Improved estimator for the estimation of sensitive variable using ORRT models

Abstract

In this study, we concern with the improved estimation of sensitive variable when there is non-response and measurement error on sensitive variable but the auxiliary variable is non sensitive in nature. For the purpose, we propose an improved estimator in the presence of non-response and measurement error using Optional randomized response technique (ORRT) under simple random sampling without replacement (SRSWOR). The properties of the estimator have been studied and the efficiency conditions are obtained in comparison to the mean estimator, ratio estimator and Zhang’s estimator. Simulation study based on hypothetical populations has been carried out to demonstrate the performance of the proposed estimator at its optimum among others. It has been observed that the proposed estimator is more efficient than other considered estimators in term of having higher Percent Relative Efficiency (PRE).

KEYWORDS:

• Sensitive variable
• auxiliary variable
• non-response
• measurement error
• ORRT

1 Introduction

Survey researchers find difficult to obtain efficient parameters due to the presence of non-response and measurement errors. If the variable of interest is sensitive in nature, one may find it more difficult to collect data from the respondents. Face-to-face interview method is more reliable to collect data but the cost associated to this method is higher in comparison to other methods. Hansen and Hurwitz (1946) suggested a procedure of taking a sub-sample from non-respondents after the first call and collecting information by personal interviews. If the variable of interest is sensitive in nature, then the respondent may not provide honest answers in face-to-face interview. To reduce the bias caused by sensitive questions, one could use Randomized response technique (RRT) models. Diana et al. (2014), Ahmed et al. (2017) and Makhdum et al. (2020) proposed estimators for a sensitive variable in the presence of non-response using RRT models. According to Collins et al. (2001), the use of auxiliary variables when combined with the variable under study help to achieve more efficient estimators. Gupta et al. (2020) studied the estimation of variance of a sensitive study variable using a highly correlated but non-sensitive auxiliary variable.

Another important cornerstone of non-sampling error is measurement error. Kumar et al. (2011), Khalil et al. (2021), and Singh et al. (2019) proposed different estimators for estimating population parameters in presence of measurement error. Azeem (2014), Kumar et al. (2015), Kumar et al. (2018), Kumar (2016), Singh and Sharma (2015), Audu et al. (2020), Singh and Vishwakarma (2019), Kumar and Chowdhary (2021), studied the problem of mean estimation in the presence of non-response and measurement error simultaneously.

Further, Khalil et al. (2018) pioneered the estimation procedure for a sensitive variable in the presence of measurement error by using optional and non-optional RRT models.

On the basis of previous studies, a researcher may think about estimating the population mean of a sensitive variable in the presence of both measurement error and non-response. This issue has received little consideration in the existing literature. The RRT models utilized in earlier studies (Ahmed et al. 2017; Diana et al. 2014; Makhdum et al. 2020) are non-optional RRT models in which all respondents need to give a scrambled response. A survey question, on the other hand, may be sensitive for one person but not for another. According to Gupta et al. (2002), if we give respondents the option of answering the sensitive question directly or providing a scrambled response, the model will be more efficient while causing no further loss of privacy Gupta et al. (2018).

2 Sampling procedure

Let $U(=U_1{U}_2\dots U_N)$ be a finite population of size N units and a sample of size n is taken from U by using simple random sampling without replacement (SRSWOR). Let Y be a sensitive study variable which cannot be observed directly and $X$ be a non sensitive auxiliary variable correlated with Y, both having unknown mean and variance i.e.,

$({\mu }_{x,}{\mu }_y)$ and (${\sigma }_{x,}^2{\sigma }_y^2)$, respectively. Suppose T and S be two scrambling variable(s) with mean $({\mu }_T=1,{\mu }_S=0)$ and variance (${\sigma }_{T,}^2{\sigma }_S^2)$, respectively. Let W be the probability that respondent find the question sensitive. If the respondents consider the question sensitive then he/she is asked to report a scrambled response and else a correct response is reported/recorded.

Further, to collect sensitive information from the respondents, the researchers find difficulty due to the occurrence of non-response. If the variable of interest is sensitive in nature, then to tackle with non-response, Hansen and Hurwitz (1946) technique has been modified by Zhang et al. (2021), Kumar and Kour (2022). In this technique, the respondent gives direct answer in first phase then ORRT model is used to get answer from a sub-group of non-respondents in the second phase.

