Estimation Through Calibration Under Stratified Sampling with Non-Response and Measurement Error Effects

Abstract

In survey sampling, the presence of non-response and measurement error often leads to biased and inefficient estimates, particularly in stratified random sampling designs. This study introduces a new calibration estimation technique for stratified sampling that effectively accounts for non-response and measurement error. By incorporating auxiliary data and optimizing calibrated weights, the proposed estimator minimizes bias and enhances efficiency. The estimator employs auxiliary information through calibrated weights derived using a chi-square-type distance function. Furthermore, the performance of the suggested calibration estimator has been compared with that of the Hansen and Hurwitz’ estimator, the separate ratio-type estimator and the Singh’s estimator. To validate the efficiency and superiority of the proposed method over traditional estimators, an empirical evaluation has been carried out using simulated datasets. The comparative assessment with existing estimators demonstrates that the proposed method provides improved precision and robustness.

1. Introduction and Literature Review

In an exploratory survey, we encounter/incur certain errors that are not related to sampling. These errors include inaccuracies in observations due to wrong reporting, recording, tabulating, processing of data, or failure to measure some units of the sample. The measurements obtained on the units for estimating the study characteristic are rarely accurate. Measurement errors or observational errors, which refer to the disparity between observed values and the true values of the study characteristic, are common in survey sampling. For instance, a lot of families in a country typically do not register their baby; hence, no birth certificate can be issued because the birth could not have been registered. Since the birth was not registered, it is likely that the respondent who was part of the sample provided an approximate age rather than the real age.

Moreover, even when a variable is clearly specified, it occasionally happens that observations can be made on closely comparable alternatives known as proxies. As an illustration, if we want to learn about someone’s financial situation but they are unwilling to answer, we can still gather the information by changing the inquiry. For example, we could inquire about their educational background rather than explicitly asking about their financial situation. This will only be an estimate, though, as having a high level of education does not automatically translate to having a successful career. The issue of measuring inaccuracies has been covered by numerous authors, such as [1,2,3,4,5,6,7,8].

Besides this, some other factors may contribute to the non-sampling errors. It is possible that the respondent could not provide the needed details; however, the right respondent was intended for the query. Such a kind of non-sampling error is termed a non-response. Non-response is a further problem that frequently occurs during survey sampling. The problem of non-response might occur for any of the following reasons: the respondent may have been absent at the time of the survey, have refused to participate or his/her inability to remember the correct response. The phenomenon of non-response has been studied by the authors including [9,10,11,12,13,14,15,16,17,18,19]. Several researchers, such as [20,21,22,23,24] have recently combined their studies on measurement error and non-response.

In this chapter, we have proposed a new calibration estimator for estimating the mean of a stratified population in the presence of both non-response and measurement error. We endeavored to examine the consequences of measurement error and non-response on the study variable. New calibration weights have been created by minimizing the chi-square-type distance function subject to some calibration constraints based on the auxiliary information. Additionally, empirical data through simulation analysis have been shown to support the effectiveness of the proposed calibration estimator. The effectiveness of the suggested calibration estimator has been elaborated over some existing estimators.

2. Sampling Design, Procedures and Notations

Suppose a finite population consists of $N$ units and it is divided into $L$ homogeneous sub-groups (called strata) such that $e^{th}$ stratum comprises $N_e$ units $\left(e=1,2,\dots ,L\right)$ and ${\displaystyle {\sum }_{e=1}^L{N}_e}=N$. Let $Y$ and $X$ be the study and auxiliary variables assuming values $y_{ei}$ and $x_{ei}$, respectively, on the $i^{th}$ unit in the $e^{th}$ stratum $\left(i=1,2,\dots ,N_e\right)$. Let $\overline{Y}$ and $\overline{X}$ be the population means of the variables $Y$ and $X$, respectively. Let ${\overline{Y}}_e$ and ${\overline{X}}_e$ be the respective population means of the variables $Y$ and $X$ in the $e^{th}$ stratum. We choose a sample of $n_e$ units from $e^{th}$ stratum using the simple random sampling without replacement (SRSWOR) method and assume that $n_{e1}$ units respond and $n_{e2}$ units do not respond on study variable $Y$. It is further assumed that the auxiliary variable $X$ is free from the non-response. We then select a sub-sample of $k_e$ units $\left(k_e={\displaystyle \frac{n_{e2}}{g_e}},g_e>1\right)$ from $n_{e2}$ non-responding units in the $e^{th}$ stratum. Here, $g_e$ indicates the inverse sampling rate.

Let $\left(y_{ei}^{*},x_{ei}^{*}\right)$ and $\left(y_{ei},x_{ei}\right)$ be the observed and actual values of the variables (X, Y) on the $i^{th}$ unit in the $e^{th}$ stratum. The measurement errors $U$ and $V$ can be defined as

$$u_{ei}=y_{ei}^{*}-y_{ei}$$

(1)

$$v_{ei}=x_{ei}^{*}-x_{ei}$$

(2)

Let $S_{Ye}^2$ and $S_{Xe}^2$ be the population variances of $Y$ and $X$ in the $e^{th}$ stratum. Let $S_{Ye\left(2\right)}^2$ and $S_{Xe\left(2\right)}^2$ be the population variances of $Y$ and $X$ for the non-responding group in the $e^{th}$ stratum. Let $S_{Ue}^2$ and $S_{Ve}^2$ be the population variances related to the measurement errors $U$ and $V$ in the $e^{th}$ stratum. Let $S_{Ue\left(2\right)}^2$ and $S_{Ve\left(2\right)}^2$ be the population variances related to the measurement errors $U$ and $V$ for the non-responding group in the $e^{th}$ stratum. Let $\left(C_{Ye},C_{Xe}\right)$ be the coefficients of variation under the variables $\left(Y,X\right)$ in the $e^{th}$ stratum. Let $\left(C_{Ye\left(2\right)},C_{Xe\left(2\right)}\right)$ be the coefficients of variation under the variables $\left(Y,X\right)$ for the non-responding group in the $e^{th}$ stratum. The mathematical interpretations of the notations are given below:

