Estimation of Means of Two Quantitative Sensitive Variables Using Randomized Response Technique

Abstract

I propose an improved randomized response model for the simultaneous estimation of population means of two quantitative sensitive variables by using blank card option that make use of one scramble response and another fake response. The properties of the proposed estimator have been analysed. To judge the performance of the proposed model, I have considered a real data set and it is to be pointed out that the proposed model is more efficient in terms of relative efficiencies and privacy protection of respondents as well. Suitable recommendations have been made to the survey practitioners.

1. Introduction

Reliability of data is compromised when sensitive topics on embarrassing or illegal acts such as students taking drugs, drunk driving, abortion, family income, tax evasion etc. are required in direct method of data collection in sample survey. Survey on human population has established the fact that the direct question about sensitive characters often results in either refusal to respond or falsification of the answer. To overcome this difficulty and ensure confidentially of respondents, Warner [1] initiated a technique which is called as randomized response technique (RRT). For estimating $\mathrm{\pi }$, the population proportion of respondents, a simple random sample of size $\mathrm{n}$ respondents selected from the population $\mathrm{N}$ with replacement. Each respondent selected in the sample has a random device which consists two statements “I belong to sensitive group $\mathrm{A}$” and “its compliment ${\mathrm{A}}^{\mathrm{c}}$”. The respondent answers of sensitive or non-sensitive questions depending on the outcome of the random device which is unobservable to the sampler. Greenberg et al. [2] adjusted the Warner [1] model with respect to efficiency and respondent’s cooperation by suggesting unrelated question randomized response model, where the sensitive question was combined with an unrelated (non-sensitive) question.

Greenberg et al. [3] extended the Greenberg et al. [2] model to estimate the population mean of quantitative sensitive variable, such as income, tax dodging etc. In their model, each respondent selected in the sample with replacement was given a random device which presents two outcomes $\mathrm{Y}$ and $\mathrm{X}$ with probabilities $\mathrm{P}$ and $\left(1‐\mathrm{P}\right)$ respectively, where $\mathrm{Y}$ is the true quantitative sensitive variable and $\mathrm{X}$ is non-sensitive independent variable. Later, Eichhorn and Hayre [4] introduced a new multiplicative randomized response model for estimating the population mean of quantitative sensitive variable.

Under simple random sampling with replacement (SRSWR) scheme, Perri [5] modified Greenberg et al. [3] technique to obtain the estimator of population mean ${\mathrm{\mu }}_{\mathrm{Y}}$ by using a blank card option, if a blank card is selected then the respondents are requested to use Greenberg et al. [3] model. In his model, the observed response ${\mathrm{\theta }}_{\mathrm{P}}$ is given by:

$${\mathrm{\theta }}_{\mathrm{P}}=\left\{\begin{array}{l}\mathrm{Y}\text{with}\text{probability}{\mathrm{P}}_1\\ \mathrm{Y}\text{with}\text{probability}{\mathrm{P}}_2\\ \mathrm{\text{Blank Card}}\text{with}\text{probability}{\mathrm{P}}_3\end{array}\right.$$E1

Perri [5] proposed an unbiased estimator of the population mean ${\mathrm{\mu }}_{\mathrm{Y}}$

$${\widehat{\mathrm{\mu }}}_{\mathrm{P}}=\frac{{\overline{\mathrm{\theta }}}_{\mathrm{P}}‐\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\mu }}_{\mathrm{X}}}{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)}$$E2

with variance

$$\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{P}}\right)=\frac{{\mathrm{\sigma }}_{{\mathrm{\theta }}_{\mathrm{P}}}^2}{\mathrm{n}{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)}^2}$$E3

where ${\overline{\mathrm{\theta }}}_{\mathrm{P}}=1/\mathrm{n}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{\theta }}_{\mathrm{Pi}}$ and

$${\mathrm{\sigma }}_{{\mathrm{\theta }}_{\mathrm{P}}}^2=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\sigma }}_{\mathrm{Y}}^2+{\mathrm{\mu }}_{\mathrm{Y}}^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\sigma }}_{\mathrm{X}}^2+{\mathrm{\mu }}_{\mathrm{X}}^2\right)‐{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\mu }}_{\mathrm{Y}}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\mu }}_{\mathrm{X}}\right]}^2$$

Many different suggestions have been made for the use of these blank cards by various authors including Bhargava and Singh [6], Singh et al. [7], Batool et al. [8], Singh [9] and Singh et al. [10, 11] among others. Furthermore in addition, the theory of randomized response technique to estimate the population parameters of sensitive characteristics was extended by Narjis and Shabbir [12, 13].

Recently, Ahmed et al. [14] have introduced the idea to estimate the means of two quantitative sensitive variables simultaneously by using one scramble response and other face response. Let ${\mathrm{Y}}_{1\mathrm{i}}$ and ${\mathrm{Y}}_{2\mathrm{i}}$ be the two values of quantitative sensitive variables with means $\left({\mathrm{\mu }}_{\mathrm{Y}1},{\mathrm{\mu }}_{\mathrm{Y}2}\right)$ and variances $\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2,{\mathrm{\sigma }}_{\mathrm{Y}2}^2\right)$ respectively connected with the ${\mathrm{i}}^{\mathrm{th}}$ unit in the population$\mathrm{N}$. The parameters of interest are $\left({\mathrm{\mu }}_{\mathrm{Y}1},{\mathrm{\mu }}_{\mathrm{Y}2}\right)$ which are to be estimated. Each respondent selected in the sample with replacement is asked to produce two fake values of scramble variables ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ from two known distributions. Let ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ be the independent scramble variables with known means $\left({\mathrm{\theta }}_1,{\mathrm{\theta }}_2\right)$ and variance $\left({\mathrm{\gamma }}_{20},{\mathrm{\gamma }}_{02}\right)$ respectively, which help to maintain the protection of respondents. Ahmed et al. [14] defined the scramble response as:

$${\mathrm{Z}}_{1\mathrm{i}}={\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}$$E4

