4	0.0001505	0.0001462	0.0001416	0.0001367
0.20	0.20	142.6426	141.6	140.7	140.0	0.0001505	0.0001462	0.0001416	0.0001367

Table 6.

For Case IV:Q₁= 9.93, Q₂=10.05, Q₃=10.06, and Q₄= 10.19. Subsequently, Bias and PRE were obtained from simulated data following a Poisson distribution.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	142.8	141.8	141.0	140.4	0.0001505	0.0001461	0.0001415	0.0001367
0.05	0.10	146.0	145.2	144.5	144.0	0.0001505	0.0001461	0.0001415	0.0001367
0.05	0.15	149.4	148.6	148.1	147.7	0.0001505	0.0001461	0.0001415	0.0001367
0.05	0.20	152.8	152.2	151.8	151.5	0.0001505	0.0001461	0.0001415	0.0001367
0.10	0.05	139.2	138.1	137.2	136.4	0.0001505	0.0001461	0.0001415	0.0001367
0.10	0.10	142.4	141.4	140.6	139.9	0.0001505	0.0001461	0.0001415	0.0001367
0.10	0.15	145.7	144.8	144.1	143.5	0.0001505	0.0001461	0.0001415	0.0001367
0.10	0.20	149.0	148.3	147.7	147.2	0.0001505	0.0001461	0.0001415	0.0001367
0.15	0.05	136.0	134.8	133.7	132.8	0.0001505	0.0001461	0.0001415	0.0001367
0.15	0.10	139.1	138.0	137.0	136.2	0.0001505	0.0001461	0.0001415	0.0001367
0.15	0.15	142.3	141.3	140.4	139.7	0.0001505	0.0001461	0.0001415	0.0001367
0.15	0.20	145.6	144.7	143.9	143.4	0.0001505	0.0001461	0.0001415	0.0001367
0.20	0.05	133.1	131.8	130.7	129.6	0.0001505	0.0001461	0.0001415	0.0001367
0.20	0.10	136.1	134.9	133.9	132.9	0.0001505	0.0001461	0.0001415	0.0001367
0.20	0.15	139.3	138.2	137.2	136.3	0.0001505	0.0001461	0.0001415	0.0001367
0.20	0.20	142.5	141.5	140.6	139.9	0.0001505	0.0001461	0.0001415	0.0001367

Table 7.

In the absence of non-response, bias and PRE are observed from simulated data following a Poisson distribution when p₁= p₂= p₃=0.

Stratum	PRE	Bias
Case I	144.5398	0.0001416434
Case II	144.5906	0.0001352958
Case III	144.5827	0.0001416345
Case IV	144.4316	0.0001415639

Table 9.

Calibrated strata weights for simulated data following a Normal distribution.

Case	Stratum	Wk	${\mathrm{\Omega }}_k^{⁎}$	${\mathrm{\Omega }}_k^{⁎⁎}$
I	1	0.2727273	0.3808514	0.2609850
	2	0.3947368	0.1446900	0.3997344
	3	0.1363636	0.1269082	0.1304925
	4	0.2045455	0.3475505	0.2087880
II	1	0.2727273	0.3558239	0.2614920
	2	0.3863636	0.1629070	0.3997344
	3	0.1363636	0.1002457	0.1299131
	4	0.2045455	0.3810234	0.2088605
III	1	0.2727273	0.3825238	0.2607414
	2	0.3863636	0.1434726	0.3997344
	3	0.1363636	0.1286899	0.1306599
	4	0.2045455	0.3453136	0.2088643
IV	1	0.2727273	0.3815501	0.2609781
	2	0.3863636	0.1441814	0.3997344
	3	0.1363636	0.1276526	0.1304788
	4	0.2045455	0.3466160	0.2088087

Table 10.

For Case I:Q₁ = 1.0, Q₂ = 1.0, Q₃ = 1.0, and Q₄ = 1.0. Subsequently, Bias and PRE were obtained from simulated data following a Normal distribution, where p₁, p₂, and p₃ represent the probabilities of non-response among the (n_k - 2), (m_k - 2), and (u_k - 2) possible values of non-responses, respectively.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	177.6	172.9	168.8	165.3	0.0004008	0.0003805	0.0003597	0.0003385
0.05	0.10	182.6	177.9	173.8	170.4	0.0004008	0.0003805	0.0003597	0.0003385
0.05	0.15	187.8	183.1	179.1	175.8	0.0004008	0.0003805	0.0003597	0.0003385
0.05	0.20	193.2	188.6	184.7	181.4	0.0004008	0.0003805	0.0003597	0.0003385
0.10	0.05	172.9	168.1	163.8	160.2	0.0004008	0.0003805	0.0003597	0.0003385
0.10	0.10	177.8	172.9	168.7	165.1	0.0004008	0.0003805	0.0003597	0.0003385
0.10	0.15	182.8	178.0	173.8	170.3	0.0004008	0.0003805	0.0003597	0.0003385
0.10	0.20	188.1	183.3	179.2	175.7	0.0004008	0.0003805	0.0003597	0.0003385
0.15	0.05	168.5	163.5	159.1	155.3	0.0004008	0.0003805	0.0003597	0.0003385
0.15	0.10	173.2	168.2	163.8	160.1	0.0004008	0.0003805	0.0003597	0.0003385
0.15	0.15	178.1	173.1	168.8	165.1	0.0004008	0.0003805	0.0003597	0.0003385
0.15	0.20	183.2	178.3	174.0	170.3	0.0004008	0.0003805	0.0003597	0.0003385
0.20	0.05	164.3	159.2	154.7	150.8	0.0004008	0.0003805	0.0003597	0.0003385
0.20	0.10	168.9	163.7	159.2	155.3	0.0004008	0.0003805	0.0003597	0.0003385
0.20	0.15	173.6	168.5	164.0	160.2	0.0004008	0.0003805	0.0003597	0.0003385
0.20	0.20	178.6	173.5	169.0	165.2	0.0004008	0.0003805	0.0003597	0.0003385

Table 11.