$\overline{Y}={\displaystyle \sum _{e=1}^L{p}_e{\overline{Y}}_e}$, $\overline{X}={\displaystyle \sum _{e=1}^L{p}_e{\overline{X}}_e}$, ${\overline{Y}}_e={\displaystyle \frac{1}{N_e}}{\displaystyle \sum _{i=1}^{N_e}{y}_{ei}}$, ${\overline{X}}_e={\displaystyle \frac{1}{N_e}}{\displaystyle \sum _{i=1}^{N_e}{x}_{ei}}$, $S_{Ye}^2={\displaystyle \frac{1}{N_e-1}}{\displaystyle \sum _{i=1}^{N_e}{\left(y_{ei}-{\overline{Y}}_e\right)}^2}$, $S_{Xe}^2={\displaystyle \frac{1}{N_e-1}}{\displaystyle \sum _{i=1}^{N_e}{\left(x_{ei}-{\overline{X}}_e\right)}^2}$, $S_{Ye(2)}^2={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}{\left(y_{ei}-{\overline{Y}}_{e2}\right)}^2}$, $S_{Xe(2)}^2={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}{\left(x_{ei}-{\overline{X}}_{e2}\right)}^2}$, $S_{XYe(2)}={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}\left(x_{ei}-{\overline{X}}_{e2}\right)}\left(y_{ei}-{\overline{Y}}_{e2}\right)$, $S_{UVe(2)}={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}\left(u_{ei}-{\overline{U}}_{e2}\right)}\left(v_{ei}-{\overline{V}}_{e2}\right)$, $S_{Ue}^2={\displaystyle \frac{1}{N_e-1}}{\displaystyle \sum _{i=1}^{N_e}{\left(u_{ei}-{\overline{U}}_e\right)}^2}$, $S_{Ve}^2={\displaystyle \frac{1}{N_e-1}}{\displaystyle \sum _{i=1}^{N_e}{\left(v_{ei}-{\overline{V}}_e\right)}^2}$, $S_{Ue(2)}^2={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}{\left(u_{ei}-{\overline{U}}_{e2}\right)}^2}$, $S_{Ve(2)}^2={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}{\left(v_{ei}-{\overline{V}}_{e2}\right)}^2}$, $C_{Ye}={\displaystyle \frac{S_{Ye}}{{\overline{Y}}_e}}$, $C_{Xe}={\displaystyle \frac{S_{Xe}}{{\overline{X}}_e}}$, $C_{Ye(2)}={\displaystyle \frac{S_{Ye(2)}}{{\overline{Y}}_{e2}}}$, $C_{Xe(2)}={\displaystyle \frac{S_{Xe(2)}}{{\overline{X}}_{e2}}}$, ${\overline{Y}}_{e2}={\displaystyle \frac{1}{N_{e2}}}{\displaystyle \sum _i^{N_{e2}}{y}_{ei}}$, ${\overline{X}}_{e2}={\displaystyle \frac{1}{N_{e2}}}{\displaystyle \sum _i^{N_{e2}}{x}_{ei}}$, ${\overline{U}}_e={\displaystyle \frac{1}{N_e}}{\displaystyle \sum _{i=1}^{N_e}{u}_{ei}}$, ${\overline{V}}_e={\displaystyle \frac{1}{N_e}}{\displaystyle \sum _{i=1}^{N_e}{v}_{ei}}$, ${\overline{U}}_{e2}={\displaystyle \frac{1}{N_{e2}}}{\displaystyle \sum _i^{N_{e2}}{u}_{ei}}$, ${\overline{V}}_{e2}={\displaystyle \frac{1}{N_{e2}}}{\displaystyle \sum _i^{N_{e2}}{v}_{ei}}$ and $P_e={\displaystyle \frac{N_e}{N}}$. $N_{e2}$ is the number of non-responding units in the $e^{th}$ stratum.

3. Existing Estimators

In stratified random sampling, ref. [15] estimator for the population mean $\overline{Y}$ under measurement error and non-response is given as

$${\overline{y}}_{st\left(HH\right)}^{\prime }={\displaystyle \sum _{e=1}^L{p}_e{\overline{y}}_e^{*}}$$

(3)

where ${\overline{y}}_e^{*}=\left({\displaystyle \frac{n_{e1}}{n_e}}\right){\overline{y}}_{e1}+\left({\displaystyle \frac{n_{e2}}{n_e}}\right){\overline{y}}_{e2}^{\prime }$. ${\overline{y}}_{e1}$ and ${\overline{y}}_{e2}^{\prime }$ are, respectively, the means based on $n_{e1}$ and $k_e$ units under study variable $Y$.

The expression for the variance (VAR) of the estimator ${\overline{y}}_{st\left(HH\right)}^{\prime }$ is represented as

$$V\left({\overline{y}}_{st\left(HH\right)}^{\prime }\right)={\displaystyle \sum _{e=1}^L{P}_e^2{A}_e}$$

(4)

where $A_e={\lambda }_{e2}\left(S_{Ye}^2+S_{Ue}^2\right)+{\theta }_{e2}\left(S_{Ye\left(2\right)}^2+S_{Ue\left(2\right)}^2\right)$, ${\lambda }_{e2}=\left({\displaystyle \frac{1}{n_e}}-{\displaystyle \frac{1}{N_e}}\right)$ and ${\theta }_{e2}={\displaystyle \frac{P_{e2}\left(g_e-1\right)}{n_e}}$. $P_{e2}$ is the non-response rate in the $e^{th}$ stratum.