Each respondent selected in the sample is also requested to draw a card from the deck which consist two types of cards, similar to Warner [1] model but has different type of outcomes. Let $\mathrm{P}$ be the probability of cards bearing the statements in the deck, “the selected respondent to report scramble response as ${\mathrm{S}}_1$” and $\left(1‐\mathrm{P}\right)$ is the probability of cards bearing the statement in the deck, “the selected respondent to report scramble response as ${\mathrm{S}}_2$”. Thus, the second response from the ${\mathrm{i}}^{\mathrm{th}}$ respondent given as:

$${\mathrm{Z}}_{\mathrm{i}}=\left\{\begin{array}{l}{\mathrm{S}}_1\text{with}\text{probability}\mathrm{P}\\ {\mathrm{S}}_2\text{with}\text{probability}\left(1‐\mathrm{P}\right)\end{array}\right.$$E5

where $\mathrm{P}\ne \frac{{\mathrm{\theta }}_1{\mathrm{\gamma }}_{20}}{{\mathrm{\theta }}_1{\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_2{\mathrm{\gamma }}_{02}}$

Ahmed et al. [14] proposed unbiased estimators of population means ${\mathrm{\mu }}_{\mathrm{Y}1}$ and ${\mathrm{\mu }}_{\mathrm{Y}2}$ respectively, and are given as:

$${\widehat{\mathrm{\mu }}}_{\mathrm{AY}1}=\frac{\left[{\mathrm{P\theta }}_1{\mathrm{\theta }}_2+\left(1‐\mathrm{P}\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\overline{\mathrm{Z}}}_1‐{\mathrm{\theta }}_2{\overline{\mathrm{Z}}}_2}{\left(1‐\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐{\mathrm{P\theta }}_2{\mathrm{\gamma }}_{20}}$$E6

and

$${\widehat{\mathrm{\mu }}}_{\mathrm{AY}2}=\frac{{\mathrm{\theta }}_1{\overline{\mathrm{Z}}}_2‐\left[\mathrm{P}\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left(1‐\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]{\overline{\mathrm{Z}}}_1}{\left(1‐\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐{\mathrm{P\theta }}_2{\mathrm{\gamma }}_{20}}$$E7

where ${\overline{\mathrm{Z}}}_1=\frac{1}{\mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Z}}_{1\mathrm{i}}$ and ${\overline{\mathrm{Z}}}_2=\frac{1}{\mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Z}}_{2\mathrm{i}}$

In follow up of above works and motivated by Ahmed et al. [14], I adopt Perri [5] method and proposes a new improved randomized response model by introducing blank card option for estimation of population means of two quantitative sensitive variables. For example, ${\mathrm{Y}}_1$ may stand for the respondents’ income and ${\mathrm{Y}}_2$ may stand for the respondents’ expenditure, ${\mathrm{Y}}_1$ denotes the import in millions and ${\mathrm{Y}}_2$ denotes the export in millions etc. I have demonstrated the efficacious performance of the proposed randomized response model over the Ahmed et al. [14] model along with privacy protection of respondents.

2. Proposed model

In the proposed model, I have considered the similar supposition as it is the case of Ahmed et al. [14] procedure with the modification that the second response of Warner [1] method is replaced with Perri [5] blank card method. Proceeding on the lines of Ahmed et al. [14] as given in their model, the first observed response is given by:

$${\mathrm{Z}}_{1\mathrm{i}}={\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}$$E8

Noted that by mixing of two quantitative sensitive variables with two scramble variables will make more comfortable to respondent about providing information because it make very hard to guess the true value of two quantitative sensitive variables to an interviewer.

Here I differ from the existing randomized response model available in the literature, in that, the second response is replaced with Perri [5] procedure but has different outcomes. Each selected respondent in the sample provided a random device which consists three type of cards bearing the statements (i) green cards with the statement: report scramble variable ${\mathrm{S}}_1$, (ii) red cards with the statement: report scramble variable ${\mathrm{S}}_2$ and (iii) yellow card with no statement (blank cards) with probabilities ${\mathrm{P}}_1$, ${\mathrm{P}}_2$ and ${\mathrm{P}}_3$ respectively such that ${\sum }_{\mathrm{i}=1}^3{\mathrm{P}}_{\mathrm{i}}=1$. Thus, the second response ${\mathrm{Z}}_{\mathrm{Ai}}$ in the proposed model from ${\mathrm{i}}^{\mathrm{th}}$ respondent is given by:

$${\mathrm{Z}}_{\mathrm{Ai}}=\left\{\begin{array}{l}{\mathrm{S}}_1\text{with}\text{probability}{\mathrm{P}}_1\\ {\mathrm{S}}_2\text{with}\text{probability}{\mathrm{P}}_2\\ \text{Blank}card\text{with}\text{probability}{\mathrm{P}}_3\end{array}\right.$$E9

If a blank card is selected, the respondents are requested to use Ahmed et al. [14] second response. Thus, the second response ${\mathrm{Z}}_{\mathrm{Ai}}$ can be rewritten as:

$${\mathrm{Z}}_{\mathrm{Ai}}=\left\{\begin{array}{l}{\mathrm{S}}_1\text{with}\text{probability}{\mathrm{P}}_1\\ {\mathrm{S}}_2\text{with}\text{probability}{\mathrm{P}}_2\\ \left\{\begin{array}{l}{\mathrm{S}}_1\text{with}\text{probability}\mathrm{P}\\ {\mathrm{S}}_2\text{with}\text{probability}\left(1‐\mathrm{P}\right)\end{array}\right\}\text{with}\text{probability}{\mathrm{P}}_3\end{array}\right.$$E10

where $\mathrm{P}\ne \frac{\left({\mathrm{P}}_2+{\mathrm{P}}_3\right){\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐{\mathrm{P}}_2{\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}}{{\mathrm{P}}_3\left({\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right)}$

Taking expectation on both sides of (Eq. (8)), I have

$$\mathrm{E}\left({\mathrm{Z}}_{1\mathrm{i}}\right)=\mathrm{E}\left({\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\right)={\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}$$E11

