Calibration Estimators with Different Types of Distance Measures Under Stratified Sampling in the Presence of Measurement Error

Abstract

The calibration method is used in stratified random sampling in the presence of measurement error to achieve optimum strata weights for better precision. In this study we attempt to analyze the behaviour of calibration estimators with different types of distance measures in the presence of measurement error for the estimation of population mean under stratified random sampling. The proposed estimators are compared with the estimators in the absence of measurement error. A simulation study has been carried out to evaluate the proposed estimators.

1. Introduction

Various fields involving survey sampling like health and epidemiological studies assume that variables are measured precisely. However, this condition is frequently violated, since, in practice, it is almost impossible for accurate values to occur. As a result, the recorded values are contaminated with error. The error is referred to as measurement error, which is the difference between the actual value and the recorded value. Measurement error may be due to data processors, interviewers, respondents, instruments used, and so on. To understand the concept of measurement errors, one needs to know the two terms that define the error; these are actual and measured value. The actual value is often impossible to find by experimental means. It may be defined as the average value of an infinite number of measured values. The measured value is a single measure of the object, meant to be as accurate as possible. There are three types of errors: random errors, systematic errors, and gross errors. Random errors are those errors which occur irregularly and, hence, are random. However, systematic errors can be understood in a better way by dividing them into subgroups such as observational errors, instrumental errors, and environmental errors. Gross errors basically take into account human oversight and other mistakes such as reading, recording, etc.

There are many books addressing measurement error models like Fuller [1], Carroll et al. [2], Cheng and Van Nes [3], Yi [4], and so on. All these authors discuss the impact of measurement errors in linear and nonlinear regression modeling. Biemer [5] and Buonaccorsi [6] address the measurement error in the context of finite population.

However, Cochran [7], Sukhatme et al. [8], Chandhok and Han [9], Rao [10], Singh [11], Gupta and Kabe [12], Thompson [13], and so on, have explored the impact of measurement errors in the context of survey sampling.

In survey sampling, measurement errors need to be accounted for in order to produce more reliable estimates. At the very least, effort should be made to minimize measurement errors. These errors may occur due to instrument error or sampling error. For example, the presence or non-existence or the intensity of disease such as hepatitis, AIDS, breast cancer, etc., are usually described through a less-than-perfect diagnostic procedure such as an imaging technique or blood test. So, in these situations, due to fallacious calibration of the instrument, the delineated values may involve some error. Many eminent researchers like Mahalanobis [14], Deming [15], Raj [16], Sukhatme et al. [8], Särndal et al. [17], Gregoire and Salas [18], Singh and Karpe [19], Vishwakarma and Singh [20], Zhang et al. [21], Priyanka et al. [22], Priyanka and Trisandhya [23] have proposed important estimation techniques to improve measurement error.

Auxiliary information is also commonly used in sampling design. This can produce not only better estimates, but also achieve better stratification of heterogeneous populations. A significant amount of work has been conducted by many authors on stratified sampling, where variables under study are directly observable. These include Kadilar and Cingi ([24,25]), Koyuncu and Kadilar [26], Shabbir and Gupta [27], Vishwakarma and Singh [28] and Tailor et al. [29]. In stratified random sampling, strata weights are optimized by utilizing calibration techniques. The original design weights are adjusted to minimize a pre-defined distance measure, while conforming to a set of constraints. Notable work on calibration has been conducted by Deville and Särndal [30], Farrell and Singh [31], Kim and Park [32], Kim and Elam [33], Koyuncu and Kadilar [34] and Särndal [35], among others.

With this motivation, we intend to modify the Horvitz Thomson [36] type estimator in the presence of measurement error under the stratified sampling design. The calibration-based estimation method helps improve the estimates by means of auxiliary information by adjusting the initial basic design weights.The aim of the present study is to propose the calibration estimators considering three different distance measures in the presence of measurement error in stratified sampling. The properties of the proposed estimator are also examined. Also, with the help of a natural population, a Monte Carlo simulation study has been carried out to compare the behaviour of the proposed estimators. The proposed estimators have been compared with the case when there is no measurement error.

2. Survey Design and Calibration Estimators in the Presence of Measurement Error

Let $U=(U_1,U_2,\dots ,U_N)$ be a finite heterogeneous population, which is partitioned into L non-overlapping strata with the $h^{th}$ stratum containing $N_h;(h=1,2,\dots ,L)$ units, such that ${\displaystyle \sum _{h=1}^L}{N}_h=N.$ A simple random sample of size $n_h$ is drawn without replacement from the $h^{th}$ stratum ($h=1,2,\dots ,L$). Let $(y_h,x_h)$ be the value of the study variable $\left(y\right)$ and the auxiliary variable $\left(x\right)$ in the $h^{th}$ stratum. Furthermore, it is assumed that $(y_{hej},x_{hej})$ be the stratified observed value, instead of stratified true values $(y_{hj},x_{hj})$ on $j^{th}$ unit $(j=1,2,\dots ,n_h;h=1,2,\dots ,L)$. Let the observational errors be denoted as $u_{hj}=(y_{hej}-y_{hj})$ and $v_{hj}=(x_{hej}-x_{hj})$, which are assumed to be normally distributed with mean 0 and variances ${\sigma }_{u_h}^2$ and ${\sigma }_{v_h}^2$, respectively. Let ${\rho }_{y_h{x}_h}$ be actual correlation coefficients and ${\rho }_{u_h{v}_h}$ be the correlation coefficient between observational errors.