For Case II, the values are Q₁ = 3.80, Q₂ = 2.53, Q₃ = 7.60, and Q₄ = 4.75. Subsequently, Bias and PRE were obtained from simulated data following a Normal distribution.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	178.7	173.4	168.8	164.9	0.0004473	0.0004280	0.000408	0.0003874
0.05	0.10	183.0	177.7	173.2	169.3	0.0004473	0.0004280	0.000408	0.0003874
0.05	0.15	187.4	182.2	177.7	173.8	0.0004473	0.0004280	0.000408	0.0003874
0.05	0.20	192.0	186.8	182.4	178.6	0.0004473	0.0004280	0.000408	0.0003874
0.10	0.05	174.7	169.3	164.6	160.5	0.0004473	0.0004280	0.000408	0.0003874
0.10	0.10	178.9	173.5	168.8	164.7	0.0004473	0.0004280	0.000408	0.0003874
0.10	0.15	183.2	177.8	173.2	169.2	0.0004473	0.0004280	0.000408	0.0003874
0.10	0.20	187.7	182.3	177.7	173.8	0.0004473	0.0004280	0.000408	0.0003874
0.15	0.05	170.9	165.4	160.5	156.2	0.0004473	0.0004280	0.000408	0.0003874
0.15	0.10	175.0	169.4	164.6	160.4	0.0004473	0.0004280	0.000408	0.0003874
0.15	0.15	179.2	173.6	168.8	164.7	0.0004473	0.0004280	0.000408	0.0003874
0.15	0.20	183.6	178.1	173.3	169.2	0.0004473	0.0004280	0.000408	0.0003874
0.20	0.05	167.3	161.6	156.6	152.2	0.0004473	0.0004280	0.000408	0.0003874
0.20	0.10	171.2	165.6	160.6	156.2	0.0004473	0.0004280	0.000408	0.0003874
0.20	0.15	175.3	169.7	164.7	160.4	0.0004473	0.0004280	0.000408	0.0003874
0.20	0.20	179.6	173.9	169.0	164.7	0.0004473	0.0004280	0.000408	0.0003874

Table 12.

For Case III:Q₁=11.764729, Q₂=9.509398, Q₃= 8.948854, and Q₄= 10.113946. Subsequently, Bias and PRE were obtained from simulated data following a Normal distribution.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	177.5	172.8	168.8	165.3	0.0003978	0.000377	0.0003566	0.0003354
0.05	0.10	182.5	177.8	173.9	170.5	0.0003978	0.000377	0.0003566	0.0003354
0.05	0.15	187.8	183.1	179.2	175.9	0.0003978	0.000377	0.0003566	0.0003354
0.05	0.20	193.3	188.7	184.8	181.6	0.0003978	0.000377	0.0003566	0.0003354
0.10	0.05	172.8	168.0	163.8	160.2	0.0003978	0.000377	0.0003566	0.0003354
0.10	0.10	177.7	172.8	168.7	165.1	0.0003978	0.000377	0.0003566	0.0003354
0.10	0.15	182.8	178.0	173.9	170.4	0.0003978	0.000377	0.0003566	0.0003354
0.10	0.20	188.1	183.3	179.3	175.9	0.0003978	0.000377	0.0003566	0.0003354
0.15	0.05	168.3	163.3	159.0	155.3	0.0003978	0.000377	0.0003566	0.0003354
0.15	0.10	173.0	168.1	163.8	160.0	0.0003978	0.000377	0.0003566	0.0003354
0.15	0.15	178.0	173.0	168.8	165.1	0.0003978	0.000377	0.0003566	0.0003354
0.15	0.20	183.2	178.2	174.0	170.4	0.0003978	0.000377	0.0003566	0.0003354
0.20	0.05	164.1	159.0	154.6	150.7	0.0003978	0.000377	0.0003566	0.0003354
0.20	0.10	168.7	163.6	159.1	155.3	0.0003978	0.000377	0.0003566	0.0003354
0.20	0.15	173.5	168.4	164.0	160.1	0.0003978	0.000377	0.0003566	0.0003354
0.20	0.20	178.5	173.4	169.0	165.2	0.0003978	0.000377	0.0003566	0.0003354

Table 13.

For Case IV:Q₁= 0.9961224, Q₂=1.0018931, Q₃=1.0054827, and Q₄= 0.9811516. Subsequently, Bias and PRE were obtained from simulated data following a Normal distribution.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	0.05	0.10	0.15	0.20
0.05	0.05	177.6	172.9	168.8	165.3	0.0003995	0.0003792	0.0003584	0.0003372
0.05	0.10	182.6	177.9	173.8	170.4	0.0003995	0.0003792	0.0003584	0.0003372
0.05	0.15	187.8	183.1	179.2	175.8	0.0003995	0.0003792	0.0003584	0.0003372
0.05	0.20	193.3	188.6	184.7	181.5	0.0003995	0.0003792	0.0003584	0.0003372
0.10	0.05	172.9	168.0	163.8	160.2	0.0003995	0.0003792	0.0003584	0.0003372
0.10	0.10	177.7	172.9	168.7	165.1	0.0003995	0.0003792	0.0003584	0.0003372
0.10	0.15	182.8	178.0	173.8	170.3	0.0003995	0.0003792	0.0003584	0.0003372
0.10	0.20	188.1	183.3	179.3	175.8	0.0003995	0.0003792	0.0003584	0.0003372
0.15	0.05	168.4	163.4	159.1	155.3	0.0003995	0.0003792	0.0003584	0.0003372
0.15	0.10	173.1	168.1	163.8	160.1	0.0003995	0.0003792	0.0003584	0.0003372
0.15	0.15	178.0	173.1	168.8	165.1	0.0003995	0.0003792	0.0003584	0.0003372
0.15	0.20	183.2	178.3	174.0	170.4	0.0003995	0.0003792	0.0003584	0.0003372
0.20	0.05	164.2	159.1	154.7	150.7	0.0003995	0.0003792	0.0003584	0.0003372
0.20	0.10	168.8	163.7	159.2	155.3	0.0003995	0.0003792	0.0003584	0.0003372
0.20	0.15	173.6	168.4	164.0	160.1	0.0003995	0.0003792	0.0003584	0.0003372
0.20	0.20	178.6	173.5	169.0	165.2	0.0003995	0.0003792	0.0003584	0.0003372

