A two-stage randomized response technique for simultaneous estimation of sensitivity and truthfulness

Abstract

Privacy protection is a critical concern when dealing with sensitive survey questions. Conventional randomized response (RR) models frequently fall short in providing respondents with adequate secrecy when assessing important parameters like the probability of success p and the probability of truthfulness T. This study proposes an improved RR technique that addresses these drawbacks by providing better privacy protections and enabling the simultaneous calculation of T and $\pi$.The advantage of the proposed model is that it applies a two-stage randomization process, which estimates both T and $\pi$ thereby offering enhanced protection for privacy. The proposed method is first initially developed using simple random sampling and builds upon a two-stage RR approach described in previous research. It is then expanded to include stratified random sampling in order to make it more applicable to survey designs that are more intricate. The methodology is derived analytically and evaluated with respect to computing efficiency and algebraic measures. The suggested model increases the overall quality and reliability of the survey data by reducing respondent reluctance and producing more accurate parameter estimations, as shown by efficiency comparisons with current methods. Additionally, without sacrificing the accuracy of the statistical estimates, the approach improves respondent participation. Simulations using different fixed values of n, $\pi$, and T show that the suggested model continuously performs better than traditional techniques in terms of reducing variance and protecting privacy under both simple and stratified sampling. These findings show that it is a useful, statistically sound method for carrying out surveys on delicate subjects, guaranteeing data quality and respondent protection.

Introduction

Accurate gathering of information on sensitive issues including substance misuse, tax evasion, or criminal activity can frequently be compromised by participants reluctance to provide honest answers. Confidentiality issues and social desirability bias are the causes of this problem. In these situations, traditional survey techniques often yield false or deceptive answers. To tackle this issue, Randomized Response Methodology (RRM) have been created as a statistically sound method to preserve respondent anonymity and promote truthful reporting. By minimizing response bias and maintaining respondent confidentiality, Random Response Methodology (RRM) is a statistical technique that aims to obtain true responses to topics that are sensitive. A novel randomized response (RRM) methodology that reduced under reporting by using a randomization device to hide individual responses was introduced in the novel work by¹.

The traditional randomized response model conceals the answer from the interviewer by having respondents respond to either the sensitive question or its complement based on a randomization mechanism². developed a two-stage randomized response (RR) model based on the¹ framework to improve estimation efficiency. In² model, a randomization device $R_1$ is used to interact with each respondent who was chosen via simple random sampling with replacement (SRSWR). The respondent proceeds to a second device $R_2$ with probability $(1 - p)$, after having said directly with probability p if they belong to the sensitive attribute A. With probability $p_0$ and $(1 - p_0)$ for selecting the sensitive and non-sensitive statements, respectively, device $R_2$ operates in the same way as the¹ model.

In this design, the probability of a yes response is as follows:

$$\begin{aligned} \theta _{MS} = p\pi + (1 - p)\left\{ p_0 \pi + (1 - p_0)(1 - \pi )\right\} , \end{aligned}$$

(1)

, where p is the probability of a direct response in the first stage, $\pi$ is the actual proportion of the population that possesses the sensitive attribute, and $p_0$ is the probability of being given the sensitive statement in the second stage. The maximum likelihood estimator of $\pi$, is defined as:

$$\begin{aligned} \hat{\pi }_{MS}=\dfrac{\hat{\theta }_{MS}-\left( 1-p\right) \left( 1-p_{0} \right) }{2p-1+2p_{o}\big (1-p\big ) }, \end{aligned}$$

(2)

where $\hat{\theta }_{MS}$ is the number of yes responses observed in the² RR technique. The variance of $\hat{\pi }_{MS}$, is given by

$$\begin{aligned} V\big (\hat{\pi }_{MS}\big )=\dfrac{\pi \big (1-\pi \big )}{n} + \dfrac{\big (1-p\big )\big (1-p_{o}\big )\big \{1-\big (1-p\big )\big (1-p_{o}\big )\big \}}{n\big \{2p-1+2p_{o}\big (1-p\big ) \big \}^2}. \end{aligned}$$

(3)

In different ways, subsequent studies developed upon the work of¹ and². For example³, suggested design changes meant to increase the reliability of responses in delicate surveys. A modified RR model was presented by⁴, which reduced variance and improved estimation efficiency⁵. improved the method even more by putting forth a substitute that struck a compromise between estimator accuracy and responder comfort. However⁶, suggested a more respondent-friendly design to diminish evasive response inclinations, while⁷created a privacy-protecting method employing unrelated question strategies. To improve respondent secrecy⁸, provided a customized methodology designed for highly stigmatized survey questions Although these developments, a common limitation still exists, many models calculate the percentage of people who have a sensitive trait ($\pi$) without also calculating the respondents degree of cooperation or truthfulness (T). The probability Tis commonly used as a sensitivity metric, but because there is currently no feasible sample analog, its practical application is still unclear. Therefore, to rectify this problem⁹, constructed an alternative survey technique which enables the joint estimation of the unknown parameters $\pi$ and T.

Some of the limitations of existing randomized response (RR) models include their use of simple randomization techniques that are unreliable in real-world sampling situations or their emphasis on estimating only the sensitive proportion ($\pi$). Using the frameworks of² and¹⁰, this study is an attempt to proposed an enhanced two-stage RR model to fill this gap. At each stage, distinct randomization devices are introduced to estimate $\pi$ and the truthfulness parameter T simultaneously. The objective of the study is to determine maximum likelihood estimators (MLE), assess their statistical characteristics under simple and stratified random sampling, and show how effective the model is and how it outperforms other approaches, such as those by^10,11,12,13, and¹⁴.

A brief description of the¹⁰ and¹¹ models is given in Sect. "Literature review of related work". The resulting estimators, along with their properties of the proposed model, and efficiency comparison with respect to the estimators¹⁰ and¹¹, under $\big (SRS\big )$, are presented in Sect. “Methodology”. The estimation of the suggested model and the comparison of efficiency with^12,13, and¹⁴ models in stratified random sampling using an optimum allocation are discussed in Sect. “Discussion”. The conclusion is given in Sect. “Conclusion”.

Literature review of related work

The randomized response (RR) methodology has been developed by a number of scholars in recent years to address current issues with data collection that preserve privacy. The work of¹⁰ and¹¹ proposed randomized response (RR) models aimed at estimating the population proportion of sensitive characteristics alongside the corresponding sensitivity parameters. In¹⁰ introduced a model in which a simple random sample with replacement (SRSWR) of size n is drawn from a finite population of size N. Each selected respondent is asked directly whether they are a member of the sensitive group A. If the response is “no,” the individual is then directed to use a randomization device RD containing two statements: (a) “I belong to group A” and (b) “I do not belong to group A” with probabilities p and (1-p), respectively. It is assumed that respondents in group A respond truthfully under the randomized response RR method, regardless of whether a direct question or the RR technique is used. However, under the standard direct response approach, they provide honest answers with a probability of T. To ensure the respondent’s privacy, interviewees should not reveal to the interviewer the specific question they answered.

The probability of receiving a ’yes’ response in the direct questioning in the mentioned approach is denoted by $\theta _{H1}$.

$$\begin{aligned} pr\big (yes\big )=\theta _{H1} =\pi T, \end{aligned}$$

(4)

and the probability of yes response $\theta _{H2}$ in the RRT under the above model, is

$$\begin{aligned} pr\big (yes\big )=\theta _{H2} = p \pi \big (1-T\big )+\big (1-p\big )\big (1-\pi \big ), \end{aligned}$$

(5)

where, T is the probability of truthful reporting in direct questioning about a sensitive trait.

The estimators of $\pi$ and T respectively, given by

$$\begin{aligned} & \hat{\pi }_{H}=\dfrac{\hat{\theta }_{H2} +p \hat{\theta }_{H1}-\big (1-p\big )}{\big (2p-1\big )} \end{aligned}$$

(6)

$$\begin{aligned} & \hat{T}_{H}=\dfrac{p \hat{\theta }_{H1}}{ \hat{\theta }_{H2}+ p\hat{\theta }_{H1}-\big (1-p\big )}, \end{aligned}$$

(7)

where, $\hat{\theta }_{Hi} ( i=1,2)$ is the proportion of yes answer.