In a stratified sampling scheme, the classical unbiased estimator of the population mean is given by ${\overline{Y}}_{st}^{me}={\displaystyle \sum _{h=1}^L}{w}_h{\overline{y}}_{he},$ where $w_h={\displaystyle \frac{N_h}{N}}$ are stratum weights. The calibrated mean estimator in the presence of measurement error is given by

$$\begin{array}{c}\hfill T_{st}^{me}={\displaystyle \sum _{h=1}^L}{w}_h^{*}{\overline{y}}_{he},\end{array}$$

(1)

where, $w_h^{*}(h=1,2,\dots ,L)$ are the calibrated weights, which can be obtained by minimizing some distance function such as

$$\begin{array}{c}\hfill D_1(w_h^{*},w_h)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{{(w_h^{*}-w_h)}^2}{w_h{q}_h}},\end{array}$$

(2)

$$\begin{array}{c}\hfill D_2(w_h^{*},w_h)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{w_h^{*}log\left(\frac{w_h^{*}}{w_h}\right)-w_h^{*}+w_h}{q_h}},\mathrm{and}\end{array}$$

(3)

$$\begin{array}{c}\hfill D_3(w_h^{*},w_h)={\displaystyle \sum _{h=1}^L}{\left({\displaystyle \frac{w_h^{*}}{w_h}}-1\right)}^2{\displaystyle \frac{1}{q_h}},\end{array}$$

(4)

subject to the constraint

$$\begin{array}{c}\hfill {\displaystyle \sum _{h=1}^L}{w}_h^{*}{\overline{x}}_{he}={\displaystyle \sum _{h=1}^L}{w}_h{\overline{X}}_h,\end{array}$$

(5)

where $q_h$ are suitably chosen weights. The form of estimator depends on the choice of $q_h.$ We now consider three cases corresponding to these three distance functions.

Case-1:

Let ${\lambda }_1$ be the Lagrange multiplier corresponding to Equation (2). Then from Equations (1) and (2), the Lagrange function can be written as

$$\begin{array}{c}\hfill L_1={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{{\left(w_h^{*}-w_h\right)}^2}{q_h{w}_h}}-2{\lambda }_1\left({\displaystyle \sum _{h=1}^L}{w}_h^{*}{\overline{x}}_{he}-{\overline{X}}_h\right).\end{array}$$

(6)

Differentiating $L_1$ with respect to calibration weight $w_h^{*}$ and equating to zero, the weight $w_h^{*}$ is given by

$$\begin{array}{c}\hfill w_h^{*}=w_h+{\lambda }_1{\overline{x}}_{he}{w}_h{q}_h.\end{array}$$

(7)

Solving Equation (7), the Lagrange multiplier ${\lambda }_1$ is given by

$$\begin{array}{c}\hfill {\lambda }_1={\displaystyle \frac{{\overline{X}}_h-{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}}{w_h{q}_h{\overline{x}}_{he}^2}}.\end{array}$$

(8)

Substituting the value of ${\lambda }_1$ in Equation (7), the calibration weight is obtained as

$$\begin{array}{c}\hfill w_h^{*}=w_h+w_h{q}_h\left[{\displaystyle \frac{\left({\overline{X}}_h-{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}\right){\overline{x}}_{he}}{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}^2{q}_h{w}_h}}\right].\end{array}$$

(9)

Substituting $w_h^{*}$ from Equation (9) into Equation (1), we get the calibrated estimator under measurement error as

$$\begin{array}{c}\hfill {\left(T_{st}^{me}\right)}_1={\displaystyle \sum _{h=1}^L}{\overline{y}}_{he}+{\displaystyle \frac{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}{\overline{y}}_{he}{w}_h{q}_h}{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}^2{w}_h{q}_h}}\left[{\overline{X}}_h-{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}\right].\end{array}$$

(10)

Case-2:

Let ${\lambda }_2$ be the Lagrange multiplier corresponding to Equation (3). Then from Equations (1) and (3), the Lagrange function can be written as

$$\begin{array}{c}\hfill L_2={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{w_h^{*}log\left(\frac{w_h^{*}}{w_h}\right)-w_h^{*}+w_h}{q_h}}-2{\lambda }_2\left({\displaystyle \sum _{h=1}^L}{w}_h^{*}{\overline{x}}_{he}-{\overline{X}}_h\right).\end{array}$$

(11)

Differentiating $L_2$ in Equation (11) with respect to calibration weight $w_h^{*}$ and equating to zero, the weight is obtained as

$$\begin{array}{c}\hfill w_h^{*}=exp\left(log{w}_h+{\lambda }_2{\overline{x}}_{he}{q}_h\right)\end{array}$$

(12)

Solving the above Equation (12), the Lagrange multiplier ${\lambda }_2$ is obtained as

$$\begin{array}{c}\hfill {\lambda }_2=log\left({\displaystyle \frac{{\overline{X}}_h}{{\displaystyle \sum _{h=1}^L}{w}_h{q}_h{\overline{x}}_{he}^2}}\right),\end{array}$$

(13)

and substituting the value of ${\lambda }_2$ in Equation (12), the calibration weight is obtained as

$$\begin{array}{c}\hfill w_h^{*}=exp\left(log{w}_h+log\left[{\displaystyle \frac{{\overline{X}}_h}{{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}^2{q}_h}}\right]{\overline{x}}_{he}{q}_h\right).\end{array}$$

(14)

Substituting $w_h^{*}$ from Equation (14) into Equation (1), the calibrated estimator under measurement error is

$$\begin{array}{c}\hfill {\left(T_{st}^{me}\right)}_2={\displaystyle \sum _{h=1}^L}{\overline{y}}_{he}\left[exp\left(log{w}_h+\left[log{\displaystyle \frac{{\overline{X}}_h}{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}^2{w}_h{q}_h}}\right]{\overline{x}}_{he}{q}_h\right)\right].\end{array}$$

(15)

Case-3:

Let ${\lambda }_3$ be the Lagrange multiplier corresponding to Equation (4). Then from Equations (1) and (4), the Lagrange function can be written as

$$\begin{array}{c}\hfill L_3={\displaystyle \sum _{h=1}^L}{\left({\displaystyle \frac{w_h^{*}}{w_h}}-1\right)}^2{\displaystyle \frac{1}{q_h}}-2{\lambda }_3\left({\displaystyle \sum _{h=1}^L}{w}_h^{*}{\overline{x}}_{he}-{\overline{X}}_h\right).\end{array}$$

(16)