With the help from Eqs. (8) and (10), I generate a new response ${\mathrm{Z}}_{2\mathrm{i}}^{\prime }$ as:

$${\mathrm{Z}}_{2\mathrm{i}}^{\prime }={\mathrm{Z}}_{1\mathrm{i}}{\mathrm{Z}}_{\mathrm{Ai}}=\left\{\begin{array}{l}{\mathrm{S}}_1^2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\text{with}\text{probability}{\mathrm{P}}_1\\ {\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}\text{with}\text{probability}{\mathrm{P}}_2\\ \left\{\begin{array}{l}{\mathrm{S}}_1^2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\text{with}\text{probability}\mathrm{P}\\ {\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}\text{with}\text{probability}\left(1‐\mathrm{P}\right)\end{array}\right\}\text{with}\text{probability}{\mathrm{P}}_3\end{array}\right.$$E12

Taking expectation on both sides of (Eq. (12)), I get

$$\mathrm{E}\left({\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left[\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right]+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left[{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right]$$E13

Using the method of moments on Eqs. (11) and (13), I have:

$${\mathrm{\theta }}_1{\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}=\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Z}}_{1\mathrm{i}}$$E14

and

$$\begin{array}{l}\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]{\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}\\ +\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}=\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Z}}_{2\mathrm{i}}^{\prime }\end{array}$$E15

Eqs. (14) and (15) can be rewritten as:

$$\left[\begin{array}{l}{\mathrm{\theta }}_1,{\mathrm{\theta }}_2\\ \left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2\\ +\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2,+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\end{array}\right]\left[\begin{array}{l}{\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}\\ {\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}\end{array}\right]=\left[\begin{array}{l}{\overline{\mathrm{Z}}}_1\\ {\overline{\mathrm{Z}}}_2^{\prime }\end{array}\right]$$E16

Applying Cramer’s rule on Eq. (16), I obtain

$$\mathrm{\Delta }=\left|\begin{array}{l}{\mathrm{\theta }}_1,{\mathrm{\theta }}_2\\ \left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2,\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\end{array}\right|$$

$$\begin{array}{l}={\mathrm{\theta }}_1\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ ‐{\mathrm{\theta }}_2\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]\end{array}$$

$$=\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}$$E17

$${\mathrm{\Delta }}_1=\left|\begin{array}{l}{\overline{\mathrm{Z}}}_1,{\mathrm{\theta }}_2\\ {\overline{\mathrm{Z}}}_2^{\prime },\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\end{array}\right|$$

$$=\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\overline{\mathrm{Z}}}_1‐{\mathrm{\theta }}_2{\overline{\mathrm{Z}}}_2^{\prime }$$E18

and

$${\mathrm{\Delta }}_2=\left|\begin{array}{l}{\mathrm{\theta }}_1{\overline{\mathrm{Z}}}_1\\ \left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2,{\overline{\mathrm{Z}}}_2^{\prime }\end{array}\right|$$

$$={\mathrm{\theta }}_1{\overline{\mathrm{Z}}}_2^{\prime }‐\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]{\overline{\mathrm{Z}}}_1$$E19

Thus, the estimators of the population mean ${\mathrm{\mu }}_{\mathrm{Y}1}$ and ${\mathrm{\mu }}_{\mathrm{Y}2}$ are respectively given by:

$${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}=\frac{{\mathrm{\Delta }}_1}{\mathrm{\Delta }}=\frac{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\overline{\mathrm{Z}}}_1‐{\mathrm{\theta }}_2{\overline{\mathrm{Z}}}_2^{\prime }}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}$$E20

and

$${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}=\frac{{\mathrm{\Delta }}_2}{\mathrm{\Delta }}=\frac{{\mathrm{\theta }}_1{\overline{\mathrm{Z}}}_2^{\prime }‐\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]{\overline{\mathrm{Z}}}_1}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}$$E21

I have the following theorems.

Theorem 1:${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ is an unbiased estimator of the population mean ${\mathrm{\mu }}_{\mathrm{Y}1}$.

$$\mathrm{E}\left({\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}\right)={\mathrm{\mu }}_{\mathrm{Y}1}$$E22

Proof: Taking expectation on both sides of Eq. (20), I have

$$\mathrm{E}\left({\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}\right)=\frac{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\mathrm{E}\left({\overline{\mathrm{Z}}}_1\right)‐{\mathrm{\theta }}_2\mathrm{E}\left({\overline{\mathrm{Z}}}_2^{\prime }\right)}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}$$

$$=\frac{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]1/\mathrm{n}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{E}\left({\mathrm{Z}}_{1\mathrm{i}}\right)‐{\mathrm{\theta }}_{2}1/\mathrm{n}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{E}\left({\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}$$

$$=\frac{\begin{array}{l}\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\sum _{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right)\\ ‐{\mathrm{\theta }}_2{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]\end{array}}{\mathrm{n}\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}$$

After simplification, I get

$$=\frac{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]{\mathrm{\mu }}_{\mathrm{Y}1}}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}={\mathrm{\mu }}_{\mathrm{Y}1}.$$

which completes the proof.

Theorem 2:${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ is an unbiased estimator of the population mean ${\mathrm{\mu }}_{\mathrm{Y}2}$.

$$\mathrm{E}\left({\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}\right)={\mathrm{\mu }}_{\mathrm{Y}2}$$E23

Proof: Taking expectation on both sides of Eq. (21), I have

$${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}=\frac{{\mathrm{\theta }}_1\mathrm{E}\left({\overline{\mathrm{Z}}}_2^{\prime }\right)‐\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]\mathrm{E}\left({\overline{\mathrm{Z}}}_1\right)}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}$$

Similarly, following the pattern as given in Theorem 1, I obtain

$$=\frac{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]{\mathrm{\mu }}_{\mathrm{Y}2}}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}={\mathrm{\mu }}_{\mathrm{Y}2}.$$

hence, it is proved.