Therefore, ORRT model in the second phase is (1) $$Z=\{\begin{array}{c}Y \text{with probability}(1-W)\hfill \\ \mathit{\text{TY}}+S \text{with probability} W\hfill \end{array}$$(1) with mean $E(Z)=E(Y)$ and variance $\mathit{\text{Var}}(Z)={\sigma }_y^2+{\sigma }_S^{2}W+{\sigma }_T^2({\sigma }_y^2+{\mu }_y^2)W$. The RRT model is (2) $$Z=(\mathit{\text{TY}}+S)J+Y(1-J)$$(2) where $J\sim \text{Bernoulli}(W)$ with $E(J)=W$ and $\mathit{\text{Var}}(J)=W(1-W)$ When W = 1, then the randomized response becomes non-optional. So, with W = 1 the mean and variance of Z is (3) $$E_R(Z)=({\mu }_{T}W+1-W)Y+{\mu }_{S}W$$(3) (4) $$V_R(Z)=\left(Y^2{\sigma }_T^2+{\sigma }_S^2\right)W$$(4)

Let us take a transformation of the randomized response be ${\widehat{y}}_i$ whose expectation under the randomization mechanism is the true response y_i and is given as (5) $${\widehat{y}}_i=\frac{Z_i-{\mu }_S}{{\mu }_{T}W+1-W}$$(5) with $E({\widehat{y}}_i)=y_i$ and $\mathit{\text{Var}}({\widehat{y}}_i)=\frac{(y_i^2{\sigma }_T^2+{\sigma }_S^2)W}{{({\mu }_{T}W+1-W)}^2}$.

Based on above discussions, we assume that only n₁ units provide response on first call and remaining $n_2=(n-n_1)$ units do not respond. Then a sub-sample of $n_s(=\frac{n_2}{f}(f>1))$ units are taken from non-responding units n₂ respectively. A modified version of Hansen and Hurwitz estimator is given by (6) $$\widehat{\overline{y}}=w_1{\overline{y}}_1+w_2{\widehat{\overline{y}}}_2,$$(6) where ${\overline{y}}_1$ is the mean of respondents in first phase and ${\widehat{\overline{y}}}_2={\sum }_{i=1}^{n_s}\left(\frac{{\widehat{y}}_i}{n_s}\right)$ is the mean of sub-sampled units in the second phase. Also $w_1=\frac{n_1}{n}$ and $w_2=\frac{n_2}{n}$.

The mean and variance of $\widehat{\overline{y}}$ is $$\begin{array}{c}E\left(\widehat{\overline{y}}\right)=\overline{Y}\text{and} \mathit{\text{Var}}\left(\widehat{\overline{y}}\right)=\theta {\sigma }_y^2+\lambda {\sigma }_{y_{(2)}}^2+G\\ \text{where} \theta =\left(\frac{N-n}{\mathit{\text{Nn}}}\right),\lambda =\frac{W_2\left(f-1\right)}{n},\\ G=\frac{W_{2}f}{n}\left[\frac{\left[\left({\sigma }_{y_{(2)}}^2+{\mu }_{y_{(2)}}^2\right){\sigma }_T^2+{\sigma }_S^2\right]W}{{\left({\mu }_{T}W+1-W\right)}^2}\right],\text{and}\\ W_2=\frac{N_2}{N}.\end{array}$$

Moreover, let $(x_i,y_i,z_i)$ be the observed values and $(X_i{Y}_i{Z}_i)$ be the true values of the variables $\mathit{\text{XY}}$ and Z, respectively. Let u be the measurement error (ME) on Y, v be the measurement error on X and p be the measurement error on Z, respectively. The ME’s on i^th observed unit are u_i = y_i – Y _i, v_i = x_i – X_i and p_i = z_i – Z_i and assumed to be uncorrelated with mean zero and variance ${\sigma }_u^2{\sigma }_v^2$ and ${\sigma }_p^2$, respectively.

In the presence of non-response and ME, the variance of ${\widehat{\overline{y}}}^{\ast }$ is given by $$\mathit{\text{Var}}\left({\widehat{\overline{y}}}^{\ast }\right)=\theta \left({\sigma }_y^2+{\sigma }_u^2\right)+\lambda \left({\sigma }_{y_{(2)}}^2+{\sigma }_p^2\right)+G.$$

3 Existing mean estimator

By using basic terminologies, as used in Section 2, suppose that population mean and variance of the auxiliary variable X are known and is denoted by ${\mu }_x=\frac{1}{N}{\sum }_{i=1}^N{x}_i$ and ${\sigma }_x^2=\frac{1}{N-1}{\sum }_{i=1}^N{(x_i-{\mu }_x)}^2$, respectively.