Differentiating $L_3$ in Equation (16) with respect to calibration weight $w_h^{*}$ and equating to zero, the weight is obtained as

$$\begin{array}{c}\hfill w_h^{*}=w_h+{\lambda }_3{\overline{x}}_{he}{w}_h^2{q}_h.\end{array}$$

(17)

Solving Equation (17), the Lagrange multiplier ${\lambda }_3$ is obtained as

$$\begin{array}{c}\hfill {\lambda }_3={\displaystyle \frac{{\overline{X}}_h-{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}}{{\displaystyle \sum _{h=1}^L}{w}_h^2{q}_h{\overline{x}}_{he}^2}}\end{array}$$

(18)

and substituting the value of ${\lambda }_3$ in Equation (17), the calibration weight is obtained as

$$\begin{array}{c}\hfill w_h^{*}=w_h+w_h^2{q}_h{\displaystyle \frac{\left({\overline{X}}_h-{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}\right){\overline{x}}_{he}}{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}^2{q}_h{w}_h^2}}.\end{array}$$

(19)

Substituting $w_h^{*}$ from Equation (19) into Equation (1), the calibrated estimator under measurement error is

$$\begin{array}{c}\hfill {\left(T_{st}^{me}\right)}_3={\displaystyle \sum _{h=1}^L}{\overline{y}}_{he}{w}_h+{\displaystyle \frac{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}{\overline{y}}_{he}{w}_h^2{q}_h}{{\displaystyle \sum _{h=1}^L}{\overline{x}}_{he}^2{w}_h^2{q}_h}}\left[{\overline{X}}_h-{\displaystyle \sum _{h=1}^L}{w}_h{\overline{x}}_{he}\right].\end{array}$$

(20)

To obtain the expression for the proposed estimator up to first order of approximations, we define the following error terms:

$$\begin{array}{cc}\hfill & \hfill e_o={\displaystyle \frac{{\overline{y}}_{he}-{\overline{Y}}_h}{{\overline{Y}}_h}},e_1={\displaystyle \frac{{\overline{x}}_{he}-{\overline{X}}_h}{{\overline{X}}_h}}\mathrm{Such}\mathrm{that}E\left(e_o\right)=0=E\left(e_1\right)\mathrm{with}\hfill \\ & \hfill E\left(e_o^2\right)={\displaystyle \frac{{C_y}_h^2+C_{uh}^2}{n_h}},E\left(e_1^2\right)={\displaystyle \frac{{C_x}_h^2+C_{vh}^2}{n_h}}\mathrm{and}E\left(e_o{e}_1\right)={\displaystyle \frac{C_1+C_2}{n_h}},\hfill \\ & \hfill \mathrm{where}{C}_{yh}={\displaystyle \frac{{\sigma }_{yh}}{{\overline{Y}}_h}},C_{xh}={\displaystyle \frac{{\sigma }_{xh}}{{\overline{X}}_h}},C_{uh}={\displaystyle \frac{{\sigma }_{uh}}{{\overline{Y}}_h}},C_{vh}={\displaystyle \frac{{\sigma }_{vh}}{{\overline{X}}_h}},\mathrm{and}{C}_1={\rho }_{xhyh}{C}_{xh}{C}_{yh}\hfill \\ & \hfill C_2={\rho }_{uhvh}{C}_{uh}{C}_{vh}.\hfill \end{array}$$

By using the large sample approximations, the variance of the proposed estimators are obtained by assuming the general choice of $q_h=1$ in ${\left(T_{st}^{me}\right)}_i;i=1,2\mathrm{and}3$ under simple random sampling design without replacement, and is given by

$$\begin{array}{c}\hfill V{\left(T_{st}^{me}\right)}_i={\displaystyle \sum _{h=1}^L}{\left(w_h^{*}\right)}^2\left[{\overline{Y}}_h^2{\displaystyle \frac{1}{n_h}}(C_{yh}^2+C_{uh}^2)\right];i=1,2\mathrm{and}3\end{array}$$

(21)

Mutual Comparison

To judge the impact of measurement error in the presence of a stratified sampling design, the estimators ${\left(T_{st}^{me}\right)}_1,{\left(T_{st}^{me}\right)}_2$ and ${\left(T_{st}^{me}\right)}_3$ are compared with the corresponding direct estimators, i.e., $T_1,T_2$ and $T_3$, in absence of measurement error in terms of Ratios, defined below:

$$\begin{array}{c}\hfill Rati{o}_i={\displaystyle \frac{V\left[T_i\right]}{V\left[{\left(T_{st}^{me}\right)}_i\right]}};i=\{1,2,3\}\end{array}$$

The results are presented in

Table 1. Simulated Efficiency Ratios of the proposed estimators in the presence of measurement error compared with the calibrated estimators in the absence of measurement error (Population data-I).