Theorem 3: The variance of the unbiased estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ is given by:

$$\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}\right)=\frac{\begin{array}{l}{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_1}^2+{\mathrm{\theta }}_2^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2\\ ‐2{\mathrm{\theta }}_2\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2^{\prime }}\end{array}}{\mathrm{n}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}^2}$$E24

where ${\mathrm{\sigma }}_{{\mathrm{Z}}_1}^2={\mathrm{\gamma }}_{20}\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)+{\mathrm{\gamma }}_{02}\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)+{\mathrm{\theta }}_1^2{\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\theta }}_2^2{\mathrm{\sigma }}_{\mathrm{Y}2}^2+2{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}$,

$$\begin{array}{l}{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2=\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{40}+4{\mathrm{\gamma }}_{30}{\mathrm{\theta }}_1+6{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1^2+{\mathrm{\theta }}_1^4\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ +\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{04}+4{\mathrm{\gamma }}_{03}{\mathrm{\theta }}_2+6{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2^2+{\mathrm{\theta }}_2^4\right)+\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ +2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\right]\\ ‐\left[\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right\}{\mathrm{\mu }}_{\mathrm{Y}1}\right.\\ {\left.+\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right\}{\mathrm{\mu }}_{\mathrm{Y}2}\right]}^2\end{array}$$

and

$$\begin{array}{l}{\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2^{\prime }}=\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\right]\\ +\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\right]\\ +2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ ‐\left({\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}12}\right)\left[\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right\}{\mathrm{\mu }}_{\mathrm{Y}1}\right.\\ \left.+\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right\}{\mathrm{\mu }}_{\mathrm{Y}2}\right]\end{array}$$

The proof is given in Appendix.

Theorem 4: The variance of the unbiased estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ is given by:

$$\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}\right)=\frac{\begin{array}{l}{\mathrm{\theta }}_1^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2+{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_1}^2\\ ‐2{\mathrm{\theta }}_1\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right]{\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2}\end{array}}{\mathrm{n}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}^2}$$E25

Proof: The proof is similar as given in Theorem 3.

In the next section, I discuss a privacy protection measure to compare the respondent’s privacy protection and efficiency for the considered model.

3. Privacy protection measure

A number of measures have been introduced in the literature to estimate the performance of competitive strategies taking into account both efficiency and respondent privacy protection. For a discussion on privacy protection measures for randomized response survey of stigmatizing character, see Lanke [15], Leyseiffer and Warner [16], Bhargava and Singh [17] and among other. These measures of privacy protection are based on the qualitative characters.

When dealing with quantitative sensitive variable, the respondent privacy is conserved by asking interviewees to algebraically scramble the true response by means of a coding mechanism. Respondent’s privacy protection measures for quantitative sensitive variable have been investigated by Diana and perri [18] and Zhimin et al. [19] which is based on the square of correlation coefficient i.e. ${\mathrm{\rho }}_{\mathrm{y\theta }}^2\in \left[0,1\right]$. Later, Diana and Perri [20] introduced the new measure of privacy protection of respondents by using auxiliary variable. These measures are normalized with zero (one) denoting maximum (minimum) privacy protection. Recently, Singh et al. [11] considered the case when no auxiliary variable is available in the procedure and studied the normalized measure of respondent privacy. This normalized measure allows researchers to attain a trade-off between efficiency and privacy. Moreover, it is worth remarking that if one procedure is more efficient than other, then it will be less protective. Thus, all the provided measures using the randomized procedure for the privacy protection, they have concluded for a measure of respondent’s privacy protection having a trade-off between these two aspects.

$$\mathrm{\tau }=1‐{\mathrm{\rho }}_{\mathrm{y\theta }}^2$$E26

The values of $\mathrm{\tau }$ closer to 1 indicates more privacy protection and greater cooperation may be expected using randomized response models while $\mathrm{\tau }$ closer to zero denotes that the privacy protection is completely violated. Now, I use this normalized measure for comparing the trade-off between efficiency and privacy protection.

In the proposed model, there are two quantitative sensitive variables ${\mathrm{Y}}_{1\mathrm{i}}$ and ${\mathrm{Y}}_{2\mathrm{i}}$ associated with the second observed response ${\mathrm{Z}}_{2\mathrm{i}}^{\prime }$. Following Section 2, I compute the square of correlation coefficients between the second observed response ${\mathrm{Z}}_{2\mathrm{i}}^{\prime }$ and quantitative sensitive variables ${\mathrm{Y}}_{1\mathrm{i}}$ and ${\mathrm{Y}}_{2\mathrm{i}}$ respectively, and are given as:

$${\mathrm{\rho }}_{{\mathrm{y}}_{1\mathrm{i}}{\mathrm{Z}}_{2\mathrm{i}}^{\prime }}^2=\frac{{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}^2+\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_2^2\right){\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}\right\}\right]}^2}{{\mathrm{\sigma }}_{\mathrm{Y}1}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2}$$E27

and

$${\mathrm{\rho }}_{{\mathrm{y}}_{2\mathrm{i}}{\mathrm{Z}}_{2\mathrm{i}}^{\prime }}^2=\frac{{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}2}^2\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_2^2\right){\mathrm{\sigma }}_{\mathrm{Y}2}^2\right\}\right]}^2}{{\mathrm{\sigma }}_{\mathrm{Y}2}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2}$$E28

where ${\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2$ is given in Theorem 3.