The variances of $\hat{\pi }_{H}$ and MSE of $\hat{T}_{H}$ respectively, defined as:

$$\begin{aligned} & V\big (\hat{\pi }_{H}\big )=\dfrac{\pi \big (1-\pi \big ) }{n}+\dfrac{p\big (1-p\big ) \big (1-\pi T\big ) }{n\big (2p-1 \big )^2 }, \end{aligned}$$

(8)

$$\begin{aligned} & MSE\big (\hat{T}_{H}\big )=\dfrac{T\big (1-T\big ) }{n \pi }+\dfrac{p\big (1-p\big ) T^2\big (1-\pi T\big ) }{n \pi ^2 \big (2p-1 \big )^2}. \end{aligned}$$

(9)

In¹¹, a sample of size n is selected by SRSWR from a finite population of size N. The sampled individual is asked a direct question about whether they belong to group A. If the answer is “no,” the respondent is provided with a randomized response RD consisting of three statements: (i) “I belong to the sensitive group,” (ii) “say yes,” and (iii) “say no,” with known probabilities of p, $(1-p)/2$, and $(1-p)/2$, respectively. It is assumed that respondents who belong to the sensitive group have no reason to lie and are therefore completely truthful in their responses, regardless of whether a direct response or the RR approach is used. Under¹¹ the probability of receiving a ’yes’ response in the direct questioning approach is

$$\begin{aligned} P\left( Yes\right) =\theta _{S1} =\pi T, \end{aligned}$$

(10)

and probability of yes response under¹¹ model in the indirect response is

$$\begin{aligned} \textit{P(Yes)}=\theta _{S2} =p \pi \big (1-T\big )+\frac{\big (1-p\big )}{2}. \end{aligned}$$

(11)

The estimators of $\pi$ and T respectively, given by

$$\begin{aligned} \hat{\pi }_{S}=\dfrac{ \hat{\theta }_{S2}+p\hat{\theta }_{S1}-\dfrac{\big (1-p\big )}{2}}{p} \end{aligned}$$

(12)

and

$$\begin{aligned} \hat{T}_{S}=\dfrac{p \hat{\theta }_{S1}}{p \hat{\theta }_{S1}+\hat{\theta }_{S2}-\dfrac{\big (1-p\big )}{2}}. \end{aligned}$$

(13)

The variance of $\hat{\pi }_{S}$, is defined as:

$$\begin{aligned} V\big (\hat{\pi }_{S}\big )=\dfrac{\pi \big (1-\pi \big ) }{n}+\dfrac{\big (1-p\big ) \big (1+p-4p \pi T\big )}{4np^2 }, \end{aligned}$$

(14)

and the MSE of $\hat{T}_S$, is given by

$$\begin{aligned} MSE\big (\hat{T}_{S}\big )=\dfrac{T\big (1-T\big )}{n \pi }+\dfrac{\big (1-p\big ) T^2\big \{1+p+4p \pi \big (1-T\big )\big \}}{4np^2\pi ^2}. \end{aligned}$$

(15)

In Addition to this, to improve estimation reliability¹⁵, created a modified RR model specifically designed for populations with high nonresponse rates. In order to improve respondent anonymity and preserve statistical efficiency¹⁶, developed a dual-stage unrelated question strategy. Likewise¹⁷, suggested a design that incorporates several sensitive qualities to increase the adaptability of RR models in real-world surveys. A study that demonstrated field-level feasibility of RR implementation in South Asian socio-demographic surveys was provided by¹⁸. In order to reduce variance in predicting sensitive proportions under stratified sampling¹⁹, proposed a corrected RR model. To minimize digital response biases²⁰, adapted RR approach for use in online surveys. By employing auxiliary data²¹, suggested a robust RR estimator that increased respondent trust and precision. Similar studies can be found^22,23. Moreover²⁴, presented a resilient estimator that can manage partial noncompliance circumstances, and²⁵created a novel two-stage RR design appropriate for low-literacy populations. Moreover, comparative research has been crucial in improving RR models as well. In a field application with income-related questions, for instance²⁶, assessed the effectiveness of multiple RR designs. The balance between estimator bias and respondent protection in modified RR models was examined by¹⁶. An estimator that stabilizes variance in stratified settings was proposed by²⁷. In a related work²⁸, evaluated the feasibility constraints of single-device RRtechniques. Expanding on this²⁹, presented a hybrid RR strategy with enhanced privacy-preservation metrics³⁰. created a response bias correction technique specific to RR models in developing-country settings, whereas³¹ emphasized on optimizing efficiency under multi-stage sampling.

Methodology

Estimation of sensitive attribute and sensitivity level in simple random sampling SRS

Proposed model under SRS

The proposed method diverges from the models introduced by¹⁰ and¹¹ in its use of the randomized response RR technique. In this method, a simple random sample SRS of size n is drawn with replacement from a finite population of size N. Each selected individual is initially asked directly whether they possess the sensitive attribute A. If the response is no, the respondent is then guided through two randomizing devices, $R_1$ and $R_2$. The first device, $R_1$, presents two possible outcomes: (i) “I belong to the sensitive group A” with probability p, and (ii/) Proceed to the random device $R_2$ with probability $(1-p)$. Device $R_2$ mirrors the design of the model proposed by¹, involving two statements with selection probabilities $p_o$ and $(1-p_o)$. It is important to note that respondents not belonging to the sensitive group have no incentive to provide false information and thus are assumed to respond truthfully, regardless of whether a direct or randomized method is used. Conversely, individuals from group A are presumed to respond truthfully under the randomized response approach, but only with probability T under the conventional direct questioning method. The respondents are restricted that he/she does not disclose their response to the interviewer.

Let T be the probability of an honest response in first direct approach. The probability of yes answer $\theta _{1}$ in the direct response approach, under the proposed model, is

$$\begin{aligned} \theta _{1}=\pi T, \end{aligned}$$

(16)

and probability of yes answer $\theta _{2}$ in the RR approach, under the proposed model, is

$$\begin{aligned} \theta _{2}=p\pi \big (1-T\big )+\big (1-p\big )\big \{p_{o} \pi +\big (1-p_{o}\big )\big (1-\pi \big )\big \} \Rightarrow \pi =\dfrac{{\theta }_{2}+p{\theta }_{1}-\big (1-p_{o}\big )\big (1-p\big )}{\big \{2p-1+2p_{o}\big (1-p\big )\big \} }, \end{aligned}$$

To use the method of moments, an estimator of population proportion $\pi$, can be defined as:

$$\begin{aligned} \hat{\pi } =\dfrac{\hat{\theta }_{2}+p\hat{\theta }_{1}-\big (1-p_{o}\big )\big (1-p \big ) }{\big \{2p-1+2p_{o}\big (1-p\big )\big \}}, \end{aligned}$$

(17)

and the estimator of T, is given by

$$\begin{aligned} \hat{T}=\dfrac{\hat{\theta }_{1}\big \{2p-1+2p_{o}\big (1-p\big )\big \} }{\hat{\theta }_{2}+p\hat{\theta }_{1}-\big (1-p_{o}\big )\big (1-p \big )}, \end{aligned}$$

(18)

where $\hat{\theta }_{j}(j=1,2),$ the observed proportion of the response yesyesyes reported by the respondents.The key properties of the estimator $\hat{\pi }$ are summarized in the theorem below:

Theorem 3.1

The estimator $\hat{\pi }$ is an unbiased estimator, and its variance is given by:

$$\begin{aligned} V\big (\hat{\pi }\big ) =\dfrac{\pi \left( 1-\pi \right) }{n}\ + \dfrac{\big (1-p \big )\big (1-p_{o}\big )\big \{1-\big (1-p \big )\big (1-p_{o}\big )\big \}}{n W^2}-\frac{p\pi T\big (1-p\big )}{n W^2}, \end{aligned}$$

where $W=\big \{2p-1+2p_{o}\big (1-p\big )\big \}$

Proof

In view of the obvious result that $E\big (\hat{\theta }_{1}\big )=\theta _{1}$ and $E\big (\hat{\theta }_{2}\big )=\theta _{2}$, we have

$$\begin{aligned} E\big (\hat{\pi }\big ) =E\bigg [\dfrac{\hat{\theta }_{2}+p\hat{\theta }_{1}-\big (1-p_{o}\big )\big (1-p \big ) }{\big \{2p-1+2p_{o}\big (1-p\big )\big \} } \bigg ]=\dfrac{{\theta }_{2}+p{\theta }_{1}-\big (1-p_{o}\big )\big (1-p \big )}{\big \{2p-1+2p_{o}\big (1-p\big )\big \} },\nonumber =\pi . \end{aligned}$$