The separate ratio-type estimator of the population mean $\overline{Y}$ in stratified random sampling under measurement error and non-response is given by

$${\overline{y}}_{st\left(R\right)}^{\prime }={\displaystyle \sum _{e=1}^L{p}_e{R}_{re}}$$

(5)

where $R_{re}={\overline{y}}_e^{*}{\displaystyle \frac{{\overline{X}}_e}{{\overline{x}}_e^{*}}}$ and ${\overline{x}}_e^{*}=\left({\displaystyle \frac{n_{e1}}{n_e}}\right){\overline{x}}_{e1}+\left({\displaystyle \frac{n_{e2}}{n_e}}\right){\overline{x}}_{e2}^{\prime }$. ${\overline{x}}_{e1}$ and ${\overline{x}}_{e2}^{\prime }$ are, respectively, the means based on $n_{e1}$ and $k_e$ units under the auxiliary variable $X$.

The expression for the mean square error (MSE) of the estimator ${\overline{y}}_{st\left(R\right)}^{\prime }$ is given as follows:

$$MSE\left({\overline{y}}_{st\left(R\right)}^{*}\right)={\displaystyle \sum _{e=1}^U{p}_e^2}\left[\begin{array}{l}\left({\displaystyle \frac{1}{n_e}}-{\displaystyle \frac{1}{N_e}}\right)\left(S_{Ye}^2+R_e^2{S}_{Xe}^2-2R_e{\rho }_{XYe}{S}_{Ye}{S}_{Xe}+S_{Ue}^2+R_e^2{S}_{Ve}^2\right)+\\ P_{e2}\left({\displaystyle \frac{g_e-1}{n_e}}\right)\left(\left(S_{Ye\left(2\right)}^2+R_e^2{S}_{Xe\left(2\right)}^2-2R_e{\rho }_{X{Y}_{e\left(2\right)}}{S}_{Ye\left(2\right)}{S}_{Xe\left(2\right)}\right)+\left(S_{Ue\left(2\right)}^2+R_e^2{S}_{Ve\left(2\right)}^2\right)\right)\end{array}\right]$$

(6)

where $R_e={\displaystyle \frac{{\overline{Y}}_e}{{\overline{X}}_e}}$. ${\rho }_{XYe}$ is the population correlation coefficient between $Y$ and $X$ in the $e^{th}$ stratum. ${\rho }_{XYe\left(2\right)}$ is the population correlation coefficient between $Y$ and $X$ for the non-response group in the $e^{th}$ stratum.

Ref. [24] has proposed a new estimator of the population mean $\overline{Y}$ under stratified random sampling in the presence of measurement error and non-response as

$${\overline{y}}_{st\left(S\right)}^{\prime }={\displaystyle \sum _{e=1}^L{P}_e\left[{\overline{y}}_e^{*}+{\alpha }_e\log \left(\frac{{\overline{x\prime }}_e^{*}}{{\overline{X}}_e}\right)\right]}$$

(7)

where ${\overline{x\prime }}_e^{*}={\displaystyle \frac{N_e{\overline{X}}_e-n_e{\overline{x}}_e^{*}}{N_e-n_e}}$ and ${\alpha }_e$ is arbitrarily chosen scaler.

The following are the approximate expressions for the bias and MSE of the estimator ${\overline{y}}_{st\left(S\right)}^{\prime }$ up to the first order of approximation:

$$Bias\left({\overline{y}}_{st\left(S\right)}^{\prime }\right)\cong -{\displaystyle \frac{1}{2}}{\displaystyle \sum _{e=1}^L\frac{P_e{\alpha }_e}{{\overline{X}}_e^2}{B}_{e\tau }}$$

(8)

$$MSE\left({\overline{y}}_{st\left(S\right)}^{\prime }\right)\cong {\displaystyle \sum _{e=1}^L{P}_e^2}\left(A_{e\tau }+{\displaystyle \frac{{\alpha }_e^2}{{\overline{X}}_e^2}}{B}_{e\tau }+2{\displaystyle \frac{{\alpha }_e}{{\overline{X}}_e}}{C}_{e\tau }\right)$$

(9)

where

$$A_{e\tau }={\lambda }_{e2}\left(S_{Ye}^2+S_{Ue}^2\right)+{\theta }_{e2}\left(S_{Ye\left(2\right)}^2+S_{Ue\left(2\right)}^2\right),$$

$$B_{e\tau }={\lambda }_{e2}\left(S_{Xe}^2+S_{Ve}^2\right)+{\theta }_{e2}\left(S_{Xe\left(2\right)}^2+S_{Ve\left(2\right)}^2\right)$$

and

$$C_{e\tau }={\lambda }_{e2}{\rho }_{XYe}{S}_{Ye}{S}_{Xe}+{\theta }_{e2}{\rho }_{XYe\left(2\right)}{S}_{Ye\left(2\right)}{S}_{Xe\left(2\right)}$$

The expression for the minimum MSE of the estimator ${\overline{y}}_{st\left(S\right)}^{\prime }$ at the optimum value of ${\alpha }_e\left(={\displaystyle \frac{C_{e\tau }{\overline{X}}_e}{B_{e\tau }}}\right)$ is given as follows:

$$MSE{\left({\overline{y}}_{st\left(S\right)}^{\prime }\right)}_{Min}\cong {\displaystyle \sum _{e=1}^L{P}_e^2}\left(A_{e\tau }-{\displaystyle \frac{C_{e\tau }^2}{B_{e\tau }}}\right)$$

(10)

4. Proposed Calibration Estimator

The ref. [15] estimator of the population total $T={\displaystyle \sum _{e=1}^L{\displaystyle \sum _{i=1}^{N_e}{y}_{ei}}}$ under stratified random sampling in the presence of measurement error and non-response is given as

$${\overline{T}}_{st\left(HH\right)}^{\prime }={\displaystyle \sum _{e=1}^L\left({\displaystyle \sum _i^{n_{e1}}{d}_{ei}{y}_{ei}^{*}+{\displaystyle \sum _i^{k_e}{d}_{ei}^{*}{y}_{ei}^{*}}}\right)}$$

(11)

where $d_{ei}\left(={\displaystyle \frac{1}{{\pi }_{ei}}}={\displaystyle \frac{N_e}{n_e}}\right)$ and $d_{ei}^{*}\left(={\displaystyle \frac{1}{{\pi }_{ei}{\pi }_{ei/n_{e2}}}}={\displaystyle \frac{N_e}{n_e}}.{\displaystyle \frac{n_{e2}}{k_e}}\right)$ are the design weights. Further, ${\pi }_{ei/n_{e2}}\left(={\displaystyle \frac{k_e}{n_{e2}}}\right)$ and ${\pi }_{ei}\left(={\displaystyle \frac{n_e}{N_e}}\right)$ are the inclusion probabilities.