Table 14.

In the absence of non-response, bias and PRE are observed from simulated data following a Normal distribution when p₁= p₂= p₃=0.

Stratum	PRE	Bias
Case I	182.8948	0.000407048
Case II	184.5569	0.0004524072
Case III	182.7551	0.0004040932
Case IV	182.8089	0.0004057739

9.2. Study based on real data

This illustrates the application of the suggested class of estimators. The data is available in the UCI machine learning repository dataset named “Gas Turbine CO and NOx Emission Data Set,” which consists of 36733 instances of 11 sensor measurements from a gas turbine in Turkey's northwestern region. These measurements were aggregated across an hour (by means of average or sum) to analyze CO and NOx (NO + NO2) flue gas emissions. For this analysis, the $g{t}_{2011}$.csv file was chosen.

The study used the following study and auxiliary variables:

• Y:Gas turbine exhaust pressure (GTEP)
• X:Air filter difference pressure (AFDP)
• Z:Turbine inlet temperature (TIT)

The strata were formed based on the Ambient temperature (AT) into three categories:

• Stratum 1:

  from 2.1163-12.707 C

• Stratum 2:

  from 12.708-21.759 C

• Stratum 3:

  from 21.760-34.532 C

To estimate the population variance in income among college graduates in a region, non-response bias may be corrected using calibrated weights in stratified successive sampling. This process involves adjusting the sampling weights based on response probabilities. The population parameters for the real data are displayed in Table 15. Calibrated weights are presented in Fig. 1 and Table 16, while the effects of controlling parameter $Q_k$ with random non-response are shown in Figure 2, Figure 3, Figure 4, Figure 5 and Table 17, Table 18, Table 19, Table 20. Table 21 and Figure 6, Figure 7 illustrate the scenario without random non-response.

Table 15.

The statistical parameters corresponding to the real data.

Stratum	Nk	nk	r1k	mk	r2k	uk	ρxy	ρyz	ρzx
Strata 1	2500	875	125	625	125	250	0.8999226	0.8283662	0.7473441
Strata 2	2600	910	130	650	130	260	0.8914957	0.9164524	0.8335361
Strata 3	2311	809	116	578	116	231	0.9689409	0.9337467	0.9209707

Figure 1.

Calibrated strata weights for real data.

Table 16.

Calibrated strata weights for real data.

Case	Stratum	Wk	${\mathrm{\Omega }}_k^{⁎}$	${\mathrm{\Omega }}_k^{⁎⁎}$
I	1	0.3373364	0.4930872	0.3101796
	2	0.3508298	0.3163187	0.4557245
	3	0.3118338	0.1905941	0.2340959
II	1	0.3373364	0.4930873	0.3137647
	2	0.3508298	0.3163187	0.4546208
	3	0.3118338	0.1905940	0.2316144
III	1	0.3373364	0.4930873	0.3100504
	2	0.3508298	0.3163187	0.4557643
	3	0.3118338	0.1905940	0.2341853
IV	1	0.3373364	0.4930873	0.3195387
	2	0.3508298	0.3163187	0.4528433
	3	0.3118338	0.1905940	0.2276180

Figure 2.

For Case I:Q₁ = 1.0, Q₂ = 1.0, and Q₃ = 1.0. Subsequently, Bias (shown by the red line) and PRE (blue line) were obtained from real data.

Figure 3.

For Case II:Q₁=2.96, Q₂= 2.85 and Q₃= 3.21. Subsequently, Bias (shown by the red line) and PRE (blue line) were obtained from real data.

Figure 4.

For Case III:Q₁=0.0009208365, Q₂= 0.0009248637, and Q₃=0.0009196857. Subsequently, Bias (shown by the red line) and PRE (blue line) were obtained from real data.

Figure 5.

For Case IV:Q₁=0.0045, Q₂= 0.0031, and Q₃= 0.0046. Subsequently, Bias (shown by the red line) and PRE (blue line) were obtained from real data.

Table 17.

For Case I:Q₁=1.0, Q₂= 1.0 and Q₃= 1.0. Subsequently, Bias and PRE were obtained from real data.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	∀ p3
0.05	0.05	143.7177	146.5793	149.5242	152.5562	0.01287393
0.05	0.10	150.4350	153.5733	156.8091	160.1470	0.01287393
0.05	0.15	157.5665	161.0128	164.5733	168.2538	0.01287393
0.05	0.20	165.1517	168.9418	172.8659	176.9313	0.01287393
0.10	0.05	135.8471	138.5520	141.3357	144.2016	0.01287393
0.10	0.10	142.1950	145.1614	148.2200	151.3750	0.01287393
0.10	0.15	148.9341	152.1916	155.5571	159.0360	0.01287393
0.10	0.20	156.1018	159.6842	163.3933	167.2359	0.01287393
0.15	0.05	128.5160	131.0749	133.7084	136.4196	0.01287393
0.15	0.10	134.5185	137.3248	140.2182	143.2030	0.01287393
0.15	0.15	140.8907	143.9722	147.1559	150.4469	0.01287393
0.15	0.20	147.6676	151.0564	154.5651	158.2001	0.01287393
0.20	0.05	121.6989	124.1220	126.6158	129.1832	0.01287393
0.20	0.10	127.3784	130.0357	132.7756	135.6019	0.01287393
0.20	0.15	133.4072	136.3251	139.3397	142.4558	0.01287393
0.20	0.20	139.8183	143.0270	146.3492	149.7909	0.01287393

Table 18.

