Estimation of finite population mean of a sensitive variable using three-stage orrt in the presence of non-response and measurement errors

1. Introduction

In a survey, it’s challenging to collect accurate information on a sensitive study variable that has a socially stigmatizing characteristic such as “Have you ever had an abortion?”, “How much money do you make?” “Have you ever been infected with sexually transmitted diseases?” or “Are you a drug addict?” among others. Obtaining correct answers to such questions in an interview involving direct questioning is difficult since the respondent’s privacy is not protected. Most respondents will either purposefully give a false answer or refuse to respond to such questions due to fear of embarrassment or loss of social status.

Randomized Response Technique (RRT) was pioneered by Warner [1], and its main objective was to reduce response bias in surveys involving a sensitive question. In RRT, a scrambled variable is used to estimate the finite population mean of a sensitive variable. The scrambled variable is assumed to be independent of the study and the non-sensitive auxiliary variable. The respondent must adequately respond to the non-sensitive additional variable and a scrambled response for the study variable.

The Optional Randomized Response Technique (ORRT) was pioneered by Chaudhuri and Mukherjee [2]. The technique involves giving the respondent an option to provide a scrambled or direct response to a sensitive question. In one-stage ORRT [3], a respondent is expected to either give a scrambled response if they feel the question is sensitive and a direct response if the question is non-sensitive.

Two-stage ORRT aims at increasing respondent privacy and participation in a survey involving a sensitive study variable [4]. In two-stage ORRT, a known proportion of respondents, \(t_h\), is requested to respond directly to a sensitive question while maintaining anonymity. The remaining proportion of respondents provide a scrambled response using an additive model. The main drawback of two-stage ORRT is that it requires a significant value of \(\ t_h\), especially when the underlying question is susceptible.

The three-stage ORRT [5] aims to promote respondent privacy and cooperation in a survey involving a sensitive question. A known predetermined proportion, \(t_h\) of respondents, is requested to give a true response to a sensitive question. Another predetermined proportion, \(f_h\), is requested to provide a scramble response. The remaining proportion is given the option of either providing a scrambled or direct response to a sensitive question. In additive three-stage ORRT, the scrambled response provided is defined as \[\label{GrindEQ__1_} Z_{hi}=\left\{ \begin{array}{c} Y_{hi}\ \ with\ pobability\ t_h+\left(1-t_h-f_h\right)\left(1-{\psi }_h\right) \\ Y_{hi}+S_{hi},\ with\ probability\ f_h+\left(1-t_h-f_h\right)\ \ \end{array} \right. \tag{1}\] , where \(Y_{hi}\ \)is the study variable, \({\psi }_h\) is the sensitivity level, and \(S_{hi}\ \) is a scrambled variable that is normally distributed with mean 0 and variance\({\ S}^2_{Sh}\). The mean and variance of \(Z_{hi}\) are given as \(E\left(Z_{hi}\right)=E\left(Y_{hi}\right)\) and

\(S^2_{Zh}=S^2_{Yh}+{\varphi }_hS^2_{Sh}\ \), respectively, where\({\ \varphi }_h=f_h+\left(1-t_h-f_h\right)\).

In the literature, researchers who have studied the estimation of finite population mean of a sensitive study variable using non-optional RRT include Eichhron and Hayre [6], Gupta and Shabbir [7], Gupta et al. [8], Sousa et al. [9], Zatezalo [10], and Mushtaq et al [11], Mushtaq et al. [12], and Mushtaq and Noor-Ul-Amin [13].

The inability of specific units to provide information due to their unwillingness to participate, illness, or absence is referred to as non-response. Non-response in a survey reduces the sample size, raising the variance of an estimator of the finite population mean. Hansen and Hurwitz [14] proposed a strategy for getting data from non-responding units in a postal survey called subsampling. The approach included more effort to gather data directly from a subsample of non-responding units. The strategy is used in this study to handle the problem of non-response.

During the data collection and recording phases of a survey, measurement errors can occur. The difference between a variable’s valid values and those reported in a survey is known as measurement error. When measurement errors occur in a survey, the data becomes contaminated, resulting in under- or overestimated parameters during analysis.

Khalil [15] discusses the issue of estimation of the finite population mean in the presence of measurement errors in simple random sampling based on non-optional RRT. The problem of estimation of the finite population mean using a non-optional RRT model in the presence of measurement errors under stratified random sampling is addressed by Khalil et al. [16]. In the presence of measurement errors, Khalil et al. [17] extended the work of Khalil [16] to estimate the finite population mean using one-stage optional RRT. Recently, Onyango et al. [18] studied the problem of estimating the finite population mean and measurement errors using the non-optional RRT in stratified two-phase sampling.