Now, we suggest a novel calibration estimator for the population total T in stratified random sampling under measurement error and non-response as follows:

$${\overline{T}}_{st\left(PR\right)}^{\prime }={\displaystyle \sum _{e=1}^L\left({\displaystyle \sum _i^{n_{e1}}{d}_{ei}{y}_{ei}^{*}+{\displaystyle \sum _i^{k_e}{\mathrm{\Omega }}_{ei}{y}_{ei}^{*}}}\right)}$$

(12)

where ${\mathrm{\Omega }}_{ei}$ is the calibrated weight for the $i^{th}$ non-responding unit in the $e^{th}$ stratum, which minimizes the chi-square-type distance function

$${\displaystyle \sum _i^{k_e}\frac{{\left({\mathrm{\Omega }}_{ei}-d_{ei}^{*}\right)}^2}{q_{ei}{d}_{ei}^{*}}}$$

(13)

subject to the calibration constraints

$${\displaystyle \sum _i^{k_e}{\mathrm{\Omega }}_{ei}{x}_{ei}^{*}={\displaystyle \sum _i^{n_{e2}}{d}_{ei}^{*}{x}_{ei}^{*}}}$$

(14)

Here, $q_{ei}$ is the tuning parameter associated with the $i^{th}$ non-responding unit in the $e^{th}$ stratum. One can derive a new calibration weight solution by minimizing the chi-square-type distance function subject to the given calibration constraint. Thus, the Lagrange function can be written as

$${\mathrm{\Psi }}_1={\displaystyle \sum _i^{k_e}\frac{{\left({\mathrm{\Omega }}_{ei}-d_{ei}^{*}\right)}^2}{q_{ei}{d}_{ei}^{*}}}-2{\lambda }_1\left({\displaystyle \sum _i^{k_e}{\mathrm{\Omega }}_{ei}{x}_{ei}^{*}-{\displaystyle \sum _i^{n_{e2}}{d}_{ei}^{*}{x}_{ei}^{*}}}\right)$$

(15)

where ${\lambda }_1$ is the Lagrange multiplier.

Differentiating the ${\mathrm{\Psi }}_1$ with respect to ${\mathrm{\Omega }}_{ei}$, we get

$${\mathrm{\Omega }}_{ei}=d_{ei}^{*}+{\lambda }_1{q}_{ei}{d}_{ei}^{*}{x}_{ei}^{*}$$

(16)

Hence, substituting the value of ${\mathrm{\Omega }}_{ei}$ from Equation (16) into Equation (14), we have

$${\lambda }_1={\displaystyle \frac{{\displaystyle \sum _i^{n_{e2}}{d}_{ei}^{*}{x}_{ei}^{*}-{\displaystyle \sum _i^{k_e}{d}_{ei}^{*}{x}_{ei}^{*}}}}{{\displaystyle \sum _i^{k_e}{q}_{ei}{d}_{ei}^{*}{x}_{ei}^{*2}}}}$$

(17)

Putting the value of ${\lambda }_1$ from Equation (17) into Equation (16) by assuming $q_{ei}={\displaystyle \frac{1}{x_{ei}^{*}}}$, we get

$${\mathrm{\Omega }}_{ei}=d_{ei}^{*}{\displaystyle \frac{{\displaystyle \sum _i^{n_{e2}}{d}_{ei}^{*}{x}_{ei}^{*}}}{{\displaystyle \sum _i^{k_e}{d}_{ei}^{*}{x}_{ei}^{*}}}}$$

(18)

Thus, by placing the value of ${\mathrm{\Omega }}_{ei}$ from Equation (18) into Equation (12), the estimator ${\overline{T}}_{st\left(PR\right)}^{\prime }$ becomes

$${\overline{T}}_{st\left(PR\right)}^{\prime }={\displaystyle \sum _{e=1}^L\left[{\displaystyle \sum _i^{n_{e1}}{d}_{ei}{y}_{ei}^{*}+\frac{{\displaystyle \sum _i^{n_{e2}}{d}_{ei}^{*}{x}_{ei}^{*}}}{{\displaystyle \sum _i^{k_e}{d}_{ei}^{*}{x}_{ei}^{*}}}{\displaystyle \sum _i^{k_e}{d}_{ei}^{*}{y}_{ei}^{*}}}\right]}$$

(19)

Substituting the values of design weights $d_{ei}$ and $d_{ei}^{*}$, the calibration estimator ${\overline{T}}_{st\left(PR\right)}^{\prime }$ of the population total T is obtained as

$${\overline{T}}_{st\left(PR\right)}^{\prime }={\displaystyle \sum _{e=1}^L{N}_e\left[w_{e1}{\overline{y}}_{e1}+w_{e2}\frac{{\overline{y}}_{e2}^{\prime }}{{\overline{x}}_{e2}^{\prime }}{\overline{x}}_{e2}\right]}$$

(20)

where $w_{e1}={\displaystyle \frac{n_{e1}}{n_e}}$, $w_{e2}={\displaystyle \frac{n_{e2}}{n_e}}$ and ${\overline{x}}_{e2}={\displaystyle \frac{1}{n_{e2}}}{\displaystyle \sum _i^{n_{e2}}{x}_{ei}^{*}}$.