Let the population mean and variance of the respondent group of size N₁ is given by

${\mu }_{x_{(1)}}=\frac{1}{N_1}{\sum }_{i\text{=1}}^{N_1}{x}_i$ and ${\sigma }_{x_{(1)}}^2=\frac{1}{N_1-1}{\sum }_{i\text{=1}}^{N_1}{(x_i-{\mu }_{x_{(1)}})}^2$, respectively and the population mean and variance of non-respondent group of size N₂is given by ${\mu }_{x_{(2)}}=\frac{1}{N_2}{\sum }_{i\text{=1}}^{N_2}{x}_i$ and ${\sigma }_{x_{(2)}}^2=\frac{1}{N_2-1}{\sum }_{i\text{=1}}^{N_2}{(x_i-{\mu }_{x_{(2)}})}^2$, respectively. Further, ${\rho }_{\mathit{\text{xy}}}=\frac{{\sigma }_{\mathit{\text{xy}}}}{{\sigma }_x{\sigma }_y}$ be the correlation coefficient between the auxiliary variable X and sensitive variable Y

Similarly, let ${\rho }_{x{y}_{(1)}}=\frac{{\sigma }_{x{y}_{(1)}}}{{\sigma }_x{\sigma }_y}$ and ${\rho }_{x{y}_{(2)}}=\frac{{\rho }_{x{y}_{(2)}}}{{\sigma }_x{\sigma }_y}$ be the correlation coefficient between auxiliary variable X and the sensitive study variable $Y$ for the respondent group and the non-respondents group, respectively.

Assuming that the population mean μ_x of X is known and non-response happened on both Y and X. Some of the existing mean estimators of ORRT model are listed below:

1. A typical mean estimator for sensitive variable in finite population under modified Hansen and Hurwitz (HH) estimator in presence of measurement error is

   ${\widehat{\mu }}_{\mathit{\text{HH}}}={\widehat{\overline{y}}}^{\ast }=w_1{\overline{y}}_1+w_2{\overline{y}}_2^{\ast },$ (7)

   where ${\overline{y}}_2^{\ast }=\frac{1}{n_s}{\sum }_{i\text{=1}}^{n_s}{z}_i$.

   In presence of measurement error, the MSE of ${\widehat{\mu }}_{\mathit{\text{HH}}}$ is given by

   $\mathit{\text{MSE}}({\widehat{\mu }}_{\mathit{\text{HH}}})=\theta ({\sigma }_y^2+{\sigma }_u^2)+\lambda ({\sigma }_{y_{(2)}}^2+{\sigma }_p^2)+G$ (8)

2. A ratio estimator corresponding to Gupta et al. (2014) estimator under modified HH in presence of measurement error is given by

   ${\widehat{\mu }}_R=\frac{{\widehat{\overline{y}}}^{\ast }}{{\overline{x}}^{\ast }}{\mu }_x={\widehat{R}}_W^{\ast }{\mu }_x,$ (9)

   where ${\widehat{\overline{y}}}^{\ast }$ and ${\overline{x}}^{\ast }$ is the ordinary mean estimator under original HH procedure.

   The MSE of ${\widehat{\mu }}_R$ is given by

   $\begin{array}{c}\mathit{\text{MSE}}({\widehat{\mu }}_R)=\theta ({\sigma }_y^2+R^2{\sigma }_x^2-2R{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x)\\ +\lambda ({\sigma }_{y_{(2)}}^2+R^2{\sigma }_{x_{(2)}}^2-2R{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}})\\ +\theta ({\sigma }_u^2+R^2{\sigma }_v^2)+\lambda ({\sigma }_p^2+R^2{\sigma }_v^2)+G\end{array}$ (10)

   where $R=\frac{{\mu }_y}{{\mu }_x}$ and ${\rho }_{z{x}_{(2)}}=\frac{{\rho }_{y{x}_{(2)}}}{\sqrt{(1+\frac{\left[{\sigma }_S^2+{\sigma }_T^2({\sigma }_{y_{(2)}}^2+{\mu }_{y_{(2)}}^2)\right]W}{{\sigma }_{y_{(2)}}^2})}}$.

3. The generalized mean estimator considered in Zhang et al. (2021) but with non-response and measurement error is given as

   ${\widehat{\mu }}_{\mathit{\text{pw}}}=\left[{\widehat{\overline{y}}}^{\ast }+k({\mu }_x-{\overline{x}}^{\ast })\right]{(\frac{\overline{D}}{\overline{d}})}^{\nu },$ (11)

   where $\overline{d}=[\varphi (\alpha {\overline{x}}^{\ast }+\beta )+(1-\varphi )(\alpha {\mu }_x+\beta )],\overline{D}=\alpha {\mu }_x+\beta ,{\overline{x}}^{\ast }={\mu }_x(1+e_1^{\ast })$ and

   ${\overline{y}}^{\ast }={\mu }_y(1+e_0^{\ast }).$ Also, k and ν are suitable chosen constants, $\varphi$ is assumed to be an unknown constant whose value is to be determined from optimality consideration, α and β are assumed to be some known parameters of the auxiliary variable X.