Now, I define the measure of respondent’s privacy protection associated with the proposed second response ${\mathrm{Z}}_{2\mathrm{i}}^{\prime }$ as:

$${\mathrm{\tau }}_{\mathrm{PJ}}=1‐{\mathrm{\rho }}_{{\mathrm{y}}_{\mathrm{Ji}}{\mathrm{Z}}_{2\mathrm{i}}^{\prime }}^2,\mathrm{J}=1,2$$E29

I also define the square of correlation coefficients for the Ahmed et al. [14] response model. In the case of Ahmed et al. [14] model, there are also two quantitative sensitive variables associated with the second observed response. Thus, the square of correlation coefficients between the second observed response and quantitative sensitive variables ${\mathrm{Y}}_{1\mathrm{i}}$ and ${\mathrm{Y}}_{2\mathrm{i}}$ are respectively given by:

$${\mathrm{\rho }}_{{\mathrm{y}}_{1\mathrm{i}}{\mathrm{Z}}_{2\mathrm{i}}}^2=\frac{{\left[\mathrm{P}\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}\right\}+\left(1‐\mathrm{P}\right)\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}^2+\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_2^2\right){\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}\right\}\right]}^2}{{\mathrm{\sigma }}_{\mathrm{Y}1}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2}^2}$$E30

$${\mathrm{\rho }}_{{\mathrm{y}}_{2\mathrm{i}}{\mathrm{Z}}_{2\mathrm{i}}}^2=\frac{{\left[\mathrm{P}\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}2}^2\right\}+\left(1‐\mathrm{P}\right)\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_2^2\right){\mathrm{\sigma }}_{\mathrm{Y}2}^2\right\}\right]}^2}{{\mathrm{\sigma }}_{\mathrm{Y}2}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2}^2}$$E31

where

$$\begin{array}{l}{\mathrm{\sigma }}_{{\mathrm{Z}}_2}^2=\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)\left[\mathrm{P}\left({\mathrm{\gamma }}_{40}+4{\mathrm{\gamma }}_{30}{\mathrm{\theta }}_1+6{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1^2+{\mathrm{\theta }}_1^4\right)+\left(1‐\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ +\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\left[\left(1‐\mathrm{P}\right)\left({\mathrm{\gamma }}_{04}+4{\mathrm{\gamma }}_{03}{\mathrm{\theta }}_2+6{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2^2+{\mathrm{\theta }}_2^4\right)+\mathrm{P}\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ +2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\left[{\mathrm{P\theta }}_2\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right)+\left(1‐\mathrm{P}\right){\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\right]\\ ‐{\left[\left\{\mathrm{P}\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left(1‐\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2\right\}{\mathrm{\mu }}_{\mathrm{Y}1}+\left\{{\mathrm{P\theta }}_1{\mathrm{\theta }}_2+\left(1‐\mathrm{P}\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right\}{\mathrm{\mu }}_{\mathrm{Y}2}\right]}^2\end{array}$$

I also define the measure of respondent’s privacy protection for Ahmed et al. [14] as:

$${\mathrm{\tau }}_{\mathrm{AJ}}=1‐{\mathrm{\rho }}_{{\mathrm{y}}_{\mathrm{Ji}}{\mathrm{Z}}_{2\mathrm{i}}}^2,\mathrm{J}=1,2$$E32

In the next section, I investigate the performance of the proposed model with respect to Ahmed et al. [14] model in terms of relative efficiency and privacy protection under different parametric situations.

4. Efficiency vs privacy protection

The relative efficiency (RE) of the proposed estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ and ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ over Ahmed et al. [14] the estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{AY}1}$ and ${\widehat{\mathrm{\mu }}}_{\mathrm{AY}2}$ are respectively given by:

$${\mathrm{RE}}_{\mathrm{J}}\left({\widehat{\mathrm{\mu }}}_{\mathrm{YJ}},{\widehat{\mathrm{\mu }}}_{\mathrm{AYJ}}\right)=\frac{\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{AYJ}}\right)}{\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{YJ}}\right)},\mathrm{J}=1,2$$E33

To have a possible trade-off between relative efficiency and privacy protection of respondents, I consider the parametric values in this manner that the relative efficiencies are maximum and expect greater privacy protection of respondents. I decided to take $\mathrm{P}=0.6$, ${\mathrm{\mu }}_{\mathrm{Y}1}=$ 25–45 with a step 5 and ${\mathrm{\mu }}_{\mathrm{Y}2}=$ 35–55 with a step 5, five values of ${\mathrm{\theta }}_1$ and ${\mathrm{\theta }}_2$, equal to 2–10 and 4–16 with a increment 2 and 3 respectively, ${\mathrm{\sigma }}_{\mathrm{Y}1}=7$, ${\mathrm{\sigma }}_{\mathrm{Y}2}=5$, ${\mathrm{\gamma }}_{20}=2$, ${\mathrm{\gamma }}_{02}=9$, ${\mathrm{\gamma }}_{30}=1.5$, ${\mathrm{\gamma }}_{03}=1.2$, ${\mathrm{\gamma }}_{40}=3.2$ and ${\mathrm{\gamma }}_{04}=3.5$. I have also chosen different values of probabilities ${\mathrm{P}}_{\mathrm{i}}\left(\mathrm{i}=1,2,3\right)$ and presented in Tables 1 and 2.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	${\mathrm{RE}}_1$	${\mathrm{\tau }}_{\mathrm{P}1}$	${\mathrm{\tau }}_{\mathrm{A}1}$
0.20	0.20	0.60	2	4	25	35	2.03	0.9762	0.9765
			4	7	30	40	1.72	0.9572	0.9570
			6	10	35	45	1.65	0.9399	0.9395
			8	13	40	50	1.62	0.9277	0.9272
			10	16	45	55	1.60	0.9206	0.9201
0.15	0.15	0.70	2	4	25	35	1.74	0.9763	0.9765
			4	7	30	40	1.52	0.9572	0.9570
			6	10	35	45	1.47	0.9398	0.9395
			8	13	40	50	1.45	0.9276	0.9272
			10	16	45	55	1.44	0.9205	0.9201
0.10	0.10	0.80	2	4	25	35	1.47	0.9764	0.9765
			4	7	30	40	1.33	0.9571	0.9570
			6	10	35	45	1.30	0.9397	0.9395
			8	13	40	50	1.29	0.9275	0.9272
			10	16	45	55	1.28	0.9204	0.9201
0.05	0.05	0.90	2	4	25	35	1.22	0.9764	0.9765
			4	7	30	40	1.16	0.9571	0.9570
			6	10	35	45	1.15	0.9396	0.9395
			8	13	40	50	1.14	0.9274	0.9272
			10	16	45	55	1.13	0.9203	0.9201

Table 1.