Now, we define the calibration estimator for the mean of the stratified population, $\overline{Y}$ under the presence of non-response and measurement error as

$${\overline{y}}_{st\left(PR\right)}^{\prime }={\displaystyle \sum _{e=1}^L{p}_e^{*}\left[w_{e1}{\overline{y}}_{e1}+w_{e2}\frac{{\overline{y}}_{e2}^{\prime }}{{\overline{x}}_{e2}^{\prime }}{\overline{x}}_{e2}\right]}$$

(21)

Here, $p_e^{*}$ is the calibrated weight for the $e^{th}$ stratum that has to be chosen to minimize the chi-square-type distance function

$${\displaystyle \sum _{e=1}^L\frac{{\left(p_e^{*}-p_e\right)}^2}{q_e{p}_e}}$$

(22)

subject to the calibration constraint

$${\displaystyle \sum _{e=1}^L{p}_e^{*}{\overline{x}}_e^{*}}={\displaystyle \sum _{e=1}^L{p}_e{\overline{X}}_e}$$

(23)

where $q_e$ is the tuning parameter for the $e^{th}$ stratum.

Let us define the Lagrange function as

$${\mathrm{\Psi }}_2={\displaystyle \sum _{e=1}^L\frac{{\left(p_e^{*}-p_e\right)}^2}{q_e{p}_e}-2{\lambda }_2\left({\displaystyle \sum _{e=1}^L{p}_e^{*}{\overline{x}}_e^{*}}-{\displaystyle \sum _{e=1}^L{p}_e{\overline{X}}_e}\right)}$$

(24)

where ${\lambda }_2$ is the Lagrange multiplier.

Differentiating Equation (24) with respect to $p_e^{*}$ and equalizing the derivative to zero, we get

$$p_e^{*}=p_e+{\lambda }_2{q}_e{p}_e{\overline{x}}_e^{*}$$

(25)

Substituting the value of $p_e^{*}$ from Equation (25) into Equation (23), we get ${\lambda }_2$ as

$${\lambda }_2={\displaystyle \frac{\left({\displaystyle \sum _{e=1}^L{p}_e{\overline{x}}_e^{*}}-{\displaystyle \sum _{e=1}^L{p}_e{\overline{X}}_e}\right)}{{\displaystyle \sum _{e=1}^L{q}_e{p}_e{\overline{x}}_e^{*2}}}}$$

(26)

Considering $q_e={\displaystyle \frac{1}{{\overline{x}}_e^{*}}}$ and value of ${\lambda }_2$ given in Equation (26), Equation (25) reduces to

$$p_e^{*}={\displaystyle \frac{\overline{X}}{{\displaystyle \sum _{e=1}^L{p}_e{\overline{x}}_e^{*}}}}{p}_e$$

(27)

Substituting the value of $p_e^{*}$ from Equation (27) into Equation (21), the calibration estimator ${\overline{y}}_{st\left(PR\right)}^{\prime }$ of the population mean $\overline{Y}$ becomes

$${\overline{y}}_{st\left(PR\right)}^{\prime }={\displaystyle \frac{{\displaystyle \sum _{e=1}^L{p}_e\left[w_{e1}{\overline{y}}_{e1}+w_{e2}\frac{{\overline{y}}_{e2}^{\prime }}{{\overline{x}}_{e2}^{\prime }}{\overline{x}}_{e2}\right]}}{{\displaystyle \sum _{e=1}^U{p}_e{\overline{x}}_e^{*}}}}\overline{X}$$

(28)

5. Properties of Proposed Calibration Estimator ${\overline{y}}_{st\left(PR\right)}^{\prime }$

Let us rewrite the proposed calibration estimator ${\overline{y}}_{st\left(PR\right)}^{\prime }$ as

$${\overline{y}}_{st\left(PR\right)}^{\prime }={\displaystyle \frac{{\overline{y}}_{strat}}{{\overline{x}}_{strat}}}\overline{X}$$

(29)

where ${\overline{y}}_{strat}={\displaystyle \sum _{e=1}^L{p}_e\left[w_{e1}{\overline{y}}_{e1}+w_{e2}\frac{{\overline{y}}_{e2}^{\prime }}{{\overline{x}}_{e2}^{\prime }}{\overline{x}}_{e2}\right]}$ and ${\overline{x}}_{strat}={\displaystyle \sum _{e=1}^L{p}_e}{\overline{x}}_e^{*}$.

In order to obtain the MSE of the proposed calibration estimator ${\overline{y}}_{st\left(PR\right)}^{\prime }$, we use the theory of large sample approximations. Let us assume

$${\overline{y}}_{strat}=\overline{Y}\left(1+{\epsilon }_0\right) \mathrm{and} {\overline{x}}_{strat}=\overline{X}\left(1+{\epsilon }_1\right)\mathrm{such} \mathrm{that}$$

$$E\left({\epsilon }_0\right)={\displaystyle \frac{Bias\left({\overline{y}}_{strat}\right)}{\overline{Y}}}; E\left({\epsilon }_1\right)=0; E\left({\epsilon }_0^2\right)={\displaystyle \frac{V\left({\overline{y}}_{strat}\right)}{{\overline{Y}}^2}}+{\displaystyle \frac{{\left[Bias\left({\overline{y}}_{strat}\right)\right]}^2}{{\overline{Y}}^2}};$$

$$E\left({\epsilon }_1^2\right)={\displaystyle \frac{V\left({\overline{x}}_{strat}\right)}{{\overline{X}}^2}}; E\left({\epsilon }_0{\epsilon }_1\right)={\displaystyle \frac{Cov\left({\overline{y}}_{strat},{\overline{x}}_{strat}\right)}{\overline{Y}\overline{X}}}.$$

Now, articulating Equation (29) in terms of ${\epsilon }_0$, ${\epsilon }_1$ and neglecting the higher order terms, we get

$${\overline{y}}_{st\left(PR\right)}^{\prime }-\overline{Y}=\overline{Y}\left({\epsilon }_0-{\epsilon }_1\right)$$

(30)