The MSE of ${\widehat{\mu }}_{\mathit{\text{pw}}}$ is given by (12) $$\begin{array}{c}\mathit{\text{MSE}}\left({\widehat{\mu }}_{\mathit{\text{pw}}}\right)=\theta \left({\sigma }_y^2+P^2{\sigma }_x^2-2P{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x\right)\\ +\lambda \left({\sigma }_{y_{(2)}}^2+P^2{\sigma }_{x_{(2)}}^2-2P{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}\right)\\ +\theta \left({\sigma }_u^2+P^2{\sigma }_v^2\right)+\lambda \left({\sigma }_p^2+P^2{\sigma }_v^2\right)+G\end{array}$$(12) where $P=\frac{\theta {\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x+\lambda {\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}}{\theta ({\sigma }_x^2+{\sigma }_v^2)+\lambda ({\sigma }_{x_{(2)}}+{\sigma }_v^2)}$.

Therefore, the MSE of ${\widehat{\mu }}_{\mathit{\text{HH}}},{\widehat{\mu }}_R$, and ${\widehat{\mu }}_{\mathit{\text{pw}}}$ without measurement error may be obtained by puting ${\sigma }_v^2={\sigma }_u^2={\sigma }_p^2=0$.

4 Proposed mean estimator

Taking the motivation from Zhang et al. (2021), we propose a generalized mean estimator using ORRT models in the presence of non-response and measurement error simultaneously as (13) $${\widehat{\tau }}_{\mathit{\text{cs}}}=\left\{{\widehat{\overline{y}}}^{\ast }+k_1\left({\mu }_x-\overline{x}\right)+k_2\left({\mu }_x-{\overline{x}}^{\ast }\right)\right\}{\left(\frac{{\mu }_x}{{\overline{x}}^{\ast }}\right)}^{{\alpha }_1}{\left(\frac{{\mu }_x}{\overline{x}}\right)}^{{\alpha }_2},$$(13) where ${\widehat{\overline{y}}}^{\ast }$ denotes the mean of the sensitive study variable in the presence of non-response and measurement error, ${\overline{x}}^{\ast }$ is the mean of the auxiliary variable in the presence of non-response and measurement error, $\overline{x}$ is the mean of auxiliary variable, α₁ and α₂ are suitable chosen constants and k₁ and $k_2$ are assumed to be unknown constants whose values are to be optimize.

To find the MSE of the estimator, we define $${\overline{y}}^{\ast }={\mu }_y\left(1+e_0^{\ast }\right),{\overline{x}}^{\ast }={\mu }_x\left(1+e_1^{\ast }\right)\text{and}\overline{x}={\mu }_x\left(1+e_1\right)$$such that $E(e_0^{\ast })=E(e_1^{\ast })=E(e_1)\text{=0}$. $$\begin{array}{c}E\left(e_0^{{\ast }^2}\right)=\frac{1}{{\mu }_y^2}\left[\theta \left({\sigma }_y^2+{\sigma }_u^2\right)+\lambda \left({\sigma }_{y_{(2)}}^2+{\sigma }_p^2\right)\right]\\ +\frac{{}_{2}f}{n}\left[\frac{\left[\left({\sigma }_{y_{(2)}}^2+{\mu }_{y_{(2)}}^2\right){\sigma }_T^2+{\sigma }_S^2\right]W}{{\left({\mu }_{T}W+1-W\right)}^2}\right],\\ E\left(e_1^{{\ast }^2}\right)=\frac{1}{{\mu }_x^2}\left[\theta \left({\sigma }_x^2+{\sigma }_v^2\right)+\lambda \left({\sigma }_{x_{\left(2\right)}}^2+{\sigma }_v^2\right)\right],\\ E\left(e_1^2\right)=\frac{1}{{\mu }_x^2}\left[\theta \left({\sigma }_x^2\right)\right],E\left(e_0^{\ast }{e}_1^{\ast }\right)=\theta {\rho }_{\mathit{\text{yx}}}\frac{{\sigma }_y{\sigma }_x}{{\mu }_y{\mu }_x}+\lambda {\rho }_{z{x}_{(2)}}\frac{{\sigma }_z{\sigma }_{x_{\left(2\right)}}}{{\mu }_z{\mu }_x},\\ E\left(e_0^{\ast }{e}_1\right)=\frac{1}{{\mu }_y{\mu }_x}\theta {\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x,E\left(e_1^{\ast }{e}_1\right)=\frac{1}{{\mu }_x^2}\theta {\sigma }_x^2.\end{array}$$ where ${\rho }_{z{x}_{(2)}}=\frac{{\rho }_{y{x}_{(2)}}}{\sqrt{(1+\frac{\left[{\sigma }_S^2+{\sigma }_T^2({\sigma }_{y_{(2)}}^2+{\mu }_{y_{(2)}}^2)\right]W}{{\sigma }_{y_{(2)}}^2})}}$.