Naeem and Shabbir [19] and Zahid and Shabbir [20] discuss the problem of estimation of the finite population mean of a sensitive variable using non-optional RRT in the presence of measurement errors and non-response simultaneously. Zhang et al. [21] studied the problem of estimating the finite population mean using one-stage RRT in the presence of measurement errors and non-response simultaneously in simple random sampling. Recently, Zhang et al. [22] studied the estimation of the finite population mean of a sensitive study variable using ORRT under measurement errors and non-response in stratified random sampling.

The present study fills the existing gap in the literature on estimating the finite population mean of a sensitive study variable using an additive three-stage ORRT model in the presence of measurement errors and non-response simultaneously.

In this paper, section two describes the population and notations used in this study. The existing estimators of population mean under three-stage ORRT models are described in section three. The proposed estimator and its properties of bias and mean square error are discussed in section four. Section five looks at the theoretical efficiency of the proposed estimator. Section six performs a numerical analysis of the proposed estimator’s performance. Finally, section seven contains the conclusions of the study.

Notations

Consider a population \(U=U_1,\ U_2,\ \dots ,\ U_N\) of size N. The population comprises a sensitive study variable, auxiliary variable, and scrambled response denoted as Y, X, and Z, respectively. Let \(\left(X_{hi},Y_{hi},\ Z_{hi}\right)\) and \(\left(x_{hi},y_{hi},\ z_{hi}\right)\) denote the \(i^{th}\) values of X, Y, and Z in the \(h^{th}\)population and sample stratum respectively. Furthermore, let \({\overline{X}}_h,{\overline{Y}}_h,\ and\ {\overline{Z}}_{h\ }\)denote the population mean of X, Y, and Z respectively in the \(h^{th}\) stratum. The variance of the scrambled response and auxiliary variables are obtained using \(S^2_{Zh}=\frac{1}{N_h-1}\sum^{N_h}_{i=1}{{\left(z_{hi}-{\overline{Z}}_h\right)}^2}\) and \(S^2_{Xh}=\frac{1}{N_h-1}\sum^{N_h}_{i=1}{{\left(x_{hi}-{\overline{X}}_h\right)}^2}\) respectively. Let \(S_{XZ}\) denote the covariance between the auxiliary variable and scrambled response. Also, let \({\rho }_{ZXh}\) denote the correlation coefficient between the scrambled response and the auxiliary variable.

Auxiliary information may be available in a survey as an attribute. Let\({\ \tau }_{hij}\) denote the value of \(j^{th}\) attribute for \(i^{th}\) unit (i=1, 2 … and j=1, 2 …) in the \(h^{th}\) stratum. The auxiliary attribute takes the values 1 and 0 if \(i^{th}\) population unit possesses and does not possess an attribute, respectively. Furthermore, let \(A_{hj}=\sum^{N_h}_h{{\tau }_{hij}}\) and\(\ \ P_h=\frac{A_{hj}}{N_h}\), be the number of units that have an attribute and proportion of units possessing an attribute in the population respectively. Additionally, let \(\ S^2_{Ph}\) denote the population variance of an auxiliary attribute. Also, let \(\left(S_{ZPh},\ and\ S_{XPh}\right)\) \(\ \) and \(\left({\ \rho }_{Zph},\ and\ {\rho }_{XPh}\right)\) \(\ \)denote the covariances and coefficient of correlations between their subscripts respectively.

In the presence of non-response, let \({\ N}_{1h}\) and \({\ N}_{2h}\)be the population sizes of the responding and non-responding units, respectively. Let \(S^2_{Zh\eqref{GrindEQ__2_}}\)denote the population variance of the sensitive variable for the non-responding units in the \(h^{th}\) stratum.