For Case II:Q₁=2.964400, Q₂= 2.850385 and Q₃= 3.206837. Subsequently, Bias and PRE were obtained from real data.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	∀ p3
0.05	0.05	143.7036	146.5620	149.5034	152.5316	0.01288324
0.05	0.10	150.4240	153.5588	156.7909	160.1249	0.01288324
0.05	0.15	157.5591	161.0018	164.5585	168.2348	0.01288324
0.05	0.20	165.1485	168.9349	172.8550	176.9159	0.01288324
0.10	0.05	135.8289	138.5306	141.3108	144.1731	0.01288324
0.10	0.10	142.1795	145.1425	148.1974	151.3487	0.01288324
0.10	0.15	148.9218	152.1757	155.5374	159.0122	0.01288324
0.10	0.20	156.0932	159.6719	163.3770	167.2153	0.01288324
0.15	0.05	128.4939	131.0497	133.6798	136.3875	0.01288324
0.15	0.10	134.4987	137.3017	140.1916	143.1726	0.01288324
0.15	0.15	140.8736	143.9517	147.1317	150.4187	0.01288324
0.15	0.20	147.6538	151.0391	154.5439	158.1746	0.01288324
0.20	0.05	121.6731	124.0933	126.5838	129.1477	0.01288324
0.20	0.10	127.3546	130.0087	132.7451	135.5678	0.01288324
0.20	0.15	133.3858	136.3003	139.3112	142.4235	0.01288324
0.20	0.20	139.7997	143.0049	146.3233	149.7609	0.01288324

Table 19.

For Case III:Q₁=0.0009208365, Q₂= 0.0009248637, and Q₃=0.0009196857. Subsequently, Bias and PRE were obtained from real data.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	∀ p3
0.05	0.05	143.7183	146.5800	149.5251	152.5571	0.01287356
0.05	0.10	150.4355	153.5739	156.8098	160.1478	0.01287356
0.05	0.15	157.5669	161.0133	164.5739	168.2546	0.01287356
0.05	0.20	165.1519	168.9422	172.8664	176.9319	0.01287356
0.10	0.05	135.8478	138.5528	141.3366	144.2026	0.01287356
0.10	0.10	142.1957	145.1622	148.2208	151.3760	0.01287356
0.10	0.15	148.9347	152.1923	155.5579	159.0369	0.01287356
0.10	0.20	156.1022	159.6848	163.3940	167.2367	0.01287356
0.15	0.05	128.5169	131.0759	133.7094	136.4208	0.01287356
0.15	0.10	134.5193	137.3257	140.2192	143.2041	0.01287356
0.15	0.15	140.8914	143.9730	147.1569	150.4480	0.01287356
0.15	0.20	147.6682	151.0571	154.5660	158.2011	0.01287356
0.20	0.05	121.6998	124.1231	126.6169	129.1845	0.01287356
0.20	0.10	127.3793	130.0367	132.7767	135.6031	0.01287356
0.20	0.15	133.4080	136.3260	139.3407	142.4570	0.01287356
0.20	0.20	139.8190	143.0279	146.3502	149.7921	0.01287356

Table 20.

For Case IV:Q₁=0.004542526, Q₂= 0.003128517, and Q₃= 0.004646731. Subsequently, Bias and PRE were obtained from real data.

		PRE				Bias
p1	p2	p3=0.05	0.10	0.15	0.20	∀ p3
0.05	0.05	143.6914	146.5429	149.4771	152.4975	0.01289368
0.05	0.10	150.4193	153.5470	156.7715	160.0972	0.01289368
0.05	0.15	157.5632	160.9985	164.5471	168.2148	0.01289368
0.05	0.20	165.1629	168.9416	172.8533	176.9052	0.01289368
0.10	0.05	135.8075	138.5026	141.2758	144.1305	0.01289368
0.10	0.10	142.1648	145.1209	148.1684	151.3116	0.01289368
0.10	0.15	148.9149	152.1616	155.5155	158.9819	0.01289368
0.10	0.20	156.0955	159.6667	163.3637	167.1931	0.01289368
0.15	0.05	128.4640	131.0134	133.6366	136.3370	0.01289368
0.15	0.10	134.4747	137.2709	140.1536	143.1268	0.01289368
0.15	0.15	140.8566	143.9276	147.1000	150.3788	0.01289368
0.15	0.20	147.6450	151.0228	154.5196	158.1418	0.01289368
0.20	0.05	121.6354	124.0492	126.5330	129.0899	0.01289368
0.20	0.10	127.3220	129.9695	132.6988	135.5139	0.01289368
0.20	0.15	133.3592	136.2668	139.2703	142.3746	0.01289368
0.20	0.20	139.7804	142.9784	146.2889	149.7181	0.01289368

Table 21.

In the absence of non-response, Bias and PRE are observed from real data when p₁= p₂= p₃=0.

Stratum	PRE	Bias
Case I	145.1036	0.01079086
Case II	145.0877	0.01079694
Case III	145.1042	0.01079061
Case IV	145.0727	0.01080245

Figure 6.