Squaring both sides of Equation (30) and then taking the expectation, we have

$$E{\left({\overline{y}}_{st\left(PR\right)}^{\prime }-\overline{Y}\right)}^2={\overline{Y}}^2\left[E\left({\epsilon }_0^2\right)+E\left({\epsilon }_1^2\right)-2E\left({\epsilon }_0{\epsilon }_1\right)\right]$$

(31)

Substituting the values of $E\left({\epsilon }_0^2\right)$, $E\left({\epsilon }_1^2\right)$ and $E\left({\epsilon }_0{\epsilon }_1\right)$ into Equation (31), we get

$$MSE\left({\overline{y}}_{st\left(PR\right)}^{\prime }\right)={\overline{Y}}^2\left[{\displaystyle \frac{V\left({\overline{y}}_{strat}\right)+{\left(Bias\left({\overline{y}}_{strat}\right)\right)}^2}{{\overline{Y}}^2}}+{\displaystyle \frac{V\left({\overline{x}}_{strat}\right)}{{\overline{X}}^2}}-{\displaystyle \frac{2Cov\left({\overline{y}}_{strat},{\overline{x}}_{strat}\right)}{\overline{Y}\overline{X}}}\right]$$

(32)

Thus, the MSE of the proposed calibration estimator ${\overline{y}}_{st\left(PR\right)}^{\prime }$ is given as follows:

$$MSE\left({\overline{y}}_{st\left(PR\right)}^{\prime }\right)=V\left({\overline{y}}_{strat}\right)+{\left(Bias\left({\overline{y}}_{strat}\right)\right)}^2+R^{2}V\left({\overline{x}}_{strat}\right)-2RCov\left({\overline{y}}_{strat},{\overline{x}}_{strat}\right)$$

(33)

where

$$V\left({\overline{y}}_{stat}\right)={\displaystyle \sum _{e=1}^L{p}_e^2}\left[\left({\displaystyle \frac{1}{n_e}}-{\displaystyle \frac{1}{N_e}}\right)\left(S_{Ye}^2+S_{Ue}^2\right)+P_{e2}\left({\displaystyle \frac{g_e-1}{n_e}}\right)\left\{\left(S_{Ye(2)}^2+R_e^2{S}_{Xe(2)}^2-2R_e{S}_{XYe(2)}\right)+\left(S_{Ue(2)}^2+R_e^2{S}_{Ve(2)}^2-2R_e{S}_{UVe(2)}\right)\right\}\right],$$

$$Bias\left({\overline{y}}_{strat}\right)={\displaystyle \sum _{e=1}^L{P}_{e2}\frac{\left(g_e-1\right)}{n_e}\left[\begin{array}{l}\left(\frac{{\overline{Y}}_{e2}{S}_{Xe(2)}^2}{{\overline{X}}_{e2}^2}-\frac{S_{XYe(2)}}{{\overline{X}}_{e2}}\right)+\frac{{\overline{Y}}_{e2}{S}_{Xe(2)}^2}{{\overline{X}}_{e2}^2}\left(\frac{1}{n_{e2}}-\frac{1}{N_{e2}}\right)\left(\begin{array}{l}3C_{Xe(2)}^2-2C_{Xe(2)}{\lambda }_{03e(2)}-\\ 2{\rho }_{XYe(2)}{C}_{Xe(2)}{C}_{Ye(2)}+C_{Ye(2)}{\lambda }_{12e(2)}\end{array}\right)-\\ \frac{S_{XYe(2)}}{{\overline{X}}_{e2}}\left(\frac{1}{n_{e2}}-\frac{1}{N_{e2}}\right)\left(C_{Xe(2)}^2-C_{Xe(2)}\frac{{\lambda }_{12e(2)}}{{\rho }_{XYe(2)}}\right)\end{array}\right]},$$

$$V\left({\overline{x}}_{strat}\right)={\displaystyle \sum _{e=1}^L{p}_e^2}\left[\left({\displaystyle \frac{1}{n_e}}-{\displaystyle \frac{1}{N_e}}\right)\left(S_{Xe}^2+S_{Ve}^2\right)+{\displaystyle \frac{P_{e2}\left(g_e-1\right)}{n_e}}\left(S_{Xe(2)}^2+S_{Ve(2)}^2\right)\right],$$

$$Cov\left({\overline{y}}_{strat},{\overline{x}}_{strat}\right)={\displaystyle \sum _{e=1}^L{p}_e^2}\left[\begin{array}{l}\left({\displaystyle \frac{1}{n_e}}-{\displaystyle \frac{1}{N_e}}\right)\left({\rho }_{YXe}{S}_{Ye}{S}_{Xe}+{\rho }_{UVe}{S}_{Ue}{S}_{Ve}\right)+\\ {\displaystyle \frac{P_{e2}\left(g_e-1\right)}{n_e}}\left\{\left({\rho }_{YXe(2)}{S}_{Ye(2)}{S}_{Xe(2)}-R_e{S}_{Xe(2)}^2\right)+\left({\rho }_{UVe(2)}{S}_{Ve(2)}{S}_{Ue(2)}-R_e{S}_{Ve(2)}^2\right)\right\}\end{array}\right],$$

$R={\displaystyle \frac{\overline{Y}}{\overline{X}}}$, ${\lambda }_{jke(2)}={\displaystyle \frac{{\mu }_{jke(2)}}{{\mu }_{20e(2)}^{\raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.{\mu }_{02e(2)}^{\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$ and ${\mu }_{jke(2)}={\displaystyle \frac{1}{N_{e2}-1}}{\displaystyle \sum _i^{N_{e2}}{\left(y_{ei}-{\overline{Y}}_{e2}\right)}^j{\left(x_{ei}-{\overline{X}}_{e2}\right)}^k}$. ${\rho }_{UVe}$ is the correlation coefficient between measurement errors $U_{ei}$ and $V_{ei}$ in the $e^{th}$ stratum. ${\rho }_{UVe(2)}$ is the correlation coefficient between measurement errors $U_{ei}$ and $V_{ei}$ for the non-response group in the $e^{th}$ stratum.