		${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=30\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=40\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=60\%$
SET	${\mathit{\rho }}_{\mathit{uv}}$	${\mathit{Ratio}}_{\mathit{1}}$	${\mathit{Ratio}}_{\mathit{2}}$	${\mathit{Ratio}}_{\mathit{3}}$	${\mathit{Ratio}}_{\mathit{1}}$	${\mathit{Ratio}}_{\mathit{2}}$	${\mathit{Ratio}}_{\mathit{3}}$	${\mathit{Ratio}}_{\mathit{1}}$	${\mathit{Ratio}}_{\mathit{2}}$	${\mathit{Ratio}}_{\mathit{3}}$
	$0.9$	$0.9192$	$0.9540$	$0.9189$	$0.9825$	$0.9875$	$0.9819$	$0.9404$	$0.9346$	$0.9402$
	$0.7$	$0.9952$	$0.9242$	$0.9953$	$0.9439$	$0.9347$	$0.9437$	$0.8678$	$0.8627$	$0.8679$
	$0.5$	$0.9463$	$0.9844$	$0.9465$	$0.9022$	$0.9974$	$0.9019$	$0.9738$	$0.9715$	$0.9740$
I	$0.0$	$0.9440$	$0.9847$	$0.9436$	$0.9332$	$0.9604$	$0.9324$	$0.9255$	$0.9159$	$0.9255$
	$-0.3$	$0.9122$	$0.9605$	$0.9121$	$0.9942$	$0.9329$	$0.9937$	$0.9486$	$0.9703$	$0.9485$
	$-0.5$	$0.9980$	$0.9394$	$0.9981$	$0.9636$	$0.9835$	$0.9633$	$0.8994$	$0.9133$	$0.8994$
	$-0.7$	$0.9458$	$0.9816$	$0.9460$	$0.9031$	$0.9974$	$0.9028$	$0.9688$	$0.9664$	$0.9640$
	$-0.9$	$0.9693$	$0.9236$	$0.9688$	$0.9322$	$0.9467$	$0.9311$	$0.9515$	$0.9443$	$0.9512$
	$0.9$	$0.909$	$0.9082$	$0.9100$	$0.9179$	$0.9159$	$0.9180$	$0.9368$	$0.9354$	$0.9369$
	$0.7$	$0.9582$	$0.9552$	$0.9578$	$0.9493$	$0.9470$	$0.9490$	$0.9203$	$0.9178$	$0.9202$
	$0.5$	$0.9669$	$0.9589$	$0.9669$	$0.9763$	$0.9652$	$0.9763$	$0.9966$	$0.9821$	$0.9964$
II	$0.0$	$0.9429$	$0.9164$	$0.9427$	$0.9460$	$0.9257$	$0.9460$	$0.9594$	$0.9517$	$0.9595$
	$-0.3$	$0.9012$	$0.9876$	$0.9018$	$0.9149$	$0.9988$	$0.9153$	$0.9295$	$0.9145$	$0.9296$
	$-0.5$	$0.9132$	$0.9366$	$0.9131$	$0.9005$	$0.9221$	$0.9003$	$0.9947$	$0.9147$	$0.9946$
	$-0.7$	$0.9048$	$0.9939$	$0.9042$	$0.9921$	$0.9824$	$0.9916$	$0.9446$	$0.9382$	$0.9445$
	$-0.9$	$0.9700$	$0.9612$	$0.9700$	$0.9931$	$0.9862$	$0.9931$	$0.9454$	$0.9350$	$0.9452$
	$0.9$	$0.9281$	$0.9162$	$0.9272$	$0.9827$	$0.9724$	$0.9826$	$0.9641$	$0.9480$	$0.9641$
	$0.7$	$0.8968$	$0.8941$	$0.8957$	$0.9658$	$0.9615$	$0.9656$	$0.9365$	$0.9206$	$0.9367$
	$0.5$	$0.9344$	$0.9585$	$0.9335$	$0.0245$	$0.0533$	$0.0241$	$0.9571$	$0.9739$	$0.9569$
III	$0.0$	$0.9259$	$0.9411$	$0.9252$	$0.9146$	$0.9329$	$0.0145$	$0.9610$	$0.9710$	$0.9611$
	$-0.3$	$0.9258$	$0.9219$	$0.9251$	$0.9971$	$0.9012$	$0.9970$	$0.9594$	$0.9601$	$0.9595$
	$-0.5$	$0.9173$	$0.9085$	$0.9160$	$0.9900$	$0.9855$	$0.9901$	$0.9967$	$0.9814$	$0.9970$
	$-0.7$	$0.8965$	$0.8925$	$0.8956$	$0.9641$	$0.9600$	$0.9638$	$0.8912$	$0.8941$	$0.8912$
	$-0.9$	$0.9723$	$0.9594$	$0.9725$	$0.9541$	$0.9521$	$0.9540$	$0.8962$	$0.8916$	$0.8963$

and

Table 2. Simulated Efficiency Ratios of the proposed estimators in the presence of measurement error compared with the calibrated estimators in the absence of measurement error (Population data-II).