In the absence of non-response, PRE is observed from real data when p₁ = p₂ = p₃ = 0.

Figure 7.

In the absence of non-response, biases are observed in real data when p₁ = p₂ = p₃ = 0.

10. Discussion and conclusions

Based on the empirical results presented above:

• 1.Table 2, Table 9, Table 16, along with Fig. 1, demonstrate that calibrated strata weights closely resemble the original weights, particularly for the fresh sample. The estimator effectively mitigates the adverse effects of non-responses, leading to negligible bias across all choices of $Q_k$. This study underscores the improved estimation of population variance in stratified successive sampling using calibrated weights in the presence of non-response.
• 2.The results from Table 3, Table 4, Table 5, Table 6, presenting simulated data, demonstrate the effectiveness of the proposed method in reducing non-response bias in practical scenarios. The negligible bias observed in the estimator, with values of ${10}^{-4}$ for simulated data from a Poisson distribution, and similarly observed in Table 10, Table 11, Table 12, Table 13 for simulated data from a Normal distribution, indicates a favorable outcome of our research in addressing non-response cases. Additionally, the findings from Table 17, Table 18, Table 19, Table 20 and Figure 2, Figure 3, Figure 4, Figure 5, depicting real data, confirm the consistency of our approach. The bias, with values of the order of ${10}^{-2}$, emphasizes the significance of our approach in minimizing bias resulting from non-response.
• 3.Table 3, Table 4, Table 5, Table 6, presenting simulated data from the Poisson distribution, and Table 10, Table 11, Table 12, Table 13, showcasing simulated data from the Normal distribution, demonstrate that the proposed estimator exhibits lower mean squared error (MSE) and higher percentage relative efficiency (PRE) compared to the standard estimator when dealing with random non-response. Similarly, Table 17, Table 18, Table 19, Table 20 and Figure 2, Figure 3, Figure 4, Figure 5, based on real data, confirm the same observation, with the percentage relative efficiency (PRE) exceeding 100. These findings indicate that the proposed method is more effective.
• 4.Table 7, Table 14, Table 21, and Figure 6, Figure 7 collectively demonstrate that the proposed estimator exhibits negligible bias and higher percentage relative efficiency (PRE) compared to the standard estimator in the absence of non-response. This observation suggests that the proposed method is more effective, even in scenarios without non-response.
• 5.In a simulation study, we may observe that as the correlation coefficient increases, the percentage relative efficiency (PRE) also increases, while the bias decreases. Conversely, when we decrease the correlation coefficient, the PRE decreases, and the bias increases.
• 6.The findings suggest that for a fixed value of $p_1$ and $p_2$, as the value of $p_3$ increases, both the bias and percentage relative efficiency (PRE) decrease for simulated data. Conversely, for real data, while the bias remains fixed, the percentage relative efficiency (PRE) increases. Similarly, for a fixed value of $p_1$ and $p_3$, increasing $p_2$ results in a constant bias but an increasing percentage relative efficiency (PRE) for both simulated and real data. Furthermore, increasing $p_1$ while keeping $p_2$ and $p_3$ fixed leads to a constant bias but a decreasing percentage relative efficiency (PRE) for both simulated and real data. Additionally, our research reveals that as the non-response rate of $p_2$ increases, the percentage relative efficiency (PRE) also increases, as observed in Table 17, Table 18, Table 19, Table 20 and Figure 2, Figure 3, Figure 4, Figure 5 (real data), as well as in Table 3, Table 4, Table 5, Table 6 and 10-13 (simulated data). These findings highlight an important outcome of our study.

The results obtained from the analysis of the “Gas Turbine CO and NOx Emission Data Set” and simulation studies demonstrate the effectiveness of the proposed methodology in correcting non-response bias and obtaining more accurate estimates. By employing the suggested class of estimators and utilizing calibrated weights, the estimation of population variance may be performed on other real-world data sets as well, such as the estimation of the variance in income among college graduates may be significantly improved. Survey statisticians are recommended to use the estimator for their practical applications in such cases.

CRediT authorship contribution statement

M.K. Pandey: Writing – review & editing, Writing – original draft, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. G.N. Singh: Visualization, Validation, Supervision. Tolga Zaman: Writing – review & editing. Aned Al Mutairi: Software, Writing – review & editing. Manahil SidAhmed Mustafa: Writing – review & editing.

Declaration of Competing Interest

The authors have no conflicts of interest to disclose.

Appendix A. Deriving calibrated strata weights

The calibrated weights are determined by minimizing the Lagrange function while satisfying the given constraints. The Lagrange function, considering the chi-square distance measure and the calibration constraints, is formulated as follows:

$$L_m=\sum _{k=1}^L\frac{{({\mathrm{\Omega }}_k^{⁎}-W_k)}^2}{Q_k{W}_k}-2l_{1m}(\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎}-1)-2l_{2m}(\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎}\log ({\overline{z}}_{n_k})-\sum _{k=1}^L{W}_k\log ({\overline{Z}}_k))-2l_{3m}(\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎}{c}_{x_{m_k-r_{2k}}}-\sum _{k=1}^L{W}_k{c}_{x_{n_k-r_{1k}}})$$ (21)

where $l_{1m}$, $l_{2m}$ and $l_{3m}$ are Lagrange multipliers.