Hence prove that the estimator ${\hat{\pi }}$ is unbiased estimator of $\pi$. $\square$

The variance of the estimator $\hat{\pi }$, is given by

$$\begin{aligned} V\big (\hat{\pi }\big )= & \dfrac{\bigg [ V\big (\hat{\theta }_{1}\big )+V\big (\hat{\theta }_{2}\big )+2 cov\big ( \hat{\theta }_{1},\hat{\theta }_{2}\big ) \bigg ] }{\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2 }\nonumber \\= & \frac{1}{\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2 }\bigg [ \dfrac{\theta _{1}\big ( 1-\theta _{1}\big )}{n} +\dfrac{\theta _{2}\big (1-\theta _{2}\big ) }{n}-\dfrac{2\theta _{1}\theta _{2}}{n}\bigg ] \nonumber \\= & \frac{1}{n\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2 }\bigg [ \theta _{1}\big (1-\theta _{1}\big ) +\theta _{2}\big ( 1-\theta _{2}\big )-2\theta _{1}\theta _{2}\bigg ]\nonumber \\= & \dfrac{\pi \big (1-\pi \big )}{n}+\dfrac{\big (1-p_{o} \big )\big (1-p\big )\big \{1-\big (1-p_{o} \big )\big (1-p\big )\big \}}{n\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}-\frac{p\pi T\big (1-p\big )}{n\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}\nonumber \\= & \dfrac{\pi \big (1-\pi \big )}{n}+\dfrac{\big (1-p_{o} \big )\big (1-p\big )\big \{1-\big (1-p_{o} \big )\big (1-p\big )\big \}}{n W^2}-\frac{p\pi T\big (1-p\big )}{n W^2}. \end{aligned}$$

(19)

This complete the proof.

Corollary 3.1

The unbiased estimator of $V\big (\hat{\pi }\big )$, is given by

$$\begin{aligned} \hat{V}\big (\hat{\pi }\big )=\dfrac{\hat{\pi }\big ( 1-\hat{\pi }\big )}{n-1} +\dfrac{\big (1-p_{o} \big )\big (1-p\big )\big \{1-\big (1-p_{o} \big )\big (1-p\big )\big \}}{n-1\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}-\frac{p \hat{\pi } T\big (1-p\big )}{n-1\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}. \end{aligned}$$

To derive MSE from the estimator $\hat{T}$, we write $d_1=\hat{\theta }_{1}\big \{2p-1+2p_{o}\big (1-p\big )\big \}$ and $d_2=\hat{\theta }_{2}+p \hat{\theta }_{1}-\left( 1-p_{o}\right) \left( 1-p\right)$, and it follows that $E\big (d_1\big )=\big \{2p-1+2p_{o}\big (1-p\big )\big \}\pi T$ and $E\big (d_2\big )=\big \{2p-1+2p_{o}\big (1-p\big )\big \}\pi$. The estimator $\hat{T}$ can be written as $\hat{T}=\dfrac{d_1}{d_2}$, and we have ${T}=\dfrac{E(d_1)}{E(d_2)}$. Furthermore, we define the following terms:

$$\begin{aligned} e_1=\dfrac{\big (d_1-E\big (d_1\big )\big )}{E\big (d_1\big )}, \end{aligned}$$

and

$$\begin{aligned} e_2=\dfrac{\big (d_2- E\big (d_2\big )\big ) }{E\big (d_2\big )}. \end{aligned}$$

Suppose that $|e_1 |< 1$, ensuring that the function $(1 + e_2)^{-1}$ is analytically expandable as a power series. Under this condition, it can be readily demonstrated that

$$\begin{aligned} & E\big (e_1^2\big )=\dfrac{\theta _{1}\big ( 1-\theta _{1}\big ) }{n\pi ^2T^2}\\ & E\big (e_2^2\big )=\dfrac{\theta _{1}\big (1-\theta _{1}\big ) +\theta _{2}\big (1-\theta _{2}\big )-2\theta _{1}\theta _{2}}{n \pi ^2\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}\\ & E\big ( e_1 e_2\big ) =\dfrac{p\theta _{1}\big ( 1-\theta _{1}\big )-\theta _{1}\theta _{2}}{n \pi ^2 T\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}. \end{aligned}$$

The estimator $\hat{T}$ produces an estimation error given by:

$$\begin{aligned} \hat{T}-T=\dfrac{T\pi \big (1+e_1\big )}{\pi \big ( 1+e_2\big )}-T \end{aligned}$$

(20)

$$\begin{aligned} \hat{T}-T= T\big (1+e_1\big )\big (1+e_2\big ) -T, \end{aligned}$$

ignoring the high- order term error component

$$\begin{aligned} \hat{T}-T= T\big (e_1-e_2\big )-T. \end{aligned}$$

(21)

Theorem 3.2

The MSE of $\hat{T}$, accurate to order $O(n^{-1})$, is expressed as:

$$\begin{aligned} MSE\big (\hat{T}\big )= \dfrac{T\big (1-T\big ) }{n\pi }+\dfrac{\big (1-p_{o}\big ) \big (1-p\big )T^2\big \{1-\big (1-p_{o}\big ) \big (1-p\big )\big \}}{n\pi ^2W^2}+\frac{\big (1-p\big )T^2\big (2p_{o}W-pT\big )}{n\pi W^2}, \end{aligned}$$

(22)

Proof

We consider

$$\begin{aligned} MSE\big (\hat{T}\big )= & E\big ( T-\hat{T}\big )^2 \\= & T^2\bigg [E\big (e_1\big )^2-2 E\big (e_1 e_2\big )+E\big (e_2\big )^2\bigg ] \\= & T^2\bigg [\dfrac{\theta _{1}\big (1-\theta _{1}\big )}{n\pi ^2T^2}-\dfrac{2 \big \{p\theta _{1}\big (1-\theta _{1}\big )-\theta _{1}\theta _{2}\big \}}{n\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}+\dfrac{\theta _{1}\big (1-\theta _{1}\big )+\theta _{2} \big (1-\theta _{2}\big )-2\theta _{1}\theta _{2}}{n\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}\bigg ]\\= & \frac{T\big (1-T\big )}{n\pi }+\frac{\big (1-p_{o}\big )\big ( 1-p\big )T^2\big \{1-\big (1-p_{o}\big )\big ( 1-p\big )\big \}}{n\pi ^2\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}+\frac{2p_{o} T^2\left( 1-p\right) }{n \pi \big \{2p-1+2p_{o}\big (1-p\big )\big \}} \\- & \frac{p\pi T^3\big (1-p\big )}{n \pi ^2 \big \{2p-1+2p_{o}\big (1-p\big )\big \}^2} \\= & \frac{T\big (1-T\big )}{n\pi }+\frac{\big (1-p_{o}\big )\big ( 1-p\big )T^2\big \{1-\big (1-p_{o}\big )\big ( 1-p\big )\big \}}{n\pi ^2\big \{2p-1+2p_{o}\big (1-p\big )\big \}^2} \\+ & \frac{\big (1-p\big )T^2\big [2p_{o}\big \{2p-1+2p_{o}\big (1-p\big )\big \}-pT\big ]}{n \pi \big \{2p-1+2p_{o}\big (1-p\big )\big \}^2}\\= & \frac{T\big (1-T\big )}{n\pi }+\frac{\big (1-p_{o}\big )\big ( 1-p\big )T^2\big \{1-\big (1-p_{o}\big )\big ( 1-p\big )\big \}}{n\pi ^2 W^2}+\frac{\big (1-p\big )T^2 \big (2p_{o}W-pT\big )}{n \pi W^2} \end{aligned}$$

$\square$

Hence proved.