Relative efficiency of the proposed estimator ${\widehat{\mathit{\mu }}}_{\mathit{Y}\mathit{1}}$ with respect to Ahmed et al. [14] estimator ${\widehat{\mathit{\mu }}}_{\mathit{AY}\mathit{1}}$ and privacy protection of the ${\mathit{\tau }}_{\mathit{P}\mathit{1}}$ and ${\mathit{\tau }}_{\mathit{A}\mathit{1}}$.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	${\mathrm{RE}}_2$	${\mathrm{\tau }}_{\mathrm{P}2}$	${\mathrm{\tau }}_{\mathrm{A}2}$
0.20	0.20	0.60	2	4	25	35	1.98	0.9762	0.9765
			4	7	30	40	1.70	0.9572	0.9570
			6	10	35	45	1.64	0.9399	0.9395
			8	13	40	50	1.62	0.9277	0.9272
			10	16	45	55	1.60	0.9206	0.9201
0.15	0.15	0.70	2	4	25	35	1.71	0.9763	0.9765
			4	7	30	40	1.51	0.9572	0.9570
			6	10	35	45	1.47	0.9398	0.9395
			8	13	40	50	1.45	0.9276	0.9272
			10	16	45	55	1.44	0.9205	0.9201
0.10	0.10	0.80	2	4	25	35	1.45	0.9764	0.9765
			4	7	30	40	1.33	0.9571	0.9570
			6	10	35	45	1.30	0.9397	0.9395
			8	13	40	50	1.29	0.9275	0.9272
			10	16	45	55	1.28	0.9204	0.9201
0.05	0.05	0.90	2	4	25	35	1.21	0.9764	0.9765
			4	7	30	40	1.16	0.9571	0.9570
			6	10	35	45	1.15	0.9396	0.9395
			8	13	40	50	1.14	0.9274	0.9272
			10	16	45	55	1.13	0.9203	0.9201

Table 2.

Relative efficiency of the proposed estimator ${\widehat{\mathit{\mu }}}_{\mathit{Y}\mathit{2}}$ with respect to Ahmed et al. [14] estimator ${\widehat{\mathit{\mu }}}_{\mathit{AY}\mathit{2}}$ and privacy protection of the ${\mathit{\tau }}_{\mathit{P}\mathit{2}}$ and ${\mathit{\tau }}_{\mathit{A}\mathit{2}}$.

Tables 1 and 2 show how the proposed model works in term of efficiency along with privacy protection. For the situation under investigate it emerges that the proposed model based on blank card method is more efficient than Ahmed et al. [14] model. Hence, the finding results, which are worth discussing, are described in the following points.

1. It is observed from Tables 1 and 2 that the proposed estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{YJ}}$$\left(\mathrm{J}=1,2\right)$ almost equally efficient in term of efficiencies and privacy protection of respondents.

2. From Tables 1 and 2, it may be observed that the values of relative efficiencies of the proposed estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{YJ}}$$\left(\mathrm{J}=1,2\right)$ with respect to Ahmed et al. [14] estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{AYJ}}$$\left(\mathrm{J}=1,2\right)$ are more than 1 for all case.

3. The behaviour of the estimators in Tables 1 and 2, indicates that the highest efficiency 2.03 attains when ${\mathrm{P}}_{\mathrm{i}}=0.20$$\left(\mathrm{i}=1,2\right)$ and ${\mathrm{P}}_{\mathrm{i}}=0.60$ with corresponding values of ${\mathrm{\theta }}_1=2$ and ${\mathrm{\theta }}_2=4$ while the minimum efficiency 1.13 attains when ${\mathrm{P}}_{\mathrm{i}}=0.05$$\left(\mathrm{i}=1,2\right)$ and ${\mathrm{P}}_{\mathrm{i}}=0.90$ with corresponding values of ${\mathrm{\theta }}_1=10$ and ${\mathrm{\theta }}_2=16$.

4. It is also seen that the values of relative efficiencies are decreasing as the values of ${\mathrm{\theta }}_{\mathrm{J}}$ and ${\mathrm{\mu }}_{\mathrm{YJ}}$$\left(\mathrm{J}=1,2\right)$ increase for the fix values of ${\mathrm{P}}_{\mathrm{i}}$$\left(\mathrm{i}=1,2,3\right)$.

5. However, it is observed that for the fix values of ${\mathrm{\theta }}_1$, ${\mathrm{\theta }}_2$, ${\mathrm{\mu }}_{\mathrm{Y}1}$ and ${\mathrm{\mu }}_{\mathrm{Y}2}$ the values of ${\mathrm{RE}}_{\mathrm{i}}$$\left(\mathrm{i}=1,2\right)$ are decreasing as the values of ${\mathrm{P}}_{\mathrm{i}}$$\left(\mathrm{i}=1,2\right)$ decrease.

6. Furthermore, it can be interpreted that with the increase in the values of ${\mathrm{P}}_{\mathrm{i}}$$\left(\mathrm{i}=1,2\right)$ while decrease in the values of ${\mathrm{P}}_3$, ${\mathrm{\theta }}_1$, ${\mathrm{\theta }}_2$, ${\mathrm{\mu }}_{\mathrm{Y}1}$ and ${\mathrm{\mu }}_{\mathrm{Y}2}$ there is a increasing pattern in the values of ${\mathrm{RE}}_{\mathrm{i}}$$\left(\mathrm{i}=1,2\right)$.

7. From Tables 1 and 2, it is clear that the measure of privacy protection of proposed randomized response model and Ahmed et al. [14] model is closer to one for all the cases, which indicate maximum privacy protection of respondents.

8. It is further observed that the degree of privacy protection of proposed randomized response model and Ahmed et al. [14] model decreasing with the values of ${\mathrm{\theta }}_{\mathrm{J}}$ and ${\mathrm{\mu }}_{\mathrm{YJ}}$$\left(\mathrm{J}=1,2\right)$ increase.