Differentiating the Lagrange function with respect to ${\mathrm{\Omega }}^{⁎}$ and setting the derivative to zero yields the calibrated weight equation:

$${\mathrm{\Omega }}_k^{⁎}=W_k+(l_{1m}+l_{2m}\log ({\overline{z}}_{n_k})+l_{3m}{c}_{x_{m_k-r_{2k}}})W_k{Q}_k$$ (22)

By substituting the derived ${\mathrm{\Omega }}^{⁎}$ values into the given calibration constraints, a matrix equation is obtained:

$$\left[\begin{array}{ccc}\hfill to{t}_{am}\hfill & \hfill to{t}_{bm}\hfill & \hfill to{t}_{cm}\hfill \\ \hfill to{t}_{bm}\hfill & \hfill to{t}_{em}\hfill & \hfill to{t}_{fm}\hfill \\ \hfill to{t}_{cm}\hfill & \hfill to{t}_{fm}\hfill & \hfill to{t}_{hm}\hfill \end{array}\right]\left[\begin{array}{c}\hfill l_{1m}\hfill \\ \hfill l_{2m}\hfill \\ \hfill l_{3m}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill to{t}_{dm}\hfill \\ \hfill to{t}_{gm}\hfill \\ \hfill to{t}_{im}\hfill \end{array}\right]$$

The solution of the matrix equation provides the Lagrange multipliers, which can be expressed as follows:

$$l_{1m}=\frac{de{t}_{\alpha m}}{de{t}_m},l_{2m}=\frac{de{t}_{\beta m}}{de{t}_m},\mathrm{\&}{l}_{3m}=\frac{de{t}_{\gamma m}}{de{t}_m},$$ (23)

where the determinants are given by:

$$de{t}_m=to{t}_{am}to{t}_{em}to{t}_{hm}-to{t}_{am}to{t}_{fm}^2-to{t}_{bm}^{2}to{t}_{hm}+2to{t}_{bm}to{t}_{cm}to{t}_{fm}-to{t}_{em}to{t}_{cm}^2$$ (24)

$$de{t}_{\alpha m}=to{t}_{dm}to{t}_{em}to{t}_{hm}-to{t}_{dm}to{t}_{fm}^2-to{t}_{bm}to{t}_{gm}to{t}_{hm}+to{t}_{bm}to{t}_{im}to{t}_{fm}+to{t}_{cm}to{t}_{gm}to{t}_{fm}-to{t}_{cm}to{t}_{im}to{t}_{em}$$ (25)

$$de{t}_{\beta m}=to{t}_{am}to{t}_{gm}to{t}_{hm}-to{t}_{am}to{t}_{im}to{t}_{fm}-to{t}_{bm}to{t}_{dm}to{t}_{hm}+to{t}_{cm}to{t}_{dm}to{t}_{fm}+to{t}_{bm}to{t}_{cm}to{t}_{im}-to{t}_{cm}^{2}to{t}_{gm}$$ (26)

$$de{t}_{\gamma m}=to{t}_{am}to{t}_{em}to{t}_{im}-to{t}_{am}to{t}_{gm}to{t}_{fm}-to{t}_{bm}^{2}to{t}_{im}+to{t}_{bm}to{t}_{cm}to{t}_{gm}+to{t}_{bm}to{t}_{dm}to{t}_{fm}-to{t}_{cm}to{t}_{dm}to{t}_{em}$$ (27)

Now, let us define the term $to{t}_{am}$, $to{t}_{bm}$, $to{t}_{cm}$, $to{t}_{dm}$, $to{t}_{em}$, $to{t}_{fm}$, $to{t}_{gm}$, $to{t}_{hm}$, $to{t}_{im}$ as follows:

totam=${\sum }_{k=1}^L{W}_k{Q}_k$	totbm=${\sum }_{k=1}^L{W}_k{Q}_k\log ({\overline{z}}_{n_k})$
totcm=${\sum }_{k=1}^L{W}_k{Q}_k{c}_{x_{m_k-r_{2k}}}$	totdm=1-${\sum }_{k=1}^L{W}_k$
totem=${\sum }_{k=1}^L{W}_k{Q}_k{(\log ({\overline{z}}_{n_k}))}^2$	totfm=${\sum }_{k=1}^L{W}_k{Q}_k\log ({\overline{z}}_{n_k})c_{x_{m_k-r_{2k}}}$
totgm=${\sum }_{k=1}^L{W}_k(\log ({\overline{Z}}_k)-\log ({\overline{z}}_{n_k}))$	tothm=${\sum }_{k=1}^L{W}_k{Q}_k{c}_{x_{m_k-r_{2k}}}^2$
totim=${\sum }_{k=1}^L{W}_k{c}_{x_{n_k-r_{1k}}}-{\sum }_{k=1}^L{W}_k{c}_{x_{m_k-r_{2k}}}$

Similarly, the calibrated weights for the $T_u$ estimator are obtained by minimizing the Lagrange function while satisfying the given calibration constraints. The Lagrange function, considering the chi-square distance measure and the calibration constraints, is expressed as follows:

$$L_u=\sum _{k=1}^L\frac{{({\mathrm{\Omega }}_k^{⁎⁎}-W_k)}^2}{Q_k{W}_k}-2l_{1u}(\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎}-1)-2l_{2u}(\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎}{c}_{{\overline{z}}_{u_k}}-C_Z)$$ (28)

where $l_{1u}$ and $l_{2u}$ are Lagrange multipliers.