		${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=30\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=40\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=60\%$
SET	${\mathit{\rho }}_{\mathit{uv}}$	${\mathit{Ratio}}_{\mathit{1}}$	${\mathit{Ratio}}_{\mathit{2}}$	${\mathit{Ratio}}_{\mathit{3}}$	${\mathit{Ratio}}_{\mathit{1}}$	${\mathit{Ratio}}_{\mathit{2}}$	${\mathit{Ratio}}_{\mathit{3}}$	${\mathit{Ratio}}_{\mathit{1}}$	${\mathit{Ratio}}_{\mathit{2}}$	${\mathit{Ratio}}_{\mathit{3}}$
	$0.9$	$0.6198$	$0.6540$	$0.6199$	$0.6825$	$0.6875$	$0.6819$	$0.6404$	$0.6346$	$0.6402$
	$0.7$	$0.6952$	$0.6242$	$0.6953$	$0.6439$	$0.6347$	$0.6437$	$0.6678$	$0.6627$	$0.6679$
	$0.5$	$0.8463$	$0.8844$	$0.8465$	$0.8022$	$0.8974$	$0.8019$	$0.8738$	$0.8715$	$0.8740$
I	$0.0$	$0.8440$	$0.8847$	$0.8436$	$0.8332$	$0.8604$	$0.8324$	$0.8255$	$0.8159$	$0.8255$
	$-0.3$	$0.5122$	$0.5605$	$0.5121$	$0.5942$	$0.5329$	$0.5937$	$0.5486$	$0.5703$	$0.5485$
	$-0.5$	$0.5980$	$0.5394$	$0.5981$	$0.5636$	$0.5835$	$0.5633$	$0.5994$	$0.5133$	$0.5994$
	$-0.7$	$0.4458$	$0.4816$	$0.4460$	$0.4031$	$0.4974$	$0.4028$	$0.4688$	$0.4664$	$0.4640$
	$-0.9$	$0.4693$	$0.4236$	$0.4688$	$0.4322$	$0.4467$	$0.4311$	$0.4515$	$0.4443$	$0.4512$
	$0.9$	$0.7900$	$0.7908$	$0.7910$	$0.7917$	$0.7915$	$0.7918$	$0.7936$	$0.7935$	$0.7936$
	$0.7$	$0.6958$	$0.6955$	$0.6957$	$0.6949$	$0.6947$	$0.6949$	$0.6920$	$0.6917$	$0.6920$
	$0.5$	$0.6966$	$0.6958$	$0.6966$	$0.6976$	$0.6965$	$0.6976$	$0.6996$	$0.6821$	$0.6964$
II	$0.0$	$0.8429$	$0.8164$	$0.8427$	$0.8460$	$0.8257$	$0.8460$	$0.8594$	$0.8517$	$0.8595$
	$-0.3$	$0.5012$	$0.5876$	$0.5018$	$0.5149$	$0.5988$	$0.5153$	$0.5295$	$0.5145$	$0.5296$
	$-0.5$	$0.4132$	$0.4366$	$0.4131$	$0.4005$	$0.4221$	$0.4003$	$0.4947$	$0.4147$	$0.4946$
	$-0.7$	$0.2048$	$0.2939$	$0.2042$	$0.29921$	$0.29824$	$0.2916$	$0.2446$	$0.2382$	$0.2445$
	$-0.9$	$0.2700$	$0.2612$	$0.2700$	$0.2931$	$0.2862$	$0.2931$	$0.2454$	$0.2350$	$0.2452$
	$0.9$	$0.9289$	$0.9169$	$0.9279$	$0.9829$	$0.9729$	$0.9829$	$0.9649$	$0.9489$	$0.9649$
	$0.7$	$0.8999$	$0.8991$	$0.8997$	$0.9698$	$0.9695$	$0.9696$	$0.9395$	$0.9296$	$0.9397$
	$0.5$	$0.9394$	$0.9595$	$0.9395$	$0.0295$	$0.0593$	$0.0291$	$0.9591$	$0.9799$	$0.9599$
III	$0.0$	$0.9299$	$0.9491$	$0.9292$	$0.9196$	$0.9399$	$0.0145$	$0.6610$	$0.6710$	$0.6616$
	$-0.3$	$0.6258$	$0.6219$	$0.6251$	$0.6971$	$0.6012$	$0.6970$	$0.6594$	$0.6601$	$0.6595$
	$-0.5$	$0.5173$	$0.5085$	$0.5160$	$0.5900$	$0.5855$	$0.5901$	$0.5967$	$0.5814$	$0.5970$
	$-0.7$	$0.5965$	$0.5925$	$0.5956$	$0.5641$	$0.5600$	$0.5638$	$0.5912$	$0.5941$	$0.5912$
	$-0.9$	$0.4723$	$0.4594$	$0.4725$	$0.4541$	$0.4521$	$0.4540$	$0.4962$	$0.4916$	$0.4963$

.

Note that $V\left[T_1\right],V\left[T_2\right]$ and $V\left[T_3\right]$ can be obtained by substituting $u=v=0$ in $V\left[{\left(T_{st}^{me}\right)}_1\right],V\left[{\left(T_{st}^{me}\right)}_2\right]$ and $V\left[{\left(T_{st}^{me}\right)}_3\right]$, respectively.

3. Simulation Study

The main objective of this study is to examine the performance of proposed estimators under measurement error in stratified sampling. We have conducted a Monte-Carlo simulation study using MATLAB software (R2024a).

Population data-I: In this study, we have drawn heterogeneous strata of sizes $N_1=11$, $N_2=19$ and $N_3=17$ with a natural population comprising $N=47$ cities of Andhra Pradesh(Northern Andhra, Eastern region of Andhra and Southern region of Andhra) of India. The related ambient air quality monitoring data of the air pollutant, specifically Particulate Matter $\le 10$ micrometers in size $\left[P{M}_{10}\right]$ and Particulate Matter $\le 2.5$ micrometers in size $\left[P{M}_{2.5}\right]$, have been obtained from manual monitoring conducted under the National Ambient Air Quality Monitoring Programme, sourced from a publicly available source.

[Data Source: https://cpcb.nic.in/manual-monitoring/, accessed on 2 September 2024], which is available online and therefore no ethical approval is required.

The variables considered in study are described as

$x_{1i},x_{2i}$ and $x_{3i}$: Annual average concentration of $P{M}_{10}$ in μg/m³ in the $i$ stratum for the year 2019.

$y_{1i},y_{2i}$ and $y_{3i}$: Annual average concentration of $P{M}_{2.5}$ in μg/m³ in the $i$ stratum for the year 2019.

The different sizes of strata are drawn and it has been assumed that the measurement errors u and v are correlated but are independent of the main study variables. The corresponding data are generated from Normal distribution and the parameters of the considered population are $N_1=11, N_2=19, N_3=17, {\overline{y}}_1=30.2727, {\overline{y}}_2=24.7368$, ${\overline{y}}_3=26.4706$, ${\overline{x}}_1=75.1818, {\overline{x}}_2=63$, and ${\overline{x}}_3=59$.

Population data-II: For this, heterogeneous strata of sizes $N_1=14, N_2=30, N_3=37$ with a natural population comprising $N=81$ districts of Southern states (Kerala, Karnataka and Tamil Nadu) of India have been considered.

[Data Source: https://www.mohfw.gov.in/#state-data, accessed on 2 September 2024]

The variables considered in study are described as

$x_{1i},x_{2i}$ and $x_{3i}$ Weekly positivity rate of COVID-19 cases in the $i$ stratum for the week 21 June to 27 June 2021;

$y_{1i},y_{2i}$ and $y_{3i}$: Weekly positivity rate of COVID-19 cases in the $i$ stratum for the week of 26 June to 2 July 2021.

It has been assumed that the measurement errors, u and v are correlated but are independent of the main study variables. The corresponding data are generated from Normal distribution and the parameters of the considered population are $N_1=14$, $N_2=30,$ $N_3=37, {\overline{y}}_1=10.5779, {\overline{y}}_2=2.1660, {\overline{y}}_3=2.8916, {\overline{x}}_1=10.5471$, ${\overline{x}}_2=2.5523$, and ${\overline{x}}_3=3.3800$.