A relatively large sample of \(n'_h\) is drawn from the \(h^{th}\) stratum population using simple random sampling without replacement (SRSWOR). The sample mean of the auxiliary variable in the first phase sample is given as\({\ \ \overline{x}}'_h=\frac{1}{n'_h}\sum^{n'_h}_{i=1}{x_{hi}}\) and the proportion of units in the first phase sample possessing an auxiliary attribute as\(\ p'_h=\frac{a_{hj}}{n'_h}\). A second phase random sample of size \({\ n}_h\) is drawn from the first phase, \({\ n}_{1h}\) units are observed to respond, and non-response is observed on the remaining \({\ n}_{2h}\) units. Let \({\overline{x}}_h=\frac{1}{{\ n}_h}\sum^{{\ n}_h}_{i=1}{x_{hi}}\) be the sample mean and \({\ p}_h=\frac{a_{hj}}{{\ n}_h}\) be the proportion of units in the second phase sample that possess an auxiliary attribute. Also, let \({\overline{z}}_{1h}=\frac{1}{{\ n}_{1h}}\sum^{{\ n}_h}_{i=1}{z_{hi}}\ \ \) be the sample mean for the responding group in the second phase sample. A sub-sample of size\(\ r_{2h}=\frac{{\ n}_{2h}}{{\ k}_{2h}}\), where \({\ k}_{2h}\) is the inverse sampling rate is drawn from the non-responding sample. Let \({\overline{z}}_{2h}=\frac{1}{{\ r}_{2h}}\sum^{{\ r}_{2h}}_{i=1}{z_{hi}}\ \ \)be the sub-sample mean for the non-responding units. The estimate of the population mean for the scrambled response is given as

\(\ {\overline{z}}^*_h=w_{1h}{\overline{z}}_{1h}+\ w_{2h}{\overline{z}}_{2h}\), where\(\ w_{1h}=\frac{n_{1h}}{{\ n}_h}\), and\(\ \ w_{2h}=\frac{n_{2h}}{{\ n}_h}\).

In the presence of measurement errors, let \(\ Z^*_{hi}\) and \(z^*_{hi}\ \)be the true and observed values, respectively, for the scrambled response. Furthermore, let\(\ T^*_{hi}=z^*_{hi}-Z^*_{hi}\) denote the measurement errors associated with the scrambled response. These measurement errors are assumed to be normally distributed with mean 0 and variance\({\ S}^2_{Th}\). Furthermore, let \(S^2_{Th\eqref{GrindEQ__2_}}\) be the variance associated with non-responding units.

In this study, the following conventional and non-conventional measures of auxiliary variables are used in developing the special cases of the proposed generalized class of estimators;

1. \(C_{Xh}=\frac{S_{Xh}}{{\overline{X}}_h}\), coefficient of variation,

2. Coefficient of correlation defined as \({\rho }_{XYh}\),

3. Coefficient of skewness defined as \({\beta }_{1h}(x)=\frac{N_h\sum^{N_h}_{i=1}{{\left(X_{hi}-{\overline{X}}_h\right)}^3}}{\left(N_h-1\right)\left(N_h-2\right)S^3_{Xh}}\)

4. Coefficient of kurtosis defined as \({\beta }_{2h}(x)=\frac{N_h\left(N_h+1\right)\sum^N_{i=1}{{\left(X_{hi}-{\overline{X}}_h\right)}^4}}{\left(N_h-1\right)\left(N_h-2\right)\left(N\_h-3\right)S^4_{Xh}}-\frac{3{\left(N_h-1\right)}^2}{\left(N_h-2\right)\left(N_h-3\right)}\)

5. Mid-range is defined as\(,\ {MR}_h(x)=\frac{x_{h\left(1\right)}+x_{h\left(Nh\right)}}{2}\).

, where \(\ x_{h\left(1\right)}\) and \(x_{\left(Nh\right)}\) are the minimum and maximum values in a data set.

1. Quartile deviation is defined as\({\ QD}_h(x)=\frac{Q_{3h}(x)-Q_{1h}(x)}{2}\).

2. Tri-mean was proposed by Turkey [23] and is defined as

\({TM}_h(x)=\frac{Q_{1h}(x)+2Q_{2h}(x)+Q_{3h}(x)}{4}\), where \(Q_{1h}(x),\ Q_{2h}(x)\ \mathrm{and}{\ Q}_{3h}(x)\) are the first, second and third quartiles respectively,

1. Hodges-Lehmann [24] estimator is defined as \({HL}_h(x)=\mathrm{Median}\left(\frac{x_{jh}+x_{kh}}{2}\right),\) \[\ 1\le jh\le kh\le N\]

2. Some Existing Estimators

3. The adapted ordinary estimator of population mean is defined as \[\label{GrindEQ__2_} {\overline{Y}}_0\mathrm{=}\sum^L_{h=1}{w_h{\overline{z}}^*_h} \tag{2}\]