Efficiency comparison of estimators under SRS

In this subsection, we derive the conditions under which the proposed estimators $\hat{\pi }$ and $\hat{T}$ outperform existing alternatives in terms of precision.

Specifically, the estimator $\hat{\pi }$ demonstrates greater efficiency compared to the estimator $\hat{\pi }_{\text {MS}}$ introduced by², provided that

$$\begin{aligned} \frac{p\pi T\big (1-p\big )}{nW^2 }>0. \end{aligned}$$

(23)

We can conclude from the inequality condition given in (23), that our proposed estimator $\hat{\pi }$ performs better than² estimator $\hat{\pi }_{MS}$ for all choices of constants p, $p_{o}$ $\pi$ and T.

The estimator $\hat{\pi }$ is preferable over¹⁰ estimator $\hat{\pi }_{H}$, if

$$\begin{aligned} \pi <\dfrac{{p W^2-\big (2p-1\big )^2\big (1-p\big )\big \{1-\big (1-p\big )\big (1-p_{o}\big )\big \}} }{p T\big (W^2-\left( 2p-1\right) ^2\big )} \end{aligned}$$

(24)

The proposed estimator $\hat{T}$ is not less efficient than the estimator¹⁰ $\hat{T}_{H}$, if

$$\begin{aligned} \pi <\dfrac{{p W^2-\big (2p-1\big )^2\big (1-p_{o}\big )\big \{1-\big (1-p\big )\big (1-p_{o}\big )\big \}}}{p T W^2+\big (2p-1\big )^2\big (2 p_{o} W-pT\big )} \end{aligned}$$

(25)

The estimator $\hat{\pi }$ will dominate¹¹ estimator $\hat{\pi }_{ST}$, if

$$\begin{aligned} \pi < \dfrac{\big (1+p\big ) W^2-4 p^2\big (1-p_{o}\big )\big \{1-\big (1-p\big )\big (1-p_{o}\big )\big \} }{4pT\big (p-W^2\big ) }, \end{aligned}$$

(26)

The proposed estimator $\hat{\pi }$ will be more efficient than¹¹ estimator $\hat{\pi }_{ST}$, if

$$\begin{aligned} \pi < \dfrac{\big (1+p\big )W^2-4 p^2\big (1-p_{o}\big )\big \{1-\big (1-p\big )\big (1-p_{o}\big )\big \} }{4p\big \{p\big (2p_{o}W-PT\big )-\big (1-T\big )W^2\big \}}. \end{aligned}$$

(27)

To determine the performance of the suggested estimators $\hat{\pi }$ and $\hat{T}$ the percent relative efficiencies PRE of the modified estimators are calculated with respect to the estimators¹⁰ and¹¹. The PRE of the proposed estimators $\hat{\pi }$ and $\hat{T}$ with the¹⁰ estimators $\hat{\pi }_{H}$ and $\hat{T}_{H}$, and¹¹ estimators $\hat{\pi }_{ST}$ and $\hat{T}_{ST}$ respectively, are computed using the following formulae,

Table 1 Comparison of percent relative efficiency (PRE) between the proposed estimator $\hat{\pi }$ and $\hat{\pi }_{H}$ (Huang) estimator.

Table 2 Comparison of percent relative efficiency (PRE) between proposed estimator $\hat{T}$ and $\hat{T}_{H}$ (H Tarray),estimator.

Table 3 Comparison of percent relative efficiency (PRE) between the proposed estimator $\hat{\pi }$ and $\hat{\pi }_{ST}$ (Singh) estimator.

Table 4 Percent Relative Error (PRE) of proposed estimator $\hat{T}$ relative to $\hat{T}_{ST}$ (Singh) estimator.

$$\begin{aligned} & PRE\big (\hat{\pi },\hat{\pi }_{H}\big )= \frac{V\big (\hat{\pi }\big )}{V\big (\hat{\pi }_{H}\big )} \times 100 \end{aligned}$$

(28)

$$\begin{aligned} & PRE\big (\hat{T},\hat{T}_{H}\big )= \frac{V\big (\hat{T}\big )}{V\big (\hat{T}_{H}\big )} \times 100 \end{aligned}$$

(29)

$$\begin{aligned} & PRE\big (\hat{\pi },\hat{\pi }_{ST}\big )= \frac{V\big (\hat{\pi }\big )}{V\big (\hat{\pi }_{ST}\big )} \times 100 \end{aligned}$$

(30)

$$\begin{aligned} & PRE\big (\hat{T},\hat{T}_{ST}\big )= \frac{V\big (\hat{T}\big )}{V\big (\hat{T}_{ST}\big )}\times 100 \end{aligned}$$

(31)

for different cases of $p, p_0, \pi$ and T. The results are displayed in Tables 1, 2, 3 and 4. From Tables 1-4, one can see that all the PRE values larger than 100, indicate that $\hat{\pi }$ is more efficient than $\hat{\pi }_{H}$ and $\hat{\pi }_{ST}$, and $\hat{T}$ is also superior to $\hat{T}_{H}$ and $\hat{T}_{ST}$. In other words, the results demonstrate the superiority of the proposed technique over the¹⁰ and¹¹ techniques. It is observed from tables 1 and 3, that PRE of the proposed estimator $\hat{\pi }$ increases as T increases for a specific combination of $p_{o}$, p and $\pi$. Moreover, the PRE values of the proposed estimator $\hat{\pi }$ increase as the values of $p_{o}$ increase keeping constant T, p, and $\pi$. Furthermore, PRE of the proposed estimator $\hat{\pi }$ is relatively higher when $\pi$ is not close to 0.5 for the specific combination of $p_{o}$, p, and T. It is observed from tables 2 and 4, that a high efficiency gain is achieved by using the proposed estimator $\hat{T}$ over the¹⁰ estimator $\hat{T}_{H}$, and the¹¹ estimator $\hat{T}_{ST}$ when $p_{o}$ and T are close to unity and $\pi$ is close to zero.

Estimation of sensitive attribute and sensitivity level in stratified random sampling

In this section, we discuss the application of the proposed randomized response (RR) model within a stratified random sampling framework, where the population is divided into L strata, each of known size $N_i$. Consider a population of size N partitioned into L mutually exclusive strata with sizes $N_i$ for $i = 1, 2, \ldots , L$. The known proportion of the population in the ith stratum is given by $w_i = \frac{N_i}{N}$. From each stratum, a sample of size $n_i$ is drawn using simple random sampling with replacement (SRSWR), such that the total sample size satisfies $\sum _{i=1}^L n_i = n$ and asked respondents to answer yes or no according to the randomization devices.

Using a proportional allocation³² developed a stratified RRtechnique¹². suggested a stratified RR technique using an optimal allocation which is more efficient and less costly than that using a proportional allocation³²approach¹³. have applied¹² stratified RR model to² two stage RRmodel, and studied the properties of their proposed model using an optimum allocation³³. have suggested a mixed randomized response model and extended this model to stratified sampling³⁴. have envisaged the stratified RR model for³⁵ unrelated question RRmodel with its properties. Furthermore¹⁴, extended the¹⁰ model in stratified random sampling using an optimal allocation, which performs better than the stratified random sampling model proposed by¹³. When researchers aim to estimate sensitive attributes, such as the proportion of HIV/AIDS positive individuals in different categories-such as rural versus urban areas, age groups, or income brackets-the extension of the randomized response (RR) technique to stratified random sampling becomes particularly valuable. Motivated by this, we investigate the proposed RR model within the stratified random sampling context. The following section presents a detailed description of the proposed RR model adapted for stratified random sampling.