The parameters chosen to generate the data are as follows:

${\sigma }_u^2\in \{1,2\}, {\sigma }_v^2\in \{1,2\}$ and ${\rho }_{uv}\in \{-0.9,-0.7,-0.5,-0.3,0.0,0.5,0.7,0.9\}$

For the above considered Population data-I and Population data-II, a simulation study has been performed based on $10,000$ replications of an independent sample to calculate the variance of the proposed estimators $V\left[{\left(T_{st}^{me}\right)}_1\right],V\left[{\left(T_{st}^{me}\right)}_2\right]$ and $V\left[{\left(T_{st}^{me}\right)}_3\right]$ in the presence and absence of measurement errors and the results are presented in

Table 3. Simulation results in the presence of measurement errors (Population data-I).

		${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=30\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=40\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=60\%$
SET	${\mathit{\rho }}_{\mathit{u}\mathit{v}}$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{1}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{2}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{3}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{1}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{2}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{3}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{1}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{2}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{3}}\right]$
	$0.9$	$0.9424$	$0.0146$	$0.9463$	$0.7322$	$0.1226$	$0.7347$	$0.2843$	$0.0191$	$0.2850$
	$0.7$	$0.8212$	$0.1244$	$0.8167$	$0.7142$	$0.1212$	$0.7123$	$0.3115$	$0.1100$	$0.3121$
	$0.5$	$0.7704$	$0.1227$	$0.7661$	$0.6641$	$0.1195$	$0.6624$	$0.2764$	$0.1088$	$0.2770$
I	$0.0$	$0.9575$	$0.1414$	$0.9610$	$0.6834$	$0.1214$	$0.6856$	$0.2641$	$0.0185$	$0.2647$
	$-0.3$	$0.5268$	$0.1214$	$0.5292$	$0.6512$	$0.1194$	$0.6490$	$0.2837$	$0.0188$	$0.2844$
	$-0.5$	$0.8443$	$0.1276$	$0.8470$	$0.5275$	$0.1156$	$0.5317$	$0.2953$	$0.0193$	$0.2960$
	$-0.7$	$0.7067$	$0.1223$	$0.7093$	$0.4341$	$0.1129$	$0.4341$	$0.2739$	$0.0187$	$0.2749$
	$-0.9$	$0.3077$	$0.1109$	$0.3090$	$0.5990$	$0.1161$	$0.6109$	$0.2561$	$0.0182$	$0.2567$
	$0.9$	$0.9619$	$0.1409$	$0.9648$	$0.6818$	$0.1210$	$0.6839$	$0.2619$	$0.0183$	$0.2625$
	$0.7$	$0.2984$	$0.1123$	$0.2975$	$0.3025$	$0.1135$	$0.3021$	$0.2925$	$0.0193$	$0.2931$
	$0.5$	$0.3019$	$0.1123$	$0.3009$	$0.3064$	$0.1134$	$0.3058$	$0.2702$	$0.0187$	$0.2709$
II	$0.0$	$0.9308$	$0.1268$	$0.9321$	$0.5536$	$0.1157$	$0.5554$	$0.2544$	$0.0182$	$0.2550$
	$-0.3$	$0.9411$	$0.1266$	$0.9421$	$0.5695$	$0.1162$	$0.5712$	$0.2618$	$0.0185$	$0.2623$
	$-0.5$	$0.8319$	$0.1277$	$0.8252$	$0.2652$	$0.0198$	$0.2650$	$0.2707$	$0.0184$	$0.2713$
	$-0.7$	$0.8456$	$0.1286$	$0.8390$	$0.2931$	$0.1111$	$0.2930$	$0.2854$	$0.0192$	$0.2861$
	$-0.9$	$0.9435$	$0.1214$	$0.9448$	$0.6169$	$0.1160$	$0.6187$	$0.2633$	$0.0175$	$0.2639$
	$0.9$	$0.9591$	$0.1295$	$0.9589$	$0.7806$	$0.1225$	$0.7810$	$0.2778$	$0.0190$	$0.2784$
	$0.7$	$0.9742$	$0.1509$	$0.9753$	$0.8017$	$0.1227$	$0.8020$	$0.2841$	$0.0192$	$0.2847$
	$0.5$	$0.9068$	$0.1380$	$0.9113$	$0.8474$	$0.1249$	$0.8469$	$0.2827$	$0.0188$	$0.2833$
III	$0.0$	$0.2707$	$0.1112$	$0.2716$	$0.3265$	$0.1146$	$0.3268$	$0.2821$	$0.0188$	$0.2827$
	$-0.3$	$0.9104$	$0.1421$	$0.9140$	$0.6997$	$0.1215$	$0.7020$	$0.2810$	$0.0189$	$0.2816$
	$-0.5$	$0.2517$	$0.0192$	$0.2513$	$0.4078$	$0.1126$	$0.4086$	$0.2695$	$0.0187$	$0.2700$
	$-0.7$	$0.9526$	$0.1434$	$0.9541$	$0.6427$	$0.1210$	$0.6471$	$0.2035$	$0.0196$	$0.2042$
	$-0.9$	$0.2527$	$0.0194$	$0.2520$	$0.4229$	$0.1120$	$0.4197$	$0.2986$	$0.0195$	$0.2993$

and

Table 4. Simulation results in the presence of measurement errors (Population data-II).