The bias of a proposed estimator up to the second order of approximation is given by (14) $$\begin{array}{c}\mathit{\text{Bias}}\left({\widehat{\tau }}_{\mathit{\text{cs}}}\right)\\ \hspace{1em}=\theta \left\{\frac{\left(A_1+A_2+A_3\right)}{{\mu }_x^2}{\sigma }_x^2+\frac{A_2}{{\mu }_x^2}{\sigma }_v^2-{\rho }_{\mathit{\text{yx}}}\frac{{\sigma }_y{\sigma }_x}{{\mu }_x}\left({\alpha }_1+{\alpha }_2\right)\right\}\\ \hspace{1em}+\lambda \left\{\frac{A_2}{{\mu }_x^2}\left({\sigma }_{x_{\left(2\right)}}^2+{\sigma }_v^2\right)-\frac{{\alpha }_1}{{\mu }_x}{\rho }_{z{x}_{\left(2\right)}}{\sigma }_z{\sigma }_{x_{(2)}}\right\},\\ \text{where}{A}_1=\left(R\frac{{\alpha }_2\left({\alpha }_2\text{+1}\right)}{2}+k_1{\alpha }_2\right){\mu }_x,A_2\\ =\left(R\frac{{\alpha }_1\left({\alpha }_1\text{+1}\right)}{2}+k_2{\alpha }_1\right){\mu }_x,\\ A_3=\left(k_1{\alpha }_1+k_2{\alpha }_2+R{\alpha }_1{\alpha }_2\right){\mu }_x,R^{\prime }=\frac{{\mu }_y}{{\mu }_z}\text{and}R=\frac{{\mu }_y}{{\mu }_x}.\end{array}$$(14)

Without measurement error, the bias of ${\widehat{\tau }}_{\mathit{\text{cs}}}$ can be obtained by taking ${\sigma }_v^2=0$ in the above equation.

The MSE of the proposed estimator is (15) $$\begin{array}{c}\mathit{\text{MSE}}\left({\widehat{\tau }}_{\mathit{\text{cs}}}\right)=\theta \{{\left[\left(k_1+{\alpha }_{2}R\right)+\left(k_2+{\alpha }_{1}R\right)\right]}^2{\sigma }_x^2+{\left(k_2+{\alpha }_{1}R\right)}^2\\ {\sigma }_v^2-2{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x\left[\left(k_1+k_2\right)+R\left({\alpha }_1+{\alpha }_2\right)\right]\\ +\left({\sigma }_y^2+{\sigma }_u^2\right)\}\\ +\lambda \{{\left(k_2+{\alpha }_{1}R\right)}^2\left({\sigma }_{x_{(2)}}^2+{\sigma }_v^2\right)\\ -2R^{\prime }\left(k_2+{\alpha }_{1}R\right){\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}+\left({\sigma }_{y_{(2)}}^2+{\sigma }_p^2\right)\}+G\end{array}$$(15)

Differentiate (15) with respect to k₁ and k₂ we get the optimum values of k₁ and $k_2$ as (16) $$k_{1_{\left(\mathit{\text{opt}}\right)}}=D-{\alpha }_{2}R\text{and}{k}_{2_{\left(\mathit{\text{opt}}\right)}}=J-{\alpha }_{1}R,$$(16) $$\begin{array}{c}\text{where} D=\frac{1}{\theta {\sigma }_v^2+\lambda \left({\sigma }_{x_{(2)}}^2+{\sigma }_v^2\right)}\{{\rho }_{\mathit{\text{yx}}}\frac{{\sigma }_y}{{\sigma }_x}[\theta \left({\sigma }_x^2+{\sigma }_v^2\right)\\ +\lambda \left({\sigma }_{x_{(2)}}^2+{\sigma }_v^2\right)]-\left[\theta {\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x+R^{\prime }\lambda {\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}\right]\}\\ \text{and} J=\frac{R^{\prime }\lambda {\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}}{\theta {\sigma }_v^2+\lambda \left({\sigma }_{x_{(2)}}^2+{\sigma }_v^2\right)}.\end{array}$$

Substitute the values of k₁ and $k_2$ from (16) in (15) we get the minimum MSE as (17) $$\begin{array}{c}{\mathit{\text{MSE}}}_{\mathrm{\text{min}}}\left({\widehat{\tau }}_{\mathit{\text{cs}}}\right)=\theta \{{\left(D+J\right)}^2{\sigma }_x^2+J^2{\sigma }_v^2-2{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x\left(D+J\right)\\ +\left({\sigma }_y^2+{\sigma }_u^2\right)\}+\lambda \{J^2\left({\sigma }_{x_{(2)}}^2+{\sigma }_v^2\right)\\ -2J{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}+\left({\sigma }_{y_{(2)}}^2+{\sigma }_p^2\right)\}+G\end{array}$$(17)