The variance of the estimator is given as \[\label{GrindEQ__3_} Var({\overline{Y}}_0)\cong \sum^L_{h=1}{W^2_h}B_h \tag{3}\]

1. The adapted Cochran [25] ratio estimators are defined as \[\label{GrindEQ__4_} {\overline{Y}}_R\mathrm{=}\sum^L_{h=1}{w_h{\overline{z}}^*_h\ \frac{{\overline{x}}'_h}{{\overline{x}}^*_h}} \tag{4}\]

The expression for the bias is given as \[\label{GrindEQ__5_} Bias({\overline{Y}}_R)\cong \sum^L_{h=1}{\frac{W_h}{{\overline{X}}_h}\left[\frac{9}{8}R_h\left(A_h-C_h\right)-\left(E_h-D_h\right)\right]} \tag{5}\]

The expression of the mean square error is given as

\[\label{GrindEQ__6_} MSE({\overline{Y}}_R)\cong \sum^L_{h=1}{W^2_h\left[B_h+R^2_h\left(A_h-C_h\right)-2R_h\left(E_h-D_h\right)\right]} \tag{6}\] , where \(A_h={\theta }_hS^2_{Xh},\ \) \(C_h={\theta }^{\ '}_hS^2_{Xh},\ \) \(D_h={\theta }^{\ '}_hS^2_{ZXh},\) \(E_h={\theta }_hS_{ZXh},\) \({\theta }'_h=\left(\frac{1}{n'_h}-\frac{1}{N_h}\right)\), and \({\theta }^{\ '}_h=\left(\frac{1}{n'_h}-\frac{1}{N_h}\right).\ \)

1. The adapted Bahl and Tuteja [26] exponential ratio-type estimator is defined as \[\label{GrindEQ__7_} t_{ER}=\sum^L_{h=1}{W_h{\overline{z}}_hexp\left(\frac{{\overline{x}}'_h-{\overline{x}}_h}{{\overline{x}}'_h+{\overline{x}}_h}\right)} \tag{7}\]

The expression for the bias is given as \[\label{GrindEQ__8_} Bias(t_{ER})\cong \sum^L_{h=1}{\frac{W_h}{2{\overline{X}}_h}\left[\frac{3}{4}R_h\left(A_h-C_h\right)-\left(E_h-D_h\right)\right]} \tag{8}\]

The expression for the mean square error is given as

\[\label{GrindEQ__9_} MSE(t_{ER})\cong \sum^L_{h=1}{W^2_h\left[B_h+\frac{1}{4}R^2_h\left(A_h-C_h\right)-R_h\left(E_h-D_h\right)\right]} \tag{9}\]

2. The Proposed Strategy of Mean Estimation

2.1. Modified Hansen and Hurwitz [14] technique

From equation (1) the expected value of the scrambled response under randomization mechanisms (Onyango et al. [27]) is defined as \[\label{GrindEQ__10_} E_R\left(Z_{hi}\right)=E_R\left[Y_{hi}\left(1-{\varphi }_h\right)+\left(Y_{hi}+S_{hi}\right){\varphi }_h\right] \tag{10}\] \[\label{GrindEQ__11_} E_R\left(Z_{hi}\right)=Y_{hi}+{\varphi }_h\ {\overline{S}}_h\ \tag{11}\] , where\(\ {\varphi }_h=f_h+{\psi }_h\left(1-t_h-f_h\right).\ \ \)