		${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=30\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=40\%$			${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3=60\%$
SET	${\mathit{\rho }}_{\mathit{uv}}$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{1}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{2}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{3}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{1}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{2}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{3}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{1}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{2}}\right]$	$\mathit{V}\left[{\left({\mathit{T}}_{\mathit{st}}^{\mathit{me}}\right)}_{\mathbf{3}}\right]$
	$0.9$	$1.3907$	$1.1119$	$1.3900$	$1.3000$	$1.1101$	$1.3009$	$1.2501$	$1.0100$	$1.2507$
	$0.7$	$1.7007$	$1.1203$	$1.7003$	$1.4301$	$0.9109$	$1.4301$	$1.2709$	$0.9007$	$1.2709$
	$0.5$	$1.8403$	$1.1206$	$1.8400$	$1.5205$	$0.9106$	$1.5307$	$1.2903$	$1.9103$	$1.2900$
I	$0.0$	$1.9505$	$1.1404$	$0.9600$	$1.6804$	$0.9205$	$1.6806$	$1.2601$	$0.9105$	$1.2607$
	$-0.3$	$1.5208$	$1.1204$	$1.5202$	$1.6502$	$0.9221$	$1.6400$	$1.2807$	$0.9106$	$1.2804$
	$-0.5$	$1.7714$	$1.1207$	$1.7601$	$1.6601$	$0.9230$	$1.6604$	$1.2704$	$0.9108$	$1.2700$
	$-0.7$	$1.8202$	$1.1204$	$1.8107$	$1.7102$	$0.9201$	$1.7103$	$1.3105$	$0.9112$	$1.3101$
	$-0.9$	$2.1404$	$1.0106$	$2.1403$	$1.7302$	$0.9302$	$1.7307$	$1.2803$	$0.9201$	$1.2800$
	$0.9$	$1.5405$	$1.1204$	$1.5408$	$1.5109$	$0.9200$	$1.5107$	$1.2303$	$0.9105$	$1.2309$
	$0.7$	$1.8406$	$1.1206$	$1.8300$	$1.2901$	$0.9201$	$1.2900$	$1.2804$	$0.90102$	$1.2801$
	$0.5$	$1.8309$	$1.1207$	$1.8202$	$1.2602$	$0.9208$	$1.2600$	$1.2717$	09104	$1.2703$
II	$0.0$	$1.9318$	$1.1208$	$1.9301$	$1.5506$	$0.9307$	$1.5504$	$1.2504$	$0.9105$	$1.2500$
	$-0.3$	$1.9401$	$1.1206$	$1.9401$	$1.5605$	$0.9202$	$1.5702$	$1.2608$	$0.9105$	$1.2603$
	$-0.5$	$1.3009$	$1.1103$	$1.3019$	$1.3004$	$0.9204$	$1.3008$	$1.2712$	$0.9103$	$1.2719$
	$-0.7$	$1.2904$	$1.1103$	$1.2905$	$1.3005$	$0.9205$	$1.3001$	$1.2905$	$0.9103$	$1.2901$
	$-0.9$	$1.9609$	$1.1419$	$1.9608$	$1.6808$	$0.9300$	$1.6809$	$1.2609$	$0.9103$	$1.2605$
	$0.9$	$1.2507$	$1.0104$	$1.2500$	$1.4209$	$0.9300$	$1.4107$	$1.2906$	$0.9205$	$1.2903$
	$0.7$	$1.9506$	$1.1404$	$1.9501$	$1.6407$	$0.9300$	$1.6401$	$1.3005$	$0.9206$	$1.3002$
	$0.5$	$1.2507$	$1.0102$	$1.2503$	$1.4008$	$0.9506$	$1.4006$	$1.2605$	$0.9307$	$1.2710$
III	$0.0$	$1.2717$	$1.1102$	$1.2706$	$1.3205$	$0.9506$	$1.3208$	$1.2801$	$0.9308$	$1.2807$
	$-0.3$	$1.9914$	$1.1401$	$1.9900$	$1.6907$	$0.9405$	$1.7000$	$1.2800$	$0.9309$	$1.2806$
	$-0.5$	$1.9008$	$1.1300$	$1.9103$	$1.8404$	$0.9509$	$1.8409$	$1.2807$	$0.9308$	$1.2803$
	$-0.7$	$1.9742$	$1.1519$	$1.9703$	$1.8007$	$0.9507$	$1.8000$	$1.2801$	$0.9302$	$1.2807$
	$-0.9$	$2.1501$	$1.1205$	$2.1509$	$1.7816$	$0.9505$	$1.7800$	$1.2708$	$0.9300$	$1.2704$

and

Table 5. Results in the absence of any measurement errors (Population data-I).

${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3$	$\mathit{V}\left[{\mathit{T}}_1\right]$	$\mathit{V}\left[{\mathit{T}}_2\right]$	$\mathit{V}\left[{\mathit{T}}_3\right]$
$30\%$	$0.6214$	$0.1257$	$0.6212$
$40\%$	$0.3546$	$0.1138$	$0.3553$
$60\%$	$0.2679$	$0.0185$	$0.2685$

Note: $V\left[T_1\right],V\left[T_2\right]$ and $V\left[T_3\right]$ denote the variances of the proposed estimators in the absence of measurement error, i.e., when $u=v=o$ in the estimators ${\left(T_{st}^{me}\right)}_1,{\left(T_{st}^{me}\right)}_2$, and ${\left(T_{st}^{me}\right)}_3$, respectively.

and

Table 6. Results in the absence of any measurement errors (Population data-II).

${\mathit{n}}_1={\mathit{n}}_2={\mathit{n}}_3$	$\mathit{V}\left[{\mathit{T}}_1\right]$	$\mathit{V}\left[{\mathit{T}}_2\right]$	$\mathit{V}\left[{\mathit{T}}_3\right]$
$30\%$	$0.7910$	$0.2357$	$0.7900$
$40\%$	$0.5540$	$0.8208$	$0.5503$
$60\%$	$0.4909$	$0.2200$	$0.4902$

Note: $V\left[T_1\right],V\left[T_2\right]$ and $V\left[T_3\right]$ denote the variances of the proposed estimators in the absence of measurement error, i.e., when $u=v=o$ in the estimators ${\left(T_{st}^{me}\right)}_1,{\left(T_{st}^{me}\right)}_2$, and ${\left(T_{st}^{me}\right)}_3$, respectively.

, respectively.