The expression for the minimized MSE of the proposed estimator without ME may be obtained by putting ${\sigma }_v^2={\sigma }_u^2={\sigma }_p^2=0$ in the above expression, we get (18) $$\begin{array}{c}{\mathit{\text{MSE}}}_{\mathrm{\text{min}}}\left({\widehat{\tau }}_{\mathit{\text{cs}}}\right)=\theta \left\{{\left(D^{\ast }+J^{\ast }\right)}^2{\sigma }_x^2-2{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x\left(D^{\ast }+J^{\ast }\right)+{\sigma }_y^2\right\}\\ +\lambda \left\{J^{{\ast }^2}{\sigma }_{x_{(2)}}^2-2J^{\ast }{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}+{\sigma }_{y_{(2)}}^2\right\}+G\\ \text{where} D^{\ast }=\frac{1}{\lambda {\sigma }_{x_{\left(2\right)}}^2}\{{\rho }_{\mathit{\text{yx}}}\frac{{\sigma }_y}{{\sigma }_x}\left[\theta {\sigma }_x^2+\lambda {\sigma }_{x_{(2)}}^2\right]\\ -\left[\theta {\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x+R^{\prime }\lambda {\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{\left(2\right)}}\right]\}\\ \text{and} J^{\ast }=\frac{R^{\prime }\lambda {\rho }_{z{x}_{\left(2\right)}}{\sigma }_z{\sigma }_{x_{\left(2\right)}}}{\lambda {\sigma }_{x_{(2)}}^2}.\end{array}$$(18)

5 Efficiencies comparison

In this section, we compare the MSE of the proposed estimator with respect to the MSE of other existing estimators mentioned in (8, 10, 12), and (17) are given as

1. ${\mathit{\text{MSE}}}_{\mathrm{\text{min}}}({\widehat{\tau }}_{\mathit{\text{cs}}})<\mathit{\text{MSE}}({\widehat{\mu }}_{\mathit{\text{HH}}})$ if

   $\begin{array}{c}\theta \left\{{(D+J)}^2{\sigma }_x^2+J^2{\sigma }_v^2-2{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x(D+J)\right\}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lambda \left\{J^2({\sigma }_{x_{(2)}}^2+{\sigma }_v^2)-2J{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}\right\}<0\end{array}$ (19)

2. ${\mathit{\text{MSE}}}_{\mathrm{\text{min}}}({\widehat{\tau }}_{\mathit{\text{cs}}})<\mathit{\text{MSE}}({\widehat{\mu }}_R)$ if

   $\begin{array}{c}\theta \{\left[{(D+J)}^2-R^2\right]{\sigma }_x^2+(J^2-R^2){\sigma }_v^2\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-2{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x(D+J-R)\}\\ +\lambda \{(J^2-R^2)({\sigma }_{x_{(2)}}^2+{\sigma }_v^2)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-2{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}(J-R)\}<0.\end{array}$ (20)

3. ${\mathit{\text{MSE}}}_{\mathrm{\text{min}}}({\widehat{\tau }}_{\mathit{\text{cs}}})<\mathit{\text{MSE}}({\widehat{\mu }}_{\mathit{\text{pw}}})$ if

   $\begin{array}{c}\theta \{\left[{(D+J)}^2-P^2\right]{\sigma }_x^2+(J^2-P^2){\sigma }_v^2\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-2{\rho }_{\mathit{\text{yx}}}{\sigma }_y{\sigma }_x(D+J-P)\}\\ +\lambda \left\{(J^2-P^2)({\sigma }_{x_{(2)}}^2+{\sigma }_v^2)-2{\rho }_{z{x}_{(2)}}{\sigma }_z{\sigma }_{x_{(2)}}(J-P)\right\}<0.\end{array}$ (21)

If the above conditions (19)–(21) hold true then the proposed estimator is always more efficient than the other considered estimators.

6 Simulation study

In this study, with the help of simulation study, we compare the performance of the proposed estimator under SRSWOR with the usual unbiased estimator and other two considered estimators.

For simulation study, data set consist of sensitive study variable Y and an auxiliary variable X is generated from a normal distribution using the model

$Y=\mathit{\text{aX}}+\mathit{\text{rnorm}}(N,{\mu }_y,{\sigma }_y^2)$ where $X=\mathit{\text{rnorm}}(N,{\mu }_x,{\sigma }_x^2)$, $({\mu }_y,{\mu }_x)\text{=}(0,0)$, $a\text{=0}.25$ and $({\sigma }_y^2,{\sigma }_x^2)$ may varies.