The variance of the response variable under randomization mechanisms is given as \[V_R\left(Z_{hi}\right)=V_R\left(Y_{hi}+{\varphi }_hS_{hi}\right)\] \[\label{GrindEQ__12_} V_R\left(Z_{hi}\right)=\varphi \left(S^2_{Sh}+{\overline{S}}^2_h\ \right)-{\varphi }^2_h{\overline{S}}^2_h \tag{12}\] \[\label{GrindEQ__13_} V_R\left(Z_{hi}\right)=\varphi S^2_{Sh} \tag{13}\] The transformed value of the randomized response is given as \[\label{GrindEQ__14_} {\hat{y}}_{hi}=z_{hi}-{\varphi }_h{\overline{S}}_h \tag{14}\] , with \(E_R\left({\hat{y}}_{hi}\right)=y_{hi}\) and,\(V_R\left({\hat{y}}_{hi}\right)=\varphi S^2_{Sh}\) where\(\ y_{hi}\) is the true response. Therefore, the modified Hansen and Hurwitz [15] technique with an additive three-stage ORRT added is defined as \[\label{GrindEQ__15_} {\widehat{\overline{y}}}_h=w_{1h}{\widehat{\overline{y}}}_{1h}+\ w_{2h}{\widehat{\overline{y}}}_{2h} \tag{15}\] \[\label{GrindEQ__16_} E\left({\hat{y}}_h\right)={\overline{Y}}_h \tag{16}\] \[\label{GrindEQ__17_} var\left({\widehat{\overline{y}}}_h\right)=E_1\left[V_2\left({\widehat{\overline{y}}}_h\right)\right]+V_1\left[E_2\left({\widehat{\overline{y}}}_h\right)\right] \tag{17}\] \[\label{GrindEQ__18_} var\left({\widehat{\overline{y}}}_h\right)=var\left({\overline{y}}_h\right)+E_1\left[\frac{n_{1h}}{n^2_h}\frac{;{o}_hS^2_{Sh}}{n_{1h}}\right]+E_1\left[\frac{n_{2h}}{n^2_h}k_{2h}{o}_h\ S^2_{Sh}\right] \tag{18}\] \[\label{GrindEQ__19_} var\left({\widehat{\overline{y}}}_h\right)=var\left({\overline{y}}_h\right)+{\mathrm{\Omega }}_h \tag{19}\] , where \({\mathrm{\Omega }}_h=\frac{\"{o}_h\ S^2_{Sh}}{n_h}\left(W_{1h}+k_{2h}W_{2h}\right)\) is the contribution of the three-stage ORRT to the variance of Hansen and Hurwitz [14] estimator.

2.2. The proposed generalized class of estimators

The suggested randomized response estimator of the finite population mean for a sensitive variable in the presence of non-response and measurement errors simultaneously is defined as \[\label{GrindEQ__20_} {\overline{Y}}_g\mathrm{=}\sum^L_{h=1}{w_h\left[{\overline{z}}^*_h+{\alpha }_h\left({\overline{x}}'_h-{\overline{x}}_h\right)+{\beta }_h\mathrm{\ }\left(p'_h-p_h\right)\right]exp\left(\frac{~a_h({\overline{x}}'_h-{\overline{x}}_h)}{a_h({\overline{x}}'_h+{\overline{x}}_h)+2b_h}\right)}, \tag{20}\] , where\(\ {\alpha }_h\), and \({\beta }_h\) are appropriately chosen constants,\({~a}_h\ and\ b_h\), are either real numbers or some known conventional and non-conventional measures of auxiliary variable. Let \[\label{GrindEQ__21_} {\sigma }_{Zh}={\overline{z}}^*_h-{\overline{Z}}_h \tag{21}\] \[\label{GrindEQ__22_} {\sigma }_{X1h}={\overline{x}}'_h-{\overline{X}}_h\ \tag{22}\] \[\label{GrindEQ__23_} {\sigma }_{P1h}=p'_h-P_h\ \tag{23}\] \[\label{GrindEQ__24_} {\sigma }_{Xh}={\overline{x}}_h-{\overline{X}}_h \tag{24}\] \[\label{GrindEQ__25_} {\sigma }_{Ph}=p_h-P_h\ \tag{25}\]