Proposed model under stratified random sampling

In the proposed model, the finite population is divided into strata, and a sample is drawn from each stratum using simple random sampling with replacement (SRSWR). To fully leverage the advantages of stratification, it is assumed that the size of each stratum is known. Each respondent selected from stratum i is asked a direct question regarding whether they possess the sensitive attribute A. If the response is no, the respondent is then directed to use two random devices, $R_{1i}$ and $R_{2i}$. $R_{1i}$ consists of two statements, $\big (i\big )$ I belong to the sensitive group A with probability $p_i$ and $\big (ii\big )$ go to random device $R_{2i}$ with probability $\big (1- p_i\big )$. The random device $R_{2i}$ is exactly the same used by¹² with probabilities of the two statements $p_{oi}$ and $\big (1-p_{oi}\big )$. A respondent belonging to the sample in different strata will perform different randomization devices, each having different preassigned probabilities. Noticed that, in stratum i the respondents in the non-sensitive group have no reason to tell a lie and thus, they are completely truthful in their response whether a direct response or randomization approach adopted by them. It is assumed that respondents in group A provide honest responses under the randomization approach, while the probability Ti follows the conventional direct response approach in stratum i.

The respondents are restricted that he/she does not disclose their response to the interviewer.

Let $T_i$ be the probability of an honest response in a first direct approach in stratum i. The probability of yes response $\theta _{1i}$ in a stratum i in the direct response approach under the proposed model is

$$\begin{aligned} \theta _{1i}=\pi _{Si} T_{i}, \end{aligned}$$

(32)

and probability of yes answer $\theta _{2i}$ in the RR approach in a stratum i under the proposed model, is

$$\begin{aligned} & \theta _{2i}=p_i\pi _{Si}\left( 1-T_i\right) + \big (1-p_i\big ) \big \{p_{oi} \pi _{Si}+\big (1-p_{oi}\big )\big (1-\pi _{Si}\big )\big \} \Rightarrow \pi _i \\ & =\dfrac{{\theta }_{2i}+p{\theta }_{1i}-\big (1-p_{oi}\big )\big (1-p_{i} \big ) }{\big \{2p_i-1+2p_{oi}\big (1-p_{i}\big )\big \}} , \end{aligned}$$

where $\pi _{Si}$ is the proportion of respondents with the sensitive attribute in the sample of a stratum i, $p_i$ is the probability that a respondent in the sample stratum i has a sensitive question (S) card in a first stage, and $p_{oi}$ is the probability that a respondent in the sample stratum i has a sensitive question (S) card in a second stage.

The proposed estimators for $\pi _{Si}$ and $T_i$ respectively, given by

$$\begin{aligned} \hat{\pi }_{Si} =\dfrac{\hat{\theta }_{2i}+p\hat{\theta }_{1i}-\big (1-p_{oi}\big )\big (1-p_i \big ) }{\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}}\,\,\,\,\,\, for (i=1,2,3....k), \end{aligned}$$

(33)

and

$$\begin{aligned} \hat{T}_{i}=\dfrac{\hat{\theta }_{1i}\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \} }{\hat{\theta }_{2i}+p_i\hat{\theta }_{1i}-\big (1-p_{oi}\big )\big (1-p_i \big )}\,\,\,\,\,\, for (i=1,2,3....k), \end{aligned}$$

(34)

where $\hat{\theta }_{ji}(j=1,2),$ the observed proportion of yes response reported by the respondents.

The estimator $\hat{\pi }_{Si}$ is unbiased with the variance given by

$$\begin{aligned} V\big (\hat{\pi }_{Si}\big ) =\dfrac{\pi _{Si}\big (1-\pi _{Si}\big )}{n}\ + \dfrac{\big (1-p_{i} \big )\big (1-p_{oi}\big )\big \{1-\big (1-p_i \big )\big (1-p_{oi}\big )\big \}}{n_i W_{i}^2}-\frac{p_i\pi _{Si} T_i\big (1-p_i\big )}{n_i W_{i}^2}. \end{aligned}$$

where $W_{i}=\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}$

Since the selection in different strata are made independently, the estimators for individual strata can be added together to obtain an estimators for the whole population. The estimator of $\pi _{S}=\sum _{i}^{k} w_i \pi _{si}$, the proportion of interviewee with the sensitive group, is

$$\begin{aligned} \hat{\pi }_{S} =\sum _{i}^{k} w_i \bigg [\dfrac{\hat{\theta }_{2i}+p\hat{\theta }_{1i}-\big (1-p_{oi}\big )\big (1-p_i \big ) }{\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}}\bigg ] \end{aligned}$$

(35)

Here, N denotes the total number of units in the entire population, $N_i$ represents the number of units in the $i_th$ stratum, and $w_i = \frac{N_i}{N}$ for $i = 1, 2, \ldots , k$, such that $\sum _{i=1}^{k} w_i = 1$

Theorem 4.1

The proposed estimator $\hat{\pi }_{S}$ provides an unbiased estimate of the population proportion $\pi _{S}$.

Proof

The unbiasedness of $\hat{\pi }_{Si}$ follows from $E\big (\hat{\theta _{ji}}\big ) =\theta _{ji}$, $(i=1,2,3...k)$. Thus the unbiasedness of $\hat{\pi }_{S}$ follows from taking the expected value of (35). $\square$

Theorem 4.2

The variance of the estimator $\hat{\pi }_{S}$ is

$$\begin{aligned} V\big (\hat{\pi }_{S}\big ) =\sum _{i}^{k} w_i^2 \bigg [\dfrac{\pi _{i}\big (1-\pi _{i}\big )}{n_i}\ + \dfrac{\big (1-p_{i} \big )\big (1-p_{oi}\big )\big \{1-\big (1-p_i \big )\big (1-p_{oi}\big )\big \}}{n_i W_{i}^2}-\frac{p_i\pi _i T_i\big (1-p_i\big )}{n_i W_{i}^2}\bigg ]. \end{aligned}$$

(36)

The variance of the¹⁴ estimator $\hat{\pi }_{HS}$, under stratified random sampling, is

$$\begin{aligned} V\big (\hat{\pi }_{TS}\big ) =\sum _{i}^{k} w_i^2 \bigg [\dfrac{\pi _i\big (1-\pi _i\big ) }{n_i}+\dfrac{p_i\big (1-p_i\big ) \big (1-\pi _i T_i\big ) }{n_i\big (2p_i-1 \big )^2 }\bigg ] \end{aligned}$$

(37)

and the variance of¹² estimator $\hat{\pi }_{KW}$, is given by

$$\begin{aligned} V\big (\hat{\pi }_{KW}\big ) =\sum _{i}^{k} w_i^2 \bigg [\dfrac{\pi _i\big (1-\pi _i\big ) }{n_i}+\dfrac{p_i\big (1-p_i\big ) }{n_i\big (2p_i-1 \big )^2 }\bigg ]. \end{aligned}$$

(38)

It is obtain from 36 and 37, such that

$$\begin{aligned} & V\big (\hat{\pi }_{TS}\big )- V\big (\hat{\pi }_{S}\big )=\bigg [\dfrac{p_i\big (1-p_i\big ) \big (1-\pi _i T_i\big ) }{n_i\big (2p_i-1 \big )^2 }-\dfrac{\big (1-p_{i} \big )\big (1-p_{oi}\big )\big \{1-\big (1-p_i \big )\big (1-p_{oi}\big )\big \}}{n_i W_{i}^2} \nonumber \\ & +\frac{p_i\pi _i T_i\big (1-p_i\big )}{n_i W_{i}^2}\bigg ]>0 \end{aligned}$$

(39)

using 36 and 38, such that

$$\begin{aligned} & V\big (\hat{\pi }_{TS}\big )- V\big (\hat{\pi }_{S}\big )=\bigg [\dfrac{p_i\big (1-p_i\big ) }{n_i\big (2p_i-1 \big )^2 }-\dfrac{\big (1-p_{i} \big )\big (1-p_{oi}\big )\big \{1-\big (1-p_i \big )\big (1-p_{oi}\big )\big \}}{n_i W_{i}^2} \nonumber \\ & +\frac{p_i\pi _i T_i\big (1-p_i\big )}{n_i W_{i}^2}\bigg ]>0 \end{aligned}$$

(40)

It follows that the proposed estimator $\hat{\pi }_{HS}$ is more efficient than the estimators $\hat{\pi }_{KW}$ of¹² and $\hat{\pi }_{TS}$ of¹⁴. An unbiased estimator of the variance $V\big (\hat{\pi }_{S}\big )$ can be readily obtained and is given by:

Corollary 4.1

An unbiased estimator of the variance $V\big (\hat{\pi }_{S}\big )$ is:

$$\begin{aligned} \hat{V}\big (\hat{\pi }_{S}\big )=\hat{V}\bigg (\sum _{i}^{k} w_i^2 \hat{\pi }_{Si}\bigg )=\sum _{i}^{k} w_i^2 \hat{V}\big (\hat{\pi }_{Si}\big ) \end{aligned}$$

(41)

where

$$\begin{aligned} & \hat{V}\big (\hat{\pi }_{Si}\big )=\dfrac{\hat{\pi }_{Si}\big (1-\hat{\pi }_{Si}\big )}{n_i} \nonumber \\ & + \dfrac{\big (1-p_{i} \big )\big (1-p_{oi}\big )\big \{1-\big (1-p_i \big )\big (1-p_{oi}\big )\big \}}{n_i W_{i}^2}-\frac{p_i \big (1-p_i\big ) \hat{\theta }_{1i}}{n_i W_{i}^2}. \end{aligned}$$

(42)

Although the values of $\pi _{Si}$ are generally unknown, the availability of prior information on $\pi _i$ and $T_i$ from previous studies allows us to derive the following optimal allocation formula:

Theorem 4.3

The optimal allocation of the total sample size n among the strata, i.e., $n_1, n_2, \ldots , n_{k-1}, n_k$, to minimize the variance of the estimator $\hat{\pi }_S$, subject to the constraint $n = \sum _{i=1}^k n_i$, is approximately given by:

$$\begin{aligned} n_i=\frac{n w_i\bigg [\pi _{Si}\left( 1-\pi _{Si}\right) \ + \dfrac{\left( 1-p_{i} \right) \left( 1-p_{oi}\right) \big \{1-\left( 1-p_i \right) \left( 1-p_{oi}\right) \big \}}{ W_{i}^2}-\frac{p_i\pi _{Si} T_i\big (1-p_i\big )}{ W_{i}^2} \bigg ]^\frac{1}{2}}{\sum _{i}^{k}w_i\bigg [\pi _{Si}\left( 1-\pi _{Si}\right) + \dfrac{\left( 1-p_{i} \right) \left( 1-p_{oi}\right) \big \{1-\left( 1-p_i \right) \left( 1-p_{oi}\right) \big \}}{ W_{i}^2}-\frac{p_i\pi _{Si} T_i\big (1-p_i\big )}{ W_{i}^2} \bigg ]^\frac{1}{2}} \end{aligned}$$

(43)

Proof

Follows from section 5.5 of³⁶. The minimal variance of the estimator $\hat{\pi }_{S}$, is given by $\square$

$$\begin{aligned} V\big (\hat{\pi }_{S}\big )=\frac{1}{n}\Bigg \{\sum _{i}^{k}w_i\bigg [\pi _{Si}\big (1-\pi _{Si}\big ) + \dfrac{\big (1-p_{i} \big )\big (1-p_{oi}\big )\big \{1-\big (1-p_i \big )\big (1-p_{oi}\big )\big \}}{ W_{i}^2}-\frac{p_i\pi _{Si} T_i\big (1-p_i\big )}{ W_{i}^2}\bigg ]^\frac{1}{2}\Bigg \}^2 \end{aligned}$$

(44)

Using (43), the unbiased minimal variance of the estimator $\hat{\pi }$ in (44) can be derived. To derive the MSE of the estimator $\hat{T}_{i}$, we write $d_{1i}=\hat{\theta }_{1i}\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}$ and $d_{2i}=\hat{\theta }_{2i}+p\hat{\theta }_{1i}-\left( 1-p_{oi}\right) \left( 1-p_i\right)$, and it follows that $E\big (d_{1i}\big )=\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}\pi _i T_i$ and $E\big (d_{1i}\big )=\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}\pi _i$. The estimator $\hat{T}_{i}$ can be written as $\hat{T }_{i}=\dfrac{d_{1i}}{d_{2i}}$ and we have ${T}_{S}=\dfrac{E\big (d_{1i}\big )}{E\big (d_{2i}\big )}$. Furthermore, we define the following terms:

$$\begin{aligned} e_{1i}=\dfrac{\big (d_{1i}-E\big (d_{1i}\big )\big ) }{E\big (d_{1i}\big )}, \end{aligned}$$

and

$$\begin{aligned} e_{2i}=\dfrac{\big ( d_{2i}- E\big (d_{2i}\big )\big ) }{E\big (d_{2i}\big )}. \end{aligned}$$

Provided that $|e_{1i} |< 1$, the expression $(1 + e_{2i})^{-1}$ admits a valid power series expansion. Consequently, it follows that

$$\begin{aligned} & E\big (e_{1i}\big )^2=\dfrac{\theta _{1i}\big ( 1-\theta _{1i}\big )}{n_i\pi _{Si}^2T_{i}^2}\\ & E\big (e_{2i}\big )^2=\dfrac{\theta _{1i}\big ( 1-\theta _{1i}\big ) +\theta _{2i}\big (1-\theta _{2i}\big ) -2\theta _{1i}\theta _{2i}}{n_i \pi _{Si}^2\big \{2p_i-1+2p_{oi}\big (1-p_i\big )\big \}^2}\\ & E\big (e_{1i} e_{2i}\big ) =\dfrac{p\theta _{1i}\big ( 1-\theta _{1i}\big )-\theta _{1i}\theta _{2i}}{n_i \pi _{Si}^2 T\big \{2p_{i}-1+2p_{oi}\big (1-p_{i}\big )\big \}^2}. \end{aligned}$$

The estimator $\hat{T}$ has an associated estimation error, which can be formulated as:

$$\begin{aligned} \hat{T}_i-T_i=\dfrac{T_i\pi _{Si}\big ( 1+e_{1i}\big )}{\pi _{Si}\big ( 1+e_{2i}\big )}-T_i \end{aligned}$$

(45)

$$\begin{aligned} \hat{T_i}-T_i= T_i\big (1+e_{1i}\big )\big (1+e_{2i}\big ) -T_i, \end{aligned}$$

ignoring the high-order term error component

$$\begin{aligned} \hat{T_i}-T_i= T_i\big (e_{1i}-e_{2i}\big )-T_i. \end{aligned}$$

(46)

We obtain the following theorem from (22)

Theorem 4.4

The MSE of $\hat{T}_i$, accurate to order $O(n^{-1})$, is expressed as:

$$\begin{aligned} MSE\big (\hat{T_i}\big )= \dfrac{T_{i}\big ( 1-T_{i}\big ) }{n_i\pi _{Si}}+\dfrac{\big (1-p_{oi} \big ) \big (1-p_{i}\big )T_{i}^2\big \{1-\big (1-p_{oi} \big ) \big (1-p_{i}\big )\big \}}{n_{i}\pi _{Si}^2 W^2}+\frac{\big (1-p_{i}\big )T_{i}^2\big (2p_{oi}W_{i}-p_{i}T_{i}\big )}{n_i\pi _{Si} W_{i}^2}, \end{aligned}$$

(47)

Let $T_S$ denote the weighted probability, defined as $T_S = \sum _{i=1}^{k} T_i w_i$, where $T_i$ represents the probability that a respondent belonging to the sensitive group in stratum i responds truthfully in direct questioning. Using (34) in $T_{S}=\sum _{i}^{k} T_{i}w{i}$, we obtain an estimate of $T_{S}$ as:

$$\begin{aligned} \hat{T}_{S}=\sum _{i}^{k} w_{i} \hat{T}_{i}=\sum _{i}^{k} w_{i}\bigg [\dfrac{\hat{\theta }_{1}\big \{2p_{i}-1+2p_{oi}\big (1-p_{i}\big )\big \} }{\hat{\theta }_{2i}+p_i\hat{\theta }_{1i}-\big (1-p_{oi}\big )\big (1-p_{i} \big )}\bigg ] \end{aligned}$$

(48)