An artificial population of size N(5000) from normal distribution and a sample of size n(850) under SRSWOR is taken. It is assumed that only $n_1(450)$ units provide response and $n_2(400)$ do not respond in the first phase. In the second phase, we take a sub-sample of size $n_s=\frac{n_2}{f}(f>1)$ from the non-respondent n₂ units by using $f=2,3,4,5$, respectively. The simulation study given in Tables 1 and 2.

Table 1 PRE of the proposed estimators with respect to existing estimators for different values of f and W using ORRT models.

W	f	Estimators
		${\widehat{\mu }}_{\mathbf{\text{HH}}}$	${\widehat{\mu }}_{\mathbf{R}}$	${\widehat{\mu }}_{\mathbf{\text{pw}}}$	${\widehat{\tau }}_{\mathbf{\text{cs}}}$
0.2	2	100.000	76.41382	100.0061	100.0119
	3	100.000	75.75772	100.0056	100.0091
	4	100.000	75.37299	100.0054	100.0074
	5	100.000	76.57487	100.0048	100.0062
0.4	2	100.000	84.91888	100.0051	100.0101
	3	100.000	84.50677	100.0047	100.0076
	4	100.000	84.19802	100.0045	100.0062
	5	100.000	85.03562	100.004	100.0051
0.6	2	100.000	92.21496	100.0043	100.0083
	3	100.000	92.04403	100.0039	100.0061
	4	100.000	91.85606	100.0036	100.0048
	5	100.000	92.34802	100.0031	100.0038
0.8	2	100.000	97.32849	100.0034	100.0064
	3	100.000	97.29573	100.0031	100.0042
	4	100.000	97.2229	100.0028	100.003
	5	100.000	97.42204	100.0023	100.0018

Table 2 PRE of the proposed estimators with respect to existing estimators for different values of f and W using ORRT models.

W	f	Estimators
		${\widehat{\mu }}_{\mathbf{\text{HH}}}$	${\widehat{\mu }}_{\mathbf{R}}$	${\widehat{\mu }}_{\mathbf{\text{pw}}}$	${\widehat{\tau }}_{\mathbf{\text{cs}}}$
0.2	2	100.000	85.42602	100.0049	100.0097
	3	100.000	85.01259	100.0046	100.0073
	4	100.000	84.72413	100.0043	100.0059
	5	100.000	85.60064	100.0038	100.0048
0.4	2	100.000	97.52038	100.0032	100.0059
	3	100.000	97.48531	100.0028	100.0039
	4	100.000	97.41648	100.0026	100.0028
	5	100.000	97.62159	100.0021	100.0016
0.6	2	100.000	99.69804	100.0018	100.0007
	3	100.000	99.70426	100.0015	99.99723
	4	100.000	99.69732	100.0013	99.99538
	5	100.000	99.72319	100.0007	99.99238
0.8	2	100.000	95.12491	100.0009	100.0022
	3	100.000	95.3436	100.0006	100.0015
	4	100.000	95.20721	100.0005	100.0012
	5	100.000	95.77384	99.9998	100.0009

Also, the scrambling variable $T$ and S are taken to be normal with mean 1 and 0, respectively and with different variances.

Further, another artificial population is used, we considered by Zhang et al. (2021) for the comparison purpose and to see the performance of the proposed estimator over other considered estimator. We have considered a population of size 5000 generated from a bivariate normal distribution with mean and covariance $(Y,X)$ as mentioned below: $$\begin{array}{c}\mu =\left[\begin{array}{c}10\\ 6\end{array}\right],\hspace{1em}\Sigma \text{=}\left[\begin{array}{cc}16& 9.051\\ 9.051& 8\end{array}\right]\hspace{1em}{\rho }_{\mathit{\text{yx}}}\text{=0}.8\\ {\mu }_x\text{=6}.0228,{\sigma }_x^2\text{=8}.1830,{\mu }_y\text{=9}.9864,{\sigma }_y^2\text{=16}.1215,{\rho }_{\mathit{\text{yx}}}\text{=0}.8024\end{array}$$

Taking sample of size $n=$ 500 using SRSWOR and in the first phase we select a sample of size $n_1(200)$ and $n_2(300)$. We take another sub-sample $(n_s=\frac{n_2}{f}\text{where}(f>1))$ from the non-respondent in the second phase by using $f=2,3,4,5.$ The simulation study based on Zhang et al. (2021) given in Table 3.

Table 3 PRE of the proposed estimators with respect to existing estimators for different values of f and W = 0.8 using ORRT models. Also $({\sigma }_v^2={\sigma }_u^2={\sigma }_p^2=1,5,10)$.