\[\label{GrindEQ__26_} E\left({\sigma }_{Zh}\right)=E\left({\sigma }_{Xh}\right)=E\left({\sigma }_{X1h}\right)=E\left({\sigma }_{P1h}\right)=E\left({\sigma }_{Ph}\right)=0\ \tag{26}\] Furthermore, let \[\label{GrindEQ__27_} E({\sigma }^2_{Xh}\mathrm{)=}~{\theta }_hS^{\mathrm{2}}_{Xh}\mathrm{=}A_h~~ \tag{27}\] \[\label{GrindEQ__28_} E({\sigma }^2_{Zh}\mathrm{)=}~~{\theta }_h~\left(S^{\mathrm{2}}_{Yh}\mathrm{+}S^{\mathrm{2}}_{Th}\right)\mathrm{+}{\theta }^{\mathrm{*}}_h\left(S^{\mathrm{2}}_{Yh\left(\mathrm{2}\right)}\mathrm{+}S^{\mathrm{2}}_{Th\left(\mathrm{2}\right)}\right)+{\mathrm{\Omega }}_h=B_h \tag{28}\] \[\label{GrindEQ__29_} E\left({\sigma }^2_{X1h}\right)\mathrm{=}{\theta }^{\mathrm{'}}_hS^{\mathrm{2}}_{Xh}\mathrm{=}C_h~ \tag{29}\] \[\label{GrindEQ__30_} \mathrm{E}\left({\sigma }_{X1h}{\sigma }_{Zh}\right)\mathrm{=}{\theta }^{\mathrm{'}}_hS_{ZXh}=D_h \tag{30}\] \[\label{GrindEQ__31_} E\left({\sigma }_{Xh}{\sigma }_{Zh}\right)\mathrm{=}{\theta }_hS_{ZXh}=E_h \tag{31}\] \[\label{GrindEQ__32_} E({\sigma }^2_{Ph}\mathrm{)=}{\theta }_hS^{\mathrm{2}}_{ph}=F_h \tag{32}\] \[\label{GrindEQ__33_} E({\sigma }^2_{P1h})\mathrm{=}{\theta }^{\mathrm{'}}_hS^{\mathrm{2}}_{Ph}=G_h \tag{33}\] \[\label{GrindEQ__34_} E\left({\sigma }_{Ph}{\sigma }_{Zh}\right)\mathrm{=}{\theta }_hS_{Zph}=H_h \tag{34}\] \[\label{GrindEQ__35_} \mathrm{E}\left({\sigma }_{P1h}{\sigma }_{Zh}\right)\mathrm{=}{\theta }^{\mathrm{'}}_hS_{Zph}=I_h \tag{35}\] \[\label{GrindEQ__36_} \mathrm{E}\left({\sigma }_{Xh}{\sigma }_{Ph}\right)\mathrm{=}{\theta }_hS_{Xph}=J_h \tag{36}\] \[\label{GrindEQ__37_} E\left({\sigma }_{P1h}{\sigma }_{Xh}\right)\mathrm{=E}\left({\sigma }_{X1h}{\sigma }_{Ph}\right)\mathrm{=E}\left({\sigma }_{X1h}{\sigma }_{P1h}\right)\mathrm{=}{\theta }^{\mathrm{'}}_hS_{XPh}=L_h \tag{37}\] , where \({\theta }'_h=\left(\frac{1}{n'_h}-\frac{1}{N_h}\right),\ {\theta }_h=\left(\frac{1}{n_h}-\frac{1}{N_h}\right)\mathrm{,\ }{\theta }^*_h=\frac{W_h\left(k_{2h}-1\right)}{n_h}\ \mathrm{and\ }W_h=\frac{N_h}{N}\ \ \)