6. Empirical Study

To assess the efficacy of the suggested calibration estimator under the influence of non-response and measurement error, it is critical to show the theoretical truth through some numerical instances. In this section, we used two different data sets that were generated intentionally to perform an empirical examination. The R programming language is used to execute the simulation analysis.

6.1. Data Set I

We created an artificial data set that defines a population of four strata with respective sizes 3000, 4500, 1200 and 2700 in order to gain some insight into effectiveness.

The data under the study variable $Y$ for each stratum are produced using Normal distribution with mean $\mu$ and standard deviation $\sigma$, i.e., $N\left(\mu ,\sigma \right)$ amid 10%, 20%, 30% and 40% weights of the non-respondent group. We followed the instructions provided by [25] to create the data associated with the auxiliary variable $X$ for each stratum under the assumption that it would have specific correlations with the study variable $Y$. Therefore, we first generate the data under a dummy variable $T$ for each stratum with the same distribution as that of $Y$. Then, we generate the data under the auxiliary variable $X$ for each stratum using the transformation $X=\rho Y+\sqrt{1-{\rho }^2}T$. Here, $\rho$ indicates the correlation coefficient between the study variable $Y$ and the auxiliary variable $X$. The particulars of the population are given in

Table 1. Particulars of population.

Stratum No. $\left(\mathit{e}\right)$	${\mathit{N}}_{\mathit{e}}$	${\mathit{n}}_{\mathit{e}}$	Distribution of Study Variable $\mathit{Y}$	Distribution of Auxiliary Variable $\mathit{X}$	${\mathit{\rho }}_{\mathit{X}{\mathit{Y}}_{\mathit{e}}}$
I	3000	526	$N\left(40, 6.5\right)$	$N\left(25, 6.5\right)$	0.89
II	4500	789	$N\left(55, 7.5\right)$	$N\left(40, 7.5\right)$	0.88
III	1200	211	$N\left(70, 8.5\right)$	$N\left(55, 8.5\right)$	0.85
IV	2700	474	$N\left(95, 9.5\right)$	$N\left(70, 9.5\right)$	0.87

.

Further, we generate data under measurement errors U and V using Normal distributions, i.e., $U\sim N\left(0,0.5\right)$ and $V\sim N\left(0,0.4\right)$. It is assumed that the measurement errors $U$ and $V$ are uncorrelated with each other. Now, we obtain the observed values as $y_{ei}^{*}=y_{ei}+u_{ei}$ and $x_{ei}^{*}=x_{ei}+v_{ei}$.

Now, we fix the sample size as $n$ = 2000. Using proportional allocation, we determine the stratum sample size and then select the sample from each stratum. Further, we compute the estimates of $\overline{Y}$, through the estimators ${\overline{y}}_{st\left(HH\right)}^{\prime },{\overline{y}}_{st\left(R\right)}^{\prime },{\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$ utilizing the sample information. There have been 1000 replications of the steps involved in selecting the sample from each stratum and computing the estimates. Finally, we have calculated the approximate VAR/MSE (AVAR/AMSE) of the estimators ${\overline{y}}_{st\left(HH\right)}^{\prime },{\overline{y}}_{st\left(R\right)}^{\prime },{\overline{y}}_{st\left(S\right)}^{\prime }\hspace{0.33em}\mathrm{and}\hspace{0.33em}{\overline{y}}_{st\left(PR\right)}^{\prime }$ using the following formulae:

$$AVAR\left({\overline{y}}_{st\left(HH\right)}^{\prime }\right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{l=1}^{1000}{\left({\overline{y}}_{st{\left(HH\right)}_l}^{\prime }-\overline{Y}\right)}^2}; l=1,2,\dots ,1000$$

$$AMSE\left({\overline{y}}_{st\left(R\right)}^{\prime }\right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{l=1}^{1000}{\left({\overline{y}}_{st{\left(R\right)}_l}^{\prime }-\overline{Y}\right)}^2}; l=1,2,\dots ,1000$$

$$AMSE\left({\overline{y}}_{st\left(S\right)}^{\prime }\right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{l=1}^{1000}{\left({\overline{y}}_{st{\left(S\right)}_l}^{\prime }-\overline{Y}\right)}^2}; l=1,2,\dots ,1000$$

$$AMSE\left({\overline{y}}_{st\left(PR\right)}^{\prime }\right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{l=1}^{1000}{\left({\overline{y}}_{st{\left(PR\right)}_l}^{\prime }-\overline{Y}\right)}^2}; l=1,2,\dots ,1000$$

Table 2. AVAR/AMSE and PRE of estimators ${\overline{y}}_{st\left(HH\right)}^{\prime }$, ${\overline{y}}_{st\left(R\right)}^{\prime }$, ${\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$.

${\mathit{g}}_{\mathit{e}}\forall \mathit{e}$	${\mathit{P}}_{\mathit{e}2} \forall \mathit{e}$ (in %)	$\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)$	$\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{R}\right)}^{\prime }\right)$	$\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{S}\right)}^{\prime }\right)$	$\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{P}\mathit{R}\right)}^{\prime }\right)$	$\mathit{P}\mathit{R}\mathit{E}=\frac{\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)}{\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{R}\right)}^{\prime }\right)}\times 100$	$\mathit{P}\mathit{R}\mathit{E}=\frac{\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)}{\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{S}\right)}^{\prime }\right)}\times 100$	$\mathit{P}\mathit{R}\mathit{E}=\frac{\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)}{\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{P}\mathit{R}\right)}^{\prime }\right)}\times 100$
2	10	0.102745	0.029351	0.018459	0.010394	350.0553	556.6057	988.4678
	20	0.108292	0.034114	0.025782	0.011077	317.4438	420.0029	977.655
	30	0.113531	0.038108	0.03597	0.012882	315.6305	297.9212	881.2957
	40	0.117689	0.052610	0.037467	0.013235	314.1183	223.6457	889.214
3	10	0.103193	0.030059	0.018659	0.010506	343.3064	553.055	982.2718
	20	0.115118	0.039727	0.026796	0.013967	289.7689	429.8067	824.2161
	30	0.122605	0.045945	0.038118	0.014729	266.851	321.6481	832.4187
	40	0.130534	0.053623	0.051708	0.017295	252.4422	243.5195	754.7496
4	10	0.106304	0.032059	0.019981	0.0122	331.5906	532.025	871.3259
	20	0.118255	0.0411	0.028799	0.015376	287.7254	410.6241	769.0829
	30	0.130514	0.038219	0.05482	0.017757	237.9387	341.4958	735.0149
	40	0.131337	0.060333	0.056648	0.020196	217.6859	231.8475	650.3204