The MSE of $\hat{T}_{S}$ to order $(n-1)$ is derived as:

$$\begin{aligned}&MSE\big (\hat{T}_{S}\big )=\frac{1}{n_i \pi _{Si}}\bigg \{\sum _{i}^{k} w_{i}^2\bigg [\dfrac{T_{i}\big ( 1-T_{i}\big ) }{n_i\pi _{Si}}+\dfrac{\big (1-p_{oi}\big ) \big (1-p_{Si}\big )T_{i}^2\big \{1-\big (1-p_{oi} \big ) \big (1-p_{i}\big )\big \}}{n_{i}\pi _{Si}^2W^2} \nonumber \\&+\frac{\big (1-p_{i}\big )T_{i}^2\big (2p_{oi}W_{i}-p_{i}T_{i}\big )}{n_i\pi _{Si} W_{i}^2}\bigg ]\bigg \} \end{aligned}$$

(49)

Theorem 4.5

The optimal allocation of the total sample size n amongs the strata, i.e., $n_1, n_2, \ldots , n_{k-1}, n_k$, to minimize the MSE of the estimator $\hat{T}_{S}$, subject to the constraint $n = \sum _{i=1}^{k} n_i$, is approximately given by:

$$\begin{aligned} n_i=\frac{n w_i\bigg [\dfrac{T_{i}\big ( 1-T_{i}\big ) }{\pi _{Si}}+\dfrac{\big (1-p_{oi} \big ) \big (1-p_{i}\big )T_{i}^2\big \{1-\big (1-p_{oi} \big ) \big (1-p_{i}\big )\big \}}{\pi _{Si}^2W_{i}^2} +\frac{\big (1-p_{i}\big )T_{i}^2\big (2p_{oi}W_{i}-p_{i}T_{i}\big )}{\pi _{Si} W^2} \bigg ]^\frac{1}{2}}{\sum _{i}^{k}w_i\bigg [\dfrac{T_{i}\big ( 1-T_{i}\big ) }{\pi _{Si}}+\dfrac{\big (1-p_{oi} \big ) \big (1-p_{i}\big )T_{i}^2\big \{1-\big (1-p_{oi} \big ) \big (1-p_{i}\big )\big \}}{\pi _{Si}^2W_{i}^2} +\frac{\big (1-p_{i}\big )T_{i}^2\big (2p_{oi}W_{i}-p_{i}T_{i}\big )}{\pi _{Si} W_{i}^2} \bigg ]^\frac{1}{2}} \end{aligned}$$

(50)

Following the results in Section 5.5 of³⁶, the minimum MSE of $\hat{T}_{S}$ is expressed as:

$$\begin{aligned} MSE\big (\hat{T}_{S}\big )=\frac{1}{n}\bigg \{\sum _{i}^{k}w_i\bigg [\dfrac{T_{i}\big ( 1-T_{i}\big ) }{\pi _{Si}}+\dfrac{\big (1-p_{oi} \big ) \big (1-p_{i}\big )T_{i}^2\big \{1-\big (1-p_{oi} \big ) \left( 1-p_{i}\right) \big \}}{\pi _{Si}^2W_{i}^2} \nonumber \\ +\frac{\big (1-p_{i}\big )T_{i}^2\big (2p_{oi}W_{i}-p_{i}T_{i}\big )}{\pi _{Si} W_{i}^2} \bigg ]^\frac{1}{2}\bigg \}^2 \end{aligned}$$

(51)

Efficiency Comparison of Estimators under Stratified Sampling

In this subsection, we made a numerically comparison of proposed estimators $\hat{\pi }_s$ and $\hat{T}_{S}$ with¹² randomized response estimator $\hat{\pi }_{KW}$¹³, $\hat{\pi }_{KE}$ and¹⁴ estimators $\hat{\pi }_{TS}$ $\hat{T}_{TS}$ in stratified randomized sampling using an optimal allocation. It is difficult to derive mathematically $V\big (\hat{\pi }_{KW}\big )/V\big (\hat{\pi }_S\big )$, $V\big (\hat{\pi }_{TS}\big )/V\big (\hat{\pi }_S\big )$ and $V\big (\hat{T}_{TS}\big )/V\big (\hat{T}_S\big )$, so we made an empirical study of the PRE $V\big (\hat{\pi }_{KW}\big )/V\big (\hat{\pi }_S\big )$, $V\big (\hat{\pi }_{TS}\big )/V\big (\hat{\pi }_S\big )$ and $V\big (\hat{T}_{TS}\big )/V\big (\hat{T}_S\big )$. We computed the PRE of the proposed estimators $\hat{\pi }_{S}$ and $\hat{T}$ with respect to¹² estimators $\hat{\pi }_{S}$, and¹⁴ estimators $\hat{\pi }_{TS}$ and $\hat{T}$ by the way of variance comparison for different values $p_o, p, p_1, p_2, T, T_1, T_2, w_1, w_2, \pi _S, \pi _1$ and $\pi _2$, assuming two strata in the population $(k=2), T=T_1=T_2$ and $p_{o1}=p_{o2}=p_{o}$, using the formulae $52-54$. Findings are shown in Table 5, 6 and 7

$$\begin{aligned} PRE\big ({\hat{\pi }}_S,\hat{\pi }_{KW}\big )=\frac{\big ( w_1 \sqrt{B_1}+w_2\sqrt{B_2}\big )^2}{\big ( w_1 \sqrt{A_1}+w_2\sqrt{A_2}\big )^2} \end{aligned}$$

(52)

Where $B_1=\pi _1 \big (1-\pi _1\big )+\frac{p_{1} \big (1-p_{1}\big )}{\big (2p_{1}-1\big )^2},$$B_1=\pi _2 \big (1-\pi _2\big )+\frac{p_{2}\big (1-p_{2}\big )}{\big (2p_{2}-1\big )^2}$ $A_1=\pi _{1}\big (1-\pi _{1}\big )+$$\dfrac{\big (1-p_{1} \big )\big (1-p_{o}\big )\big \{1-\big (1-p_{1} \big )\big (1-p_{o}\big )\big \}}{ W_{i}^2}-\frac{p_{1}\pi _{1} T\big (1-p_{1}\big )}{W_{i}^2}$$A_2=\pi _1\left( 1-\pi _1\right) +$$\dfrac{\big (1-p_{2} \big )\big (1-p_{o}\big )\big \{1-\big (1-p_{2} \big )\left( 1-p_{o}\right) \big \}}{ W_{i}^2}-\frac{p_{2}\pi _1 T\big (1-p_{2}\big )}{W_{i}^2}$ and $\pi _{S}={w_{1}} {\pi _{1}}+{w_{2}} {\pi _{2}}$

$$\begin{aligned} PRE\big ({\hat{\pi }}_S,\hat{\pi }_{KW}\big )=\frac{\big ( w_1 \sqrt{C_1}+w_2\sqrt{C_2}\big )^2}{\big ( w_1 \sqrt{A_1}+w_2\sqrt{A_2}\big )^2} \end{aligned}$$

(53)

Where $C_1=\pi _1 \big (1-\pi _1\big )+\frac{p_{1}\big (1-p_{1}\big )\big (1-\pi _1 T\big )}{\big (2p_{1}-1\big )^2}, C_2=\pi _2 \big (1-\pi _2\big )+\frac{p_{2}\big (1-p_{2}\big )\big (1-\pi _2 T\big )}{\big (2p_{2}-1\big )^2}$

$$\begin{aligned} PRE\big (\hat{T}_S,\hat{\pi }_{KW}\big )=\frac{\big ( w_1 \sqrt{C_1^*}+w_2\sqrt{C_2^*}\big )^2}{\big ( w_1 \sqrt{A_1^*}+w_2\sqrt{A_2^*}\big )^2} \end{aligned}$$

(54)

Where $C_1^*=T \big (1-T\big )+\frac{p_{1}\big (1-p_{1}\big ) T^2\big (1-\pi _1 T\big )}{\big (2p_{1}-1\big )^2},$ $C_2^*=T \big (1-T\big )+\frac{p_{2}\big (1-p_{2}\big ) T^2 \big (1-\pi _2 T\big )}{\big (2p_{2}-1\big )^2},$$A_1^*=T\big (1-T\big )+ \dfrac{\big (1-p_{1} \big )\big (1-p_{o}\big ) T^2\big \{1-\big (1-p_{1} \big )\big (1-p_{o}\big )\big \}}{ \pi _1 W_{i}^2}+\frac{\big (1-p_{2}\big )T^2\big (2p_o W-p_{1}T\big )}{ W_{i}^2},$

$A_1^*=T\big (1-T\big )+\dfrac{\big (1-p_{2} \big )\big (1-p_{o}\big ) T^2\big \{1-\big (1-p_{2} \big )\big (1-p_{o}\big )\big \}}{\pi _2 W_{i}^2}+\frac{\big (1-p_{2}\big )T^2\big (2p_o W_{i}-pT\big )}{ W_{i}^2},$

Table 5 (PRE) of the proposed estimator $\hat{\pi }_{S}$ with $\hat{\pi }_{KW}$ (Kim) estimator in stratified random sampling using an optimal allocation.