Substituting equations (21)-(25) in (20) and simplifying while ignoring terms of order greater than two to obtain the approximation for the bias as \[\label{GrindEQ__38_} Bias({\overline{Y}}_g)\cong \sum^L_{h=1}{\frac{W_h{\lambda }_h}{2}}\left[{\frac{3}{4}\lambda }_h{\overline{Z}}_h~\left(A_h-C_h\right)+{\alpha }_h\left(A_h-C_h\right)-~\left(E_h-D_h\right)+{\beta }_h\left(J_h-L_h\right)\right] \tag{38}\] \[where\ {\lambda }_h=\frac{a_h}{a_h{\overline{X}}_h+b_h}\] The approximation for the MSE is given as \[\label{GrindEQ__39_} MSE({\overline{Y}}_g)\cong \sum^L_{h=1}{W^2_h}\left[B_h+{\vartheta }_{1h}+{\alpha }^2_h{\vartheta }_{2h}+{\beta }^2_h{\vartheta }_{3h}+{\beta }_h{\vartheta }_{4h}+{\alpha }_h{\vartheta }_{5h}+2{\alpha }_h{\beta }_h{\vartheta }_{6h}\right] \tag{39}\] , where \({\vartheta }_{1h}=\frac{1}{4}{\lambda }^2_h{\overline{Y}}^2_h\left(A_h-C_h\right)-{\lambda }_h{\overline{Y}}_h\left(E_h-D_h\right)\) \[{\vartheta }_{2h}=\left(A_h-C_h\right),\] \[{\vartheta }_{3h}=\left(F_h-G_h\right),\] \[{\vartheta }_{4h}={\overline{Y}}_h{\lambda }_h\left(J_h-L_h\right)-2\left(H_h-I_h\right)\] \[{\vartheta }_{5h}={\overline{Y}}_h{\lambda }_h\left(A_h-C_h\right)-2\left(E_h-D_h\right)\] \[{\vartheta }_{6h}=\left(J_h-L_h\right)\] The optimum values for \({\alpha }_h\) and\({\ \beta }_h\) are given as \[\label{GrindEQ__40_} {\alpha }^{(opt)}_h=\frac{{\mathrm{\vartheta }}_{4h}{\mathrm{\vartheta }}_{6h}-{\mathrm{\vartheta }}_{5h}{\mathrm{\vartheta }}_{3h}}{2\left({\mathrm{\vartheta }}_{2h}{\mathrm{\vartheta }}_{3h}-{\mathrm{\vartheta }}^2_{6h}\right)} \tag{40}\] , and \[\label{GrindEQ__41_} {\beta }^{(opt)}_h=\frac{{\mathrm{\vartheta }}_{5h}{\mathrm{\vartheta }}_{6h}-{\mathrm{\vartheta }}_{4h}{\mathrm{\vartheta }}_{2h}}{2\left({\mathrm{\vartheta }}_{2h}{\mathrm{\vartheta }}_{3h}-{\mathrm{\vartheta }}^2_{6h}\right)} \tag{41}\] Substitute equations (40) and (41) in (39) to obtain the minimum MSE as \[\label{GrindEQ__42_} {MSE({\overline{Y}}_g)}_{min}\cong \sum^L_{h=1}{W^2_h}\left[{B_h+\mathrm{\vartheta }}_{1h}-\frac{{\mathrm{\vartheta }}^2_{4h}}{4{\mathrm{\vartheta }}_{3h}}-\frac{{\left({\mathrm{\vartheta }}_{5h}{\mathrm{\vartheta }}_{3h}-{\mathrm{\vartheta }}_{4h}{\mathrm{\vartheta }}_{6h}\right)}^2}{4{\mathrm{\vartheta }}_{3h}\left({\mathrm{\vartheta }}_{2h}{\mathrm{\vartheta }}_{3h}-{\mathrm{\vartheta }}^2_{6h}\right)}\right] \tag{42}\] Table 1 shows some special cases of the proposed generalized class of estimators.

The expression for the biases and mean square errors for the members of the proposed generalized class are obtained by substituting the values of \({\alpha }_h\ and\ {\beta }_h\) in equations (38) and (42) respectively.

3. Efficiency Comparison

The proposed estimators performs better than other estimators when the following conditions are satisfied

1. From equations (3) and (42), \(MSE{\left({\overline{Y}}_g\right)}_{min}<Var\left({\overline{Y}}_0\right)\) if \[\label{GrindEQ__43_} \left[{\vartheta }_{1h}-\frac{{\vartheta }^2_{4h}}{4{\vartheta }_{3h}}-\frac{{\left({\vartheta }_{5h}{\vartheta }_{3h}-{\vartheta }_{4h}{\vartheta }_{6h}\right)}^2}{4{\vartheta }_{3h}\left({\vartheta }_{2h}{\vartheta }_{3h}-{\vartheta }^2_{6h}\right)}-B_h\right]<0 \tag{43}\]

2. From equations (6) and (42), \(MSE{\left({\overline{Y}}_g\right)}_{min}<MSE\left({\overline{Y}}_R\right)\) if \[\label{GrindEQ__44_} \left[{\vartheta }_{1h}-\frac{{\vartheta }^2_{4h}}{4{\vartheta }_{3h}}-\frac{{\left({\vartheta }_{5h}{\vartheta }_{3h}-{\vartheta }_{4h}{\vartheta }_{6h}\right)}^2}{4{\vartheta }_{3h}\left({\vartheta }_{2h}{\vartheta }_{3h}-{\vartheta }^2_{6h}\right)}-B_h-R^2_h\left(A_h-C_h\right)+2R_h\left(E_h-D_h\right)\ \right]<0 \tag{44}\]

3. From equations (9) and (42), \(MSE{\left({\overline{Y}}_g\right)}_{min}-MSE\left({\overline{Y}}_{ER}\right)<0\) if \[\label{GrindEQ__45_} \left[{\vartheta }_{1h}-\frac{{\vartheta }^2_{4h}}{4{\vartheta }_{3h}}-\frac{{\left({\vartheta }_{5h}{\vartheta }_{3h}-{\vartheta }_{4h}{\vartheta }_{6h}\right)}^2}{4{\vartheta }_{3h}\left({\vartheta }_{2h}{\vartheta }_{3h}-{\vartheta }^2_{6h}\right)}-B_h-{\frac{1}{4}R}^2_h\left(A_h-C_h\right)-R_h\left(E_h-D_h\right)\ \right]<0 \tag{45}\]