Differentiating the Lagrange function with respect to ${\mathrm{\Omega }}^{⁎⁎}$ and setting the derivative to zero gives the following calibrated weight equation:

$${\mathrm{\Omega }}_k^{⁎⁎}=W_k+(l_{1u}+l_{2u}{c}_{{\overline{z}}_{u_k}})W_k{Q}_k$$ (29)

By substituting the derived ${\mathrm{\Omega }}^{⁎⁎}$ values into the given calibration constraints, a matrix equation is formed:

$$\left[\begin{array}{cc}\hfill to{t}_{au}\hfill & \hfill to{t}_{bu}\hfill \\ \hfill to{t}_{bu}\hfill & \hfill to{t}_{du}\hfill \end{array}\right]\left[\begin{array}{c}\hfill l_{1u}\hfill \\ \hfill l_{2u}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill to{t}_{cu}\hfill \\ \hfill to{t}_{eu}\hfill \end{array}\right]$$

Solving this matrix equation provides the Lagrange multipliers, given by:

$$l_{1u}=\frac{de{t}_{\alpha u}}{de{t}_u},\mathrm{\&}{l}_{2u}=\frac{de{t}_{\beta u}}{de{t}_u}$$ (30)

where the determinants are calculated as follows:

$$de{t}_u=to{t}_{au}to{t}_{du}-to{t}_{bu}^2$$ (31)

$$de{t}_{\alpha u}=to{t}_{cu}to{t}_{du}-to{t}_{bu}to{t}_{eu}$$ (32)

$$de{t}_{\beta u}=to{t}_{au}to{t}_{eu}-to{t}_{bu}to{t}_{cu}$$ (33)

Now, let us define the term $to{t}_{au}$, $to{t}_{bu}$, $to{t}_{cu}$, $to{t}_{du}$, $to{t}_{eu}$ as follows:

totau=${\sum }_{k=1}^L{W}_k{Q}_k$	totbu=${\sum }_{k=1}^L{W}_k{Q}_k{c}_{{\overline{z}}_{u_k}}$
totcu=1-${\sum }_{k=1}^L{W}_k$	totdu=${\sum }_{k=1}^L{W}_k{Q}_k{c}_{{\overline{z}}_{u_k}}^2$
toteu=$C_Z-{\sum }_{k=1}^L{W}_k{c}_{{\overline{z}}_{u_k}}$

The calibrated weights for the estimators of population variance, $T_m$ and $T_u$, are determined using these Lagrange multipliers and the initial weights $W_k$.

Appendix B. Deriving bias and MSE of the proposed estimator

The expectations obtained after applying the transformations specified in section 7 are as follows:

E(${ϵ}_{lk}$)=0 and $|{ϵ}_{lk}|\le$1, ∀ l= 0, 1,...,7.

E(${ϵ}_{0k}^2$)=f3k(λ400k − 1)	E(${ϵ}_{1k}^2$)=f8k(λ004k − 1)	E(${ϵ}_{2k}^2$)=f10k(λ400k − 1)
E(${ϵ}_{3k}^2$)=f9k(λ004k − 1)	E(${ϵ}_{4k}^2$)=f3k(λ040k − 1)	E(ϵ0kϵ2k)=f3k(λ400k − 1)
E(ϵ0kϵ4k)=f3k(λ220k − 1)	E(ϵ2kϵ4k)=f3k(λ220k − 1)	E(ϵ3kϵ4k)=f3k(λ022k − 1)
E(ϵ1kϵ4k)=f8k(λ022k − 1)	E(ϵ0kϵ3k)=f3k(λ202k − 1)	E(ϵ2kϵ3k)=f2k(λ202k − 1)
E(ϵ0kϵ1k)=f8k(λ202k − 1)	E(ϵ1kϵ2k)=f8k(λ202k − 1)	E(ϵ1kϵ3k)=f8k(λ004k − 1)
E(${ϵ}_{5k}^2$)=f6k(λ040k − 1)	E(${ϵ}_{6k}^2$)=f7k(λ004k − 1)	E(ϵ5kϵ6k)=f7k(λ022k − 1)
E(${ϵ}_{7k}^2$)=f10k(λ040k − 1)

Where the notations were previously defined in section 7.

$$f_{1k}=(\frac{1}{n_k{q}_1+2p_1}-\frac{1}{n_k}),f_{2k}=(\frac{1}{m_k}-\frac{1}{n_k}),f_{3k}=(\frac{1}{n_k{q}_1+2p_1}-\frac{1}{N_k})f_{4k}=(\frac{1}{n_k{q}_1+2p_1}-\frac{1}{m_k}),f_{5k}=(\frac{1}{m_k{q}_2+2p_2}-\frac{1}{n_k{q}_1+2p_1}),f_{6k}=(\frac{1}{u_k{q}_3+2p_3}-\frac{1}{N_k})f_{7k}=(\frac{1}{u_k}-\frac{1}{N_k}),f_{8k}=(\frac{1}{n_k}-\frac{1}{N_k}),f_{9k}=(\frac{1}{m_k}-\frac{1}{N_k})f_{10k}=(\frac{1}{m_k{q}_2+2p_2}-\frac{1}{N_k})$$

and

$${\lambda }_{\alpha \beta \gamma k}=\frac{{\mu }_{\alpha \beta \gamma k}}{\sqrt{{\mu }_{200k}^{\alpha }{\mu }_{020k}^{\beta }{\mu }_{002k}^{\gamma }}},{\mu }_{\alpha \beta \gamma k}=\frac{1}{N_k-1}\sum _{j=1}^{N_k}{(Y_{kj}-{\overline{Y}}_k)}^{\alpha }{(X_{kj}-{\overline{X}}_k)}^{\beta }{(Z_{kj}-{\overline{Z}}_k)}^{\gamma }$$

Remark

When the data are free of random non-response, or for $p_1$=0 and $p_2$=0, the above assumptions agree with the typical results.