9. From Tables 1 and 2, it may also be seen that the value of respondent’s privacy protection are showing an increasing trend with the increase in the values of ${\mathrm{P}}_{\mathrm{i}}$$\left(\mathrm{i}=1,2\right)$ while decrease in the values of ${\mathrm{P}}_3$, ${\mathrm{\theta }}_1$, ${\mathrm{\theta }}_2$, ${\mathrm{\mu }}_{\mathrm{Y}1}$ and ${\mathrm{\mu }}_{\mathrm{Y}2}$.

10. However, it is visible that the proposed model is more efficient than Ahmed et al. [14] model but less protective only when ${\mathrm{\theta }}_1=2$ and ${\mathrm{\theta }}_2=4$. The model which provides more efficiency yields less privacy protection. Hence, I conclude that

$$\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{AYJ}}\right)>\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{YJ}}\right),\mathrm{J}=1,2$$

and

$${\mathrm{\tau }}_{\mathrm{AJ}}>{\mathrm{\tau }}_{\mathrm{PJ}},\mathrm{J}=1,2$$

Hence, I conclude that small difference in efficiency may procure substantial improvement in privacy protection of respondent. Thus, our comparisons underline the good performance, in terms of efficiency and respondent’s privacy protection.

1. 11.

   Therefore, the proposed randomized response model under the blank card method may be declared to be best for estimating the mean of two quantitative sensitive variables and thus may be recommended to the survey practitioners whenever they deal with extremely sensitive characteristics.

To judge the performance of the proposed model, I consider a real data CO124 of N = 124 units of Sarndal et al. [21]. A random sample of size n = 30 units are drawn from the CO124 population. Let ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$ and $\mathrm{X}$ be the import, export and military expenditure in the state of U.S. during the year 1983, 1983 and 1981 respectively. The parametric ranges of quantitative sensitive variables ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$ and non-sensitive variable $\mathrm{X}$ have been found by using t–test and chi–square test, which are terms as ${\mathrm{\mu }}_{\mathrm{Y}1}\in \left(331.60,567.66\right)$, ${\mathrm{\mu }}_{\mathrm{Y}2}\in \left(242.56,440.30\right)$, ${\mathrm{\mu }}_{\mathrm{X}}\in \left(43,171.16\right)$, ${\mathrm{\sigma }}_{\mathrm{Y}1}\in \left(256.03,432.17\right)$, ${\mathrm{\sigma }}_{\mathrm{Y}2}\in \left(214.47,362.02\right)$ and ${\mathrm{\sigma }}_{\mathrm{X}}\in \left(138.90,234.46\right)$. The relative efficiencies have been computed for these parameters combinations and presented in Tables 3–7.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\sigma }}_{\mathrm{Y}1}$	${\mathrm{\sigma }}_{\mathrm{Y}2}$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	$\mathrm{P}$
									0.1	0.2	0.3	0.4	0.5
0.20	0.7	0.10	4	1	260	220	340	250	5.90	5.48	5.10	4.77	4.46
							390	300	5.87	5.45	5.08	4.75	4.45
							440	350	5.86	5.45	5.08	4.76	4.46
							490	400	5.86	5.46	5.10	4.78	4.49
							540	440	5.94	5.54	5.18	4.86	4.56
					310	270	340	250	5.29	4.93	4.61	4.32	4.06
							390	300	5.31	4.95	4.63	4.34	4.08
							440	350	5.35	4.99	4.67	4.38	4.12
							490	400	5.40	5.04	4.72	4.43	4.17
							540	440	5.51	5.14	4.82	4.53	4.26
					360	320	340	250	4.84	4.53	4.24	3.99	3.76
							390	300	4.89	4.57	4.29	4.03	3.80
							440	350	4.95	4.63	4.35	4.09	3.86
							490	400	5.02	4.70	4.41	4.15	3.92
							540	440	5.14	4.81	4.52	4.25	4.01
0.15	0.75	0.10	4	1	260	220	340	250	9.17	8.31	7.57	6.94	6.39
							390	300	9.03	8.20	7.49	6.87	6.34
							440	350	8.93	8.13	7.44	6.84	6.32
							490	400	8.87	8.09	7.41	6.83	6.31
							540	440	8.92	8.15	7.48	6.90	6.39
					310	270	340	250	8.04	7.32	6.70	6.16	5.70
							390	300	8.02	7.31	6.70	6.18	5.72
							440	350	8.03	7.33	6.73	6.21	5.75
							490	400	8.06	7.37	6.77	6.25	5.80
							540	440	8.17	7.48	6.89	6.37	5.91
					360	320	340	250	7.23	6.60	6.07	5.60	5.20
							390	300	7.26	6.65	6.11	5.65	5.25
							440	350	7.32	6.71	6.18	5.71	5.31
							490	400	7.40	6.78	6.25	5.79	5.38
							540	440	7.54	6.92	6.38	5.92	5.50
0.10	0.80	0.10	4	1	260	220	340	250	17.10	14.77	12.92	11.42	10.20
							390	300	16.59	14.38	12.62	11.19	10.02
							440	350	16.16	14.06	12.38	11.01	9.88
							490	400	15.81	13.80	12.19	10.88	9.79
							540	440	15.69	13.75	12.18	10.90	9.83
					310	270	340	250	14.60	12.68	11.16	9.92	8.90
							390	300	14.40	12.54	11.06	9.86	8.86
							440	350	14.24	12.44	11.00	9.83	8.85
							490	400	14.12	12.38	10.97	9.82	8.86
							540	440	14.16	12.45	11.06	9.92	8.97
					360	320	340	250	12.80	11.18	9.89	8.83	7.96
							390	300	12.75	11.17	9.89	8.85	7.99
							440	350	12.74	11.18	9.93	8.90	8.04
							490	400	12.75	11.22	9.98	8.96	8.11
							540	440	12.88	11.36	10.13	9.11	8.26
0.05	0.85	0.10	4	1	260	220	340	250	49.10	37.34	29.54	24.09	20.10
							390	300	46.49	35.56	28.29	23.17	19.43
							440	350	44.15	33.98	27.18	22.38	18.85
							490	400	42.09	32.61	26.23	21.71	18.37
							540	440	40.76	31.78	25.71	21.38	18.16
					310	270	340	250	40.46	31.02	24.73	20.30	17.06
							390	300	39.09	30.12	24.14	19.89	16.77
							440	350	37.84	29.32	23.59	19.53	16.53
							490	400	36.70	28.60	23.13	19.23	16.34
							540	440	36.03	28.23	22.94	19.16	16.34
					360	320	340	250	34.24	26.47	21.26	17.58	14.86
							390	300	34.55	26.05	21.00	17.42	14.77
							440	350	32.92	25.67	20.79	17.30	14.72
							490	400	32.34	25.35	20.61	17.22	14.69
							540	440	32.07	25.27	20.63	17.30	14.81
0.10	0.85	0.05	4	1	260	220	340	250	18.50	17.10	15.86	14.77	13.79
							390	300	17.92	16.59	15.42	14.38	13.45
							440	350	17.42	16.16	15.05	14.06	13.18
							490	400	17.01	15.81	14.75	13.80	12.96
							540	440	16.85	15.69	14.67	13.75	12.93
					310	270	340	250	15.75	14.60	13.58	12.68	11.88
							390	300	15.51	14.40	13.42	12.54	11.76
							440	350	15.31	14.24	13.29	12.44	11.69
							490	400	15.16	14.12	13.20	12.38	11.64
							540	440	15.18	14.16	13.26	12.45	11.72
					360	320	340	250	13.76	12.80	11.94	11.18	10.50
							390	300	13.70	12.75	11.91	11.17	10.50
							440	350	13.67	12.74	11.92	11.18	10.52
							490	400	13.66	12.75	11.94	11.22	10.57
							540	440	13.79	12.88	12.08	11.36	10.71