Theorem 4.6

Assume that there are two strata in the population, $n=n_1+n_2$ $p=p_1=p_2$, $T=T_1=T_2$ and $po=p_{o1}=p_{o1}$. The proposed estimator $\hat{\pi }_{S}$ will be more efficient than¹³ estimator $\hat{\pi }_{KE}$, when $\pi _{S1} \ne \pi _{S2}$ under the following condition:

$$\begin{aligned} \frac{p T\big (1-p\big )}{nW^2 }>0. \end{aligned}$$

(55)

We can check the relative efficiency $\hat{\pi }_{KE}/\hat{\pi }_{S}$, if the prior information on $\pi _{S1}$, $\pi _{S1}$, $w_1, w_2, n$ are available and $p=p_1=p_2$, $T=T_1=T_2$ and $po=p_{o1}=p_{o1}$ are set by the researcher. Equation (55) shows that the proposed estimator $\hat{\pi }_{S}$ performs better than the¹³ estimator $\hat{\pi }_{KE}$ for all possible combinations of $\pi _{S1}, \pi _{S2}, w_1, w_2, w_1, w_2, n, p=p_1=p_2, T=T_1=T_2$, so there is no need to present the numerical results.

Table 6 PRE of the proposed estimator $\hat{\pi }_{S}$ with $\hat{\pi }_{TS}$¹⁴ estimator in stratified random sampling using an optimal allocation.

The Tables 5, 6 and 7 illustrate that PRE of the proposed estimators $\hat{\pi }_{S}$ and $\hat{T}_{S}$ are larger than 100, which indicate the superiority of the proposed estimators $\hat{\pi }_{S}$ and $\hat{T}_{S}$ over¹² estimator $\hat{\pi }_{KW}$, and¹⁴ estimators $\hat{\pi }_{TS}$ and $\hat{T}_{TS}$. It is seen from Table 5, that PRE the proposed estimator increases as increases $p_o$ and $\pi$. Furthermore, PRE the proposed estimator $\hat{\pi }_{S}$ is observed high when $p_1$ and $p_2$ are not close to zero and unity. It is obtained from Table 6, that PRE the proposed estimator $\hat{\pi }_{S}$ is high when $p_1$ and $p_2$ are close to 0.5.

Table 7 PRE of the proposed estimator $\hat{T}_{S}$ with Tarry estimator $\hat{T}_{TS}$ in stratified random sampling using an optimal allocation.

In table 7, it is observed that there is a high gain in efficiency when using the proposed estimators $\hat{T}_{S}$ when $p_{1}$ and $p_{1}$ are close to 0.5, and T is close to 1.

Discussion

The proposed model has been evaluated both numerically and theoretically. Equation (23) demonstrates that the proposed estimation is theoretically more efficient under the same conditions compared to the estimators presented in². Equations (24) and (26) show that $\hat{\pi }$ outperforms $\hat{\pi }_H$ and $\hat{\pi }_{ST}$, while Equations (27) and (28) indicate that $\hat{T}$ performs better than $\hat{T}_H$ and $\hat{T}_{ST}$ in simple random sampling.

From Tables 1, 2, 3, 4, it is clear that all the PRE values greater than 100 indicate that the proposed estimator $\hat{\pi }$ is more efficient than both $\hat{\pi }_H$ and $\hat{\pi }_{ST}$. Similarly, the proposed estimator $\hat{T}$ is superior to $\hat{T}_H$ and $\hat{T}_{ST}$. These results highlight the superiority of the proposed technique over the methods proposed by¹⁰ and¹¹. From Tables 1 and 3, it is observed that the PRE of the proposed estimator $\hat{\pi }$ increases as the sample size n increases, for a given combination of p, $p_o$, and $\pi$. Additionally, the PRE values for $\hat{\pi }$ also increase as the values of $p_o$ increase, while keeping T, p, and $\pi$ constant. The PRE of $\hat{\pi }$ is relatively higher when $\pi$ is not close to 0.5 for specific combinations of p, $p_o$, and T. Furthermore, from Tables 2 and 4, it is evident that a significant efficiency gain is achieved by using the proposed estimator $\hat{T}$ compared to the estimators $\hat{T}_H$ from¹⁰ and $\hat{T}_{ST}$ from¹¹, particularly when p, $p_o$, and T are close to unity and $\pi$ is near zero. Equation (55) demonstrates that the proposed estimator $\hat{\pi }_S$ performs better than the estimator $\hat{\pi }_{KE}$ from¹³ for all possible combinations of $\pi _{S1}$, $\pi _{S2}$, $w_1$, $w_2$, n, and $p = p_1 = p_2$, thus negating the need for additional numerical results in the stratified random sampling case. In Tables 5, 6, and 7, it is shown that the PRE values for the proposed estimators $\hat{\pi }_S$ and $\hat{T}_S$ are greater than 100, which indicates the superior performance of these estimators over $\hat{\pi }_{KW}$ from¹² and $\hat{\pi }_{TS}$ and $\hat{T}_{TS}$ from¹⁴. From Table 5, it is observed that the PRE of the proposed estimator increases as p, $p_o$, and $\pi$ increase. Moreover, the PRE of $\hat{\pi }_S$ is higher when $p_1$ and $p_2$ are not near 0 or 1. Table 6 reveals that the PRE of $\hat{\pi }_S$ is high when $p_1$ and $p_2$ are close to 0.5. Finally, from Table 7, it is evident that a high efficiency gain is achieved when using the proposed estimator $\hat{T}_{ST}$, especially when $p_1$ and $p_2$ are near 0.5 and T is close to 1. This demonstrates the robust performance and efficiency of the proposed estimators in various sampling scenarios. The results from both the simple and stratified random sampling methods indicate that the proposed estimators $\hat{\pi }$ and $\hat{T}$ exhibit superior efficiency, particularly in comparison to existing methods. The PRE analysis consistently shows that the proposed techniques outperform the estimators of^10,11,12, and¹⁴, with substantial improvements in efficiency across different parameter settings.

Conclusion

The Model of¹ was the first to address sensitive topics such as drug addiction and abortion, offering solutions to these issues. Utilizing an enhanced two-stage randomized response (RR) method, this work extends the models of^1,2, and¹⁰ to jointly estimate the sensitive proportion ($\pi$) and sensitivity level (T). Under both simple and stratified random sampling, the approach improves estimation efficiency and responder privacy by combining two randomization devices. In comparison to current methods, the suggested estimators consistently produce lower mean squared error and bias, according to theoretical analysis and simulation findings. In stratified designs, the model also does well under optimal allocation, an area in which many conventional RR models are inadequate. In contrast to previous approaches that presume complete compliance or estimate only one parameter, our strategy encourages honest responses while supporting joint estimation. For sensitive data collecting in various survey scenarios, it provides an all over useful and effective response. The suggested model performs well in simulation experiments, however these points are notable, the results are derived from controlled simulations, and respondents’ comprehension and adherence to the two-stage randomization process may have an impact on their real-world application. Future studies might concentrate on empirical validation with actual survey datasets, investigating the model’s resilience to different degrees of non-compliance, and expanding the methodology to take into account longitudinal data or more intricate sample methods.