The following combination of parameters are used for the simulation:

$$\begin{array}{cc}\hfill \mathrm{SET-I}:& \hfill {\sigma }_u^2=1,{\sigma }_v^2=1\hfill \\ \hfill \mathrm{SET-II}:& \hfill {\sigma }_u^2=1,{\sigma }_v^2=2\hfill \\ \hfill \mathrm{SET-III}:& \hfill {\sigma }_u^2=2,{\sigma }_v^2=1\hfill \end{array}$$

Further, our sampling fractions are

Population data-I

$$\begin{array}{cc}\hfill 30\%:& \hfill n_1=3,n_2=6,n_3=5\hfill \\ \hfill 40\%:& \hfill n_1=4,n_2=8,n_3=7\hfill \\ \hfill 60\%:& \hfill n_1=7,n_2=11,n_3=10\hfill \end{array}$$

Population data-II

$$\begin{array}{cc}\hfill 30\%:& \hfill n_1=4,n_2=9,n_3=11\hfill \\ \hfill 40\%:& \hfill n_1=5,n_2=12,n_3=15\hfill \\ \hfill 60\%:& \hfill n_1=8,n_2=18,n_3=22\hfill \end{array}$$

4. Discussion of Results

1.

While conducting sample surveys, the performance of any estimator is examined with the natural real data to demonstrate the practical applicability of the proposed estimators. So, in order to analyze the impact of measurement error under the stratified sampling approach, we have obtained three different calibration estimators and their variances through simulation study for two natural population data sets.

Some noteworthy results from Table 3 and Table 4 are

(i)

In the presence of measurement error, it can be seen that the variance of proposed calibration estimators ${\left(T_{st}^{me}\right)}_1$ and ${\left(T_{st}^{me}\right)}_3$ perform better than the proposed calibration estimator ${\left(T_{st}^{me}\right)}_2$ for all the considered sample sizes under the stratified approach for both the Population data-I and Population data-II.

(ii)

As sample size increases, the variance decreases for all the considered estimators in the presence of measurement error under the stratified approach for both the considered Population data-I and Population data-II.

(iii)

It is to be noted that under Population data-II the variances are better than the considered population data-I for all the calibration estimators in the presence of measurement error under the stratified approach.

(iv)

For Population data-I, as ${\rho }_{uv}$ changes from $0.9$ to $-0.9$ for SET-I, the variance of the proposed calibration estimator $V{\left(T_{st}^{me}\right)}_3>V{\left(T_{st}^{me}\right)}_1>V{\left(T_{st}^{me}\right)}_2$. In addition, for SET-II and SET-III, as ${\rho }_{uv}$ changes from $0.9$ to $0.5$ and $-0.5$ to $-0.9$, the calibrated estimators satisfy the inequality $V{\left(T_{st}^{me}\right)}_3>V{\left(T_{st}^{me}\right)}_1>V{\left(T_{st}^{me}\right)}_2$ in the presence of measurement error under the stratified approach for the considered sample size of $60\%.$ And for the remaining sample sizes, moderate values can be seen for the variances of the proposed calibrated estimators when ${\rho }_{uv}$ changes from $0.9$ to $-0.9$ for all the considered SET-I, SET-II and SET-III. This indicates a significant impact of the correlation between the study error and auxiliary variables on the properties of the estimators.

(v)

However, for population data-II, as ${\rho }_{uv}$ changes from $0.9$ to $-0.9$, the variance of the proposed calibration estimator ${\left(T_{st}^{me}\right)}_1,{\left(T_{st}^{me}\right)}_2$, and ${\left(T_{st}^{me}\right)}_3$ increases for the considered SET-I, SET-II and SET-III, respectively. In this case, the variance of negative values of ${\rho }_{uv}$ is higher as compared to positive values of ${\rho }_{uv}.$ Also, the same pattern can be observed for the considered estimator ${\left(T_{st}^{me}\right)}_3$, and moderate values can be seen for the estimator ${\left(T_{st}^{me}\right)}_2$ in the presence of measurement error under the stratified approach.

2.

When there is no measurement error in the data, we intend to examine the performance of the proposed calibration estimators and their variances.

The following is the interpretation of the results based on Table 1, Table 2, Table 5 and Table 6:

(i)

From Table 1, it is clear that $Rati{o}_i\le 1;i=1,2,3$ for all choices of ${\rho }_{uv}$ and varying values of ${\sigma }_u^2$ and ${\sigma }_v^2$ when the considered calibration estimators in the absence of measurement error are compared with respect to the calibration estimators in the presence of measurement error. This shows that measurement error significantly affects the calibration estimators. The $Rati{o}_i$ increases significantly for the majority of the combinations of parameters when the same estimator is studied in the absence of measurement error.

(ii)

In Table 2, if ${\rho }_{uv}$$\in [0.9,0.0]$ the value of $Rati{o}_i,i=1,2,3$ increases, and if ${\rho }_{uv}$$\in [0.0,-0.9]$ the value of $Rati{o}_i<0.6,i=1,2,3$, and for the majority of the combination of parameters, it is greater than $0.6.$ This shows that the presence of measurement error degrades the performance of the estimator.

(iii)

From Table 5 and Table 6, the results show that, in the absence of any measurement error, the variance of the proposed three calibration estimators do not demonstrate improved performance for either the considered Population data-I or Population data-II. This shows that the presence of measurement errors do affect the statistical properties of the estimators. As sample size increases, the variance decreases for all the considered estimators in the absence of measurement error.

5. Concluding Remarks

From the results obtained, it is concluded that, the proposed calibration estimators under the stratified approach are significantly impacted by the presence of measurement error for both Population data-I and Population data-II. The proposed calibrated estimators ${\left(T_{st}^{me}\right)}_1$ and ${\left(T_{st}^{me}\right)}_3$ show superiority over the proposed estimator ${\left(T_{st}^{me}\right)}_2$ in the presence of measurement error under all the cases. For Population data-II, the calibrated estimators perform better compared to population data-I. Loss in precision is also observed in the absence of measurement error for the proposed estimators under both population data sets. Although the presence of measurement error deteriorates the performance of the calibrated estimators, in the event that they creep into the survey, then the classical additive measurement error model is proved to be feasible in handling the measurement errors situation with calibrated estimators for the estimation of the population mean under a stratified approach. Finally, looking at the nice behaviours of the proposed calibrated estimators in the presence of measurement error under a stratified approach, one may recommend them to the survey statisticians and practitioners for practical applications.