4. Empirical study

The efficiency of the proposed estimator is compared to adapted estimators in a numerical study. The real data for numerical analysis is COVID-19 obtained from www.worldometer.com and Rosner [28]. For data simulation and coding, the R programming language is used. Each population unit is subjected to measurement errors, which are normally distributed with mean 2 and variance 5. Using the least variance and percent relative efficiency (PRE) methods, the efficiency of the proposed estimator is compared to adapted estimators. The percent relative efficiency (PRE) of estimators of population mean is calculated using the formula; \[\label{GrindEQ__46_} PRE({\overline{Y}}_g)=\frac{var({\overline{Y}}_0)}{MSE({\overline{Y}}_g)}\times 100 \tag{46}\] , where\(\ g={\overline{Y}}_R,\ {\overline{Y}}_{ER},\ 1,\ 2,\ \dots ,\ 11\). An estimator with the highest value of PRE is considered the most efficient than others. The values of PREs are obtained at 20% and 80% sensitivity levels of the survey question. Also, the PREs are obtained at 20% and 30% non-response rates.

4.1. COVID-19 data (www.worldometer.com )

The data consist of six strata: the African Region (\(N_1\)=31200), the American region (\(N_2\)=34944), the Eastern Mediterranean Region (\(N_3\)=13728), the European Region (\(N_4\)=38688), the South-East Asia Region (\(N_5\)=6864), and the Western Pacific Region (\(N_6\)=21840). X is the number of new cases, Y is the number of deaths recorded in a given day, and P is the number of deaths less than one in a given day. Scrambled responses are generally distributed with mean 0 and variance 2 generated for each value of Y. Table 2 shows a summary of statistics for the responding units and Table 3 for the non-responding units.

Tables 4 and 5 represent the values of PREs of estimators of population mean in the cases without and with measurement errors, respectively. From the tables, the values of PREs decrease with an increase in inverse sampling rates and non-response rates. Additionally, the values of PREs decrease in the presence of non-response and measurement errors simultaneously. Also, the values of PREs decrease with an increase in the sensitivity levels of the survey question. The proposed estimator \({\overline{Y}}_6\) has the highest PRE compared to all other estimators in this study. Generally, the proposed estimators perform better than the adapted estimator.

4.2. Rosner [28] data

The data consist of two strata of sizes\(\ N_1=480\ and\ N_2=174\), with Y as forced expiratory volume, X as age (in years), and gender as an auxiliary attribute. The scrambling variable is taken to be smoking (Yes=1, No=0) and is used in generation of the response variable. The study variable, auxiliary attribute, and variable all have a positive bi-serial correlation. Tables 6 and 7 represents the population statistics for different data sets used in this study

Tables 8 and 9 shows summary results for the PREs in the cases for without and with measurement errors respectively at different sensitivity levels. From the tables, the values of PREs decrease with an increase in inverse sampling rates and non-response rates. Additionally, the values of PREs decline in the presence of non-response and measurement errors simultaneously. For example, at 20% non-response,\(\ {\mathrm{k}}_{\mathrm{2h}}\mathrm{=2}\), and \({\mathrm{\psi up }}_{\mathrm{h}}\mathrm{=0.2\ }\) the value of PRE for \({\overline{\mathrm{Y}}}_{\mathrm{10}}\) is 114.2515 in the case for without measurement errors and decreases to 100.2539 in the presence of non-response and measurement errors simultaneously.

Furthermore, the values of PREs decreased with an increase in sensitivity level in the case for without measurement errors. The proposed estimators perform better than other adapted estimators in both cases for without and with measurement errors.

5. Conclusion

This study addresses the challenge of estimating the finite population mean in the presence of non-response and measurement errors simultaneously on a sensitive study variable. A general class of estimators is proposed using auxiliary attributes and variables. Up to the first degree of approximation, the bias and mean squared error (MSE) for the suggested estimator are appropriately computed. The proposed estimator outperforms the adapted ordinary estimator, ratio estimator, and exponential ratio-type estimator in numerical tests. Furthermore, when the non-response rate and inverse sampling rate grow, so do the mean squared errors of the proposed estimators. Finally, when non-response and measurement errors are present simultaneously, the efficiency of estimators of population mean decreases.