Estimators	f	$({\mathit{\sigma }}_{\mathbf{v}}^{\mathbf{2}}={\mathit{\sigma }}_{\mathbf{u}}^{\mathbf{2}}={\mathit{\sigma }}_{\mathbf{p}}^{\mathbf{2}})$
		1	5	10
${\widehat{\mu }}_{\text{HH}}$	2	100.000	100.000	100.000
	3	100.000	100.000	100.000
	4	100.000	100.000	100.000
	5	100.000	100.000	100.000
${\widehat{\mu }}_{\mathrm{R}}$	2	145.0267	78.80299	57.34765
	3	145.0254	78.80271	57.34753
	4	145.0241	78.80243	57.34742
	5	145.0228	78.80215	57.34730
${\widehat{\mu }}_{\text{pw}}$	2	217.4953	143.8934	121.7771
	3	217.4937	143.8931	121.7769
	4	217.4921	143.8928	121.7768
	5	217.4904	143.8924	121.7767
${\widehat{\tau }}_{\text{cs}}$	2	253.9614	196.6296	165.9375
	3	253.9603	196.6291	165.9372
	4	253.9591	196.6285	165.9369
	5	253.9578	196.6280	165.9366

Coding for simulation was done in R software. The Percent Relative Efficiency (PRE) of the proposed estimator $({\widehat{\tau }}_{\mathit{\text{cs}}})$ with respect to usual unbiased estimator $({\widehat{\mu }}_{\mathit{\text{HH}}})$ and two considered estimators $({\widehat{\mu }}_R,{\widehat{\mu }}_{\mathit{\text{pw}}})$ is defined as $$\mathit{\text{PRE}}=\left(\frac{{\mathit{\text{MSE}}}^{\ast }({\widehat{\mu }}_{\mathit{\text{HH}}})}{{\mathit{\text{MSE}}}^{\ast }({\widehat{\mu }}_i)}\right)\ast 100,$$

where ${\widehat{\mu }}_i={\widehat{\mu }}_{\mathit{\text{HH}}},{\widehat{\mu }}_R,{\widehat{\mu }}_{\mathit{\text{pw}}}$ and ${\widehat{\tau }}_{\mathit{\text{cs}}}$

Also ${\sigma }_T^2={\sigma }_S^2\text{=0}.5$.

Also ${\sigma }_T^2={\sigma }_S^2\text{=1}$.

From Tables 1 and 2, we will compare the performance of the proposed estimator with respect to usual unbiased estimator and two considered estimators. In Table 1, when ${\sigma }_T^2={\sigma }_S^2=0.5$, we see that our proposed estimator decreases with the increase in f and W, where f = 2 to 5 and W = 0.2 to 0.8. But in Table 2, when ${\sigma }_T^2={\sigma }_S^2=1$, we see that our proposed estimator $({\widehat{\tau }}_{\mathit{\text{cs}}})$ performs better than the other considered estimators (${\widehat{\mu }}_{\mathit{\text{HH}}},{\widehat{\mu }}_R,{\widehat{\mu }}_{\mathit{\text{pw}}})$ for different values of $f$ and W except that W = 0.6, in this case PRE of the proposed estimator is less than the considered estimators (${\widehat{\mu }}_{\mathit{\text{HH}}},{\widehat{\mu }}_R,{\widehat{\mu }}_{\mathit{\text{pw}}})$.

It is noted from Table 3, that the proposed estimator (${\widehat{\tau }}_{\mathit{\text{cs}}})$ is more efficient than the usual unbiased estimator $({\widehat{\mu }}_{\mathit{\text{HH}}}),$ Gupta et al. (2014) ratio estimator $({\widehat{\mu }}_R)$ under the setup of Hansen and Hurwitz $({\widehat{\mu }}_R)$ and the generalized estimator of Zhang et al. (2021) (${\widehat{\mu }}_{\mathit{\text{pw}}})$ in terms of having higher PRE. Also, the ratio estimator not performing well at $({\sigma }_v^2={\sigma }_u^2={\sigma }_p^2=5,10)$.

For $W=0.8$, the values of PRE of estimators decreases with the increase in the value of f i.e., $f=2$ to 5.

7 Conclusion

In this paper, we studied the improved mean estimation of sensitive variable by suggested generalized mean estimator using ORRT model. The properties of the proposed estimator have been studied and the conditions are obtained where the proposed estimator is more efficient than the existing estimators. A simulation study is also supporting the theoretical results except the situation when the probability of sensitive question is moderately high (i.e., W = 0.6), under this situation the Zhang et al. (2021) estimator (${\widehat{\mu }}_{\mathit{\text{pw}}}$) is more efficient. Based on the results obtained, we recommend the use of the suggested estimator by the researchers and practitioners in future.

Acknowledgments

The authors express very sincere gratitude to the reviewers for their constructive suggestions which helped improve the presentation of the paper.

Data availability statement

No real data is used in the paper.

Disclosure statement

The authors declares that they have no conflicts of interest.