reveals the AVAR/AMSE of the estimators ${\overline{y}}_{st\left(HH\right)}^{\prime }$, ${\overline{y}}_{st\left(R\right)}^{\prime }$, ${\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$. The percentage relative efficiency (PRE) of the estimators ${\overline{y}}_{st\left(R\right)}^{\prime }$, ${\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$ with respect to the estimator ${\overline{y}}_{st\left(HH\right)}^{\prime }$ is also revealed.

6.2. Data Set II

We have generated another data set to strengthen the performance of the proposed calibration estimator. As per Section 6.1, the procedure of data generation and finalization of results was followed.

Table 3. Description of population.

Stratum No. $\left(\mathit{e}\right)$	${\mathit{N}}_{\mathit{e}}$	${\mathit{n}}_{\mathit{e}}$	Distribution of Study Variable $\mathit{Y}$	Distribution of Auxiliary Variable $\mathit{X}$
I	3000	563	$N\left(120, 4\right)$	$N\left(110, 4\right)$
II	1500	281	$N\left(135, 5\right)$	$N\left(125, 5\right)$
III	750	141	$N\left(150, 6\right)$	$N\left(140, 6\right)$
IV	1850	346	$N\left(165, 7\right)$	$N\left(155, 7\right)$
V	900	169	$N\left(180, 8\right)$	$N\left(170, 8\right)$

describes the particulars of the population.

Table 4. AVAR/AMSE and PRE of estimators ${\overline{y}}_{st\left(HH\right)}^{\prime }$, ${\overline{y}}_{st\left(R\right)}^{\prime }$, ${\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$.

${\mathit{g}}_{\mathit{e}}\forall \mathit{e}$	${\mathit{P}}_{\mathit{e}2}\forall \mathit{e}$ (in %)	$\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)$	$\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{R}\right)}^{\prime }\right)$	$\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{S}\right)}^{\prime }\right)$	$\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{P}\mathit{R}\right)}^{\prime }\right)$	$\mathit{P}\mathit{R}\mathit{E}=\frac{\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)}{\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{R}\right)}^{\prime }\right)}\times 100$	$\mathit{P}\mathit{R}\mathit{E}=\frac{\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)}{\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{S}\right)}^{\prime }\right)}\times 100$	$\mathit{P}\mathit{R}\mathit{E}=\frac{\mathit{A}\mathit{V}\mathit{A}\mathit{R}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{H}\mathit{H}\right)}^{\prime }\right)}{\mathit{A}\mathit{M}\mathit{S}\mathit{E}\left({\overline{\mathit{y}}}_{\mathit{s}\mathit{t}\left(\mathit{P}\mathit{R}\right)}^{\prime }\right)}\times 100$
2	10	0.046784	0.021893	0.014879	0.007855	214.6799	316.5576	595.6192
	20	0.048505	0.023145	0.01669	0.008826	210.4741	290.6266	555.8719
	30	0.048903	0.025269	0.019536	0.00909	193.6023	250.4525	543.9642
	40	0.049923	0.027112	0.024421	0.011597	184.1373	204.4239	430.4638
3	10	0.050241	0.022215	0.015877	0.008942	228.2089	316.438	561.8874
	20	0.050872	0.026484	0.018536	0.009114	192.3742	274.4542	564.3931
	30	0.05472	0.028544	0.019685	0.009841	193.0554	279.3968	556.0156
	40	0.056756	0.031071	0.026426	0.011683	182.7848	214.7699	485.8166
4	10	0.051982	0.024336	0.016787	0.009185	214.485	309.6545	566.5659
	20	0.05259	0.029532	0.019888	0.009386	178.1353	264.4309	560.2934
	30	0.055495	0.032645	0.021689	0.009988	169.9952	255.8705	556.181
	40	0.064398	0.036373	0.027424	0.013587	178.0275	234.8218	477.1321

depicts the AVAR/AMSE of the estimators ${\overline{y}}_{st\left(HH\right)}^{\prime }$, ${\overline{y}}_{st\left(R\right)}^{\prime }$, ${\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$. The PRE of the estimators ${\overline{y}}_{st\left(R\right)}^{\prime }$, ${\overline{y}}_{st\left(S\right)}^{\prime }$ and ${\overline{y}}_{st\left(PR\right)}^{\prime }$ with respect to the estimator ${\overline{y}}_{st\left(HH\right)}^{\prime }$ is also depicted.

7. Concluding Remarks

A new calibration estimator of the population mean under stratified random sampling in the presence of non-response and measurement error has been pioneered out. The expression for the MSE of the suggested calibration estimator has been derived up to the first order of approximation. To determine the effectiveness of the suggested calibration estimator, a simulation analysis through some artificially generated data has been carried out. In simulation analysis, the AVAR/AMSE has been utilized as a tool to assess the suggested calibration estimator’s accuracy relative to ref. [15] estimator, the separate ratio-type estimator and [24] estimator. Table 2 and Table 4 reveal that the suggested calibration estimator provides the best PRE as compared to the other estimators. According to this study, the proposed calibration estimator produces better results than the current ones, and, hence, it can be very useful in the circumstances that arise in practice.