By applying these transformations to $s_{x_{n_k}}^{⁎2}$ and $s_{x_{m_k}}^{⁎2}$, we obtain:

$$s_{x_{n_k}}^{⁎2}=S_{X_{N_k}}^2(1+{ϵ}_{01})+|-{ϵ}_{1k}|-\frac{|-{ϵ}_{1k}^2|}{2}+\frac{|-{ϵ}_{1k}^3|}{3}+...$$ (34)

$$s_{x_{m_k}}^{⁎2}=S_{X_{N_k}}^2(1+{ϵ}_{02})+|-{ϵ}_{3k}|-\frac{|-{ϵ}_{3k}^2|}{2}+\frac{|-{ϵ}_{3k}^3|}{3}+...$$ (35)

By utilizing Equations (34) and (35), we may transform Equation (1) into:

$$T_{m_k}=S_{Y_{N_k}}^2(1+{ϵ}_{4k})+a_k\left[\begin{array}{c}\hfill S_{X_{N_k}}^2({ϵ}_{0k}-{ϵ}_{2k})+({ϵ}_{1k}-{ϵ}_{3k})-\frac{({ϵ}_{1k}^2-{ϵ}_{3k}^2)}{2}-\frac{S_{X_{N_k}}^4{({ϵ}_{0k}-{ϵ}_{2k})}^2}{2}\hfill \\ \hfill -\frac{{({ϵ}_{3k}-{ϵ}_{1k})}^2}{2}-\frac{S_{X_{N_k}}^2({ϵ}_{0k}{ϵ}_{1k}-{ϵ}_{0k}{ϵ}_{3k}-{ϵ}_{2k}{ϵ}_{1k}+{ϵ}_{2k}{ϵ}_{3k})}{2}\hfill \end{array}\right]$$ (36)

We obtain, based on equations (3) and (36), the following

$$T_m=\sum _{k=1}^L{\mathrm{\Omega }}^{⁎2}{S}_{Y_{N_k}}^2+\sum _{k=1}^L{\mathrm{\Omega }}^{⁎2}{S}_{Y_{N_k}}^2{ϵ}_{4k}+\sum _{k=1}^L{a}_k{\mathrm{\Omega }}^{⁎2}\left[\begin{array}{c}\hfill S_{X_{N_k}}^2({ϵ}_{0k}-{ϵ}_{2k})+({ϵ}_{1k}-{ϵ}_{3k})-\frac{({ϵ}_{1k}^2-{ϵ}_{3k}^2)}{2}-\frac{S_{X_{N_k}}^4{({ϵ}_{0k}-{ϵ}_{2k})}^2}{2}\hfill \\ \hfill -\frac{{({ϵ}_{3k}-{ϵ}_{1k})}^2}{2}-\frac{S_{X_{N_k}}^2({ϵ}_{0k}{ϵ}_{1k}-{ϵ}_{0k}{ϵ}_{3k}-{ϵ}_{2k}{ϵ}_{1k}+{ϵ}_{2k}{ϵ}_{3k})}{2}\hfill \end{array}\right]$$ (37)

Given that the calibrated weight ${\mathrm{\Omega }}_k^{⁎}$ is in such close proximity to the strata weight $W_k$, it is reasonable to make the assumption that...

$$\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎2}{S}_{Y_{N_k}}^2\approx \sum _{k=1}^L{W}_k^2{S}_{Y_{N_k}}^2={\sigma }_Y^2$$

In order for Equation (37) to be transformed into

$$T_m={\sigma }_Y^2+\sum _{k=1}^L{\mathrm{\Omega }}^{⁎2}{S}_{Y_{N_k}}^2{ϵ}_{4k}+\sum _{k=1}^L{a}_k{\mathrm{\Omega }}^{⁎2}\left[\begin{array}{c}\hfill S_{X_{N_k}}^2({ϵ}_{0k}-{ϵ}_{2k})+({ϵ}_{1k}-{ϵ}_{3k})-\frac{({ϵ}_{1k}^2-{ϵ}_{3k}^2)}{2}-\frac{S_{X_{N_k}}^4{({ϵ}_{0k}-{ϵ}_{2k})}^2}{2}\hfill \\ \hfill -\frac{{({ϵ}_{3k}-{ϵ}_{1k})}^2}{2}-\frac{S_{X_{N_k}}^2({ϵ}_{0k}{ϵ}_{1k}-{ϵ}_{0k}{ϵ}_{3k}-{ϵ}_{2k}{ϵ}_{1k}+{ϵ}_{2k}{ϵ}_{3k})}{2}\hfill \end{array}\right]$$ (38)

In the same way, when we apply the transformations to Equation (2), we obtain:

$$T_{u_k}=S_{Y_{N_k}}^2(1+{ϵ}_{5k})+b_k|-{ϵ}_{6k}|-b_k\frac{|-{ϵ}_{6k}^2|}{2}+...$$ (39)

By utilizing Equation (4) and (39), we obtain:

$$T_u=\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎2}{S}_{Y_{N_k}}^2+\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎2}\left[\begin{array}{c}\hfill S_{Y_{N_k}}^2{ϵ}_{5k}+b_k{ϵ}_{6k}-\frac{b_k{ϵ}_{6k}^2}{2}+...\hfill \end{array}\right]$$ (40)

Given that the calibrated weight ${\mathrm{\Omega }}_k^{⁎⁎}$ is in such close proximity to the strata weight $W_k$, it is reasonable to make the assumption that...

$$\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎2}{S}_{Y_{N_k}}^2\approx \sum _{k=1}^L{W}_k^2{S}_{Y_{N_k}}^2={\sigma }_Y^2$$

In order for Equation (40) to be transformed into

$$T_u={\sigma }_Y^2+\sum _{k=1}^L{\mathrm{\Omega }}_k^{⁎⁎2}\left[\begin{array}{c}\hfill S_{Y_{N_k}}^2{ϵ}_{5k}+b_k{ϵ}_{6k}-\frac{b_k{ϵ}_{6k}^2}{2}+...\hfill \end{array}\right]$$ (41)

The equations for the bias of the proposed estimator $T_m$ and $T_u$, as well as the mean squared error (MSE) of $T_m$ and $T_u$, are derived by taking expectations on both sides of Equations (38) and (41).

Data availability statement

Data will be made available on request.