Table 3.

Relative efficiency of the proposed estimator ${\widehat{\mathit{\mu }}}_{\mathit{Y}\mathit{1}}$ with respect to Perri [5] estimator ${\widehat{\mathit{\mu }}}_{\mathit{P}}$ when ${\mathit{\theta }}_{\mathit{1}}=\mathit{4}$ and ${\mathit{\theta }}_{\mathit{2}}=\mathit{1}$.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\sigma }}_{\mathrm{Y}1}$	${\mathrm{\sigma }}_{\mathrm{Y}2}$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	$\mathrm{P}$
									0.1	0.2	0.3	0.4	0.5
0.20	0.70	0.10	4	1	260	220	340	250	1.58	1.44	1.31	1.20	1.10
							390	300	1.49	1.36	1.24	1.14	1.05
							440	350	1.43	1.31	1.20	1.10	1.01
							490	400	1.39	1.27	1.16	1.07	0.98
							540	440	1.35	1.23	1.13	1.04	0.96
					310	270	340	250	1.56	1.42	1.31	1.20	1.11
							390	300	1.49	1.36	1.25	1.15	1.06
							440	350	1.43	1.31	1.20	1.11	1.02
							490	400	1.40	1.28	1.17	1.08	1.00
							540	440	1.36	1.24	1.14	1.05	0.97
					360	320	340	250	1.54	1.41	1.30	1.20	1.11
							390	300	1.48	1.36	1.25	1.15	1.07
							440	350	1.43	1.32	1.21	1.12	1.04
							490	400	1.40	1.29	1.18	1.09	1.01
							540	440	1.36	1.25	1.15	1.07	0.99
0.15	0.75	0.10	4	1	260	220	340	250	2.69	2.39	2.14	1.93	1.74
							390	300	2.52	2.25	2.02	1.82	1.64
							440	350	2.40	2.14	1.93	1.74	1.57
							490	400	2.31	2.07	1.86	1.68	1.53
							540	440	2.22	2.00	1.83	1.63	1.48
					310	270	340	250	2.59	2.32	2.08	1.88	1.71
							390	300	2.46	2.20	1.98	1.79	1.63
							440	350	2.36	2.11	1.91	1.73	1.57
							490	400	2.28	2.05	1.85	1.68	1.53
							540	440	2.21	1.99	1.80	1.63	1.48
					360	320	340	250	2.51	2.25	2.04	1.85	1.68
							390	300	2.40	2.16	1.95	1.77	1.62
							440	350	2.32	2.09	1.89	1.72	1.57
							490	400	2.26	2.03	1.84	1.67	1.53
							540	440	2.19	1.98	1.79	1.63	1.49
0.10	0.80	0.10	4	1	260	220	340	250	5.46	4.64	3.99	3.47	3.04
							390	300	5.05	4.30	3.71	3.24	2.85
							440	350	4.73	4.05	3.50	3.06	2.70
							490	400	4.49	3.85	3.35	2.93	2.59
							540	440	4.27	3.68	3.20	2.82	2.49
					310	270	340	250	5.11	4.37	3.78	3.31	2.92
							390	300	4.79	4.11	3.57	3.12	2.76
							440	350	4.55	3.91	3.40	2.99	2.64
							490	400	4.35	3.75	3.27	2.88	2.55
							540	440	4.17	3.60	3.15	2.78	2.47
					360	320	340	250	4.81	4.14	3.60	3.17	2.81
							390	300	4.57	3.94	3.44	3.03	2.69
							440	350	4.38	3.78	3.30	2.91	2.59
							490	400	4.22	3.66	3.20	2.83	2.52
							540	440	4.07	3.53	3.10	2.74	2.44
0.05	0.85	0.10	4	1	260	220	340	250	16.98	12.71	9.90	7.95	6.53
							390	300	15.32	11.54	9.04	7.28	6.01
							440	350	14.00	10.61	8.36	6.77	5.61
							490	400	12.95	9.88	7.82	6.37	5.30
							540	440	12.03	9.23	7.35	6.01	5.03
					310	270	340	250	15.28	11.54	9.06	7.33	6.06
							390	300	14.06	10.67	8.42	6.83	5.67
							440	350	13.06	9.97	7.90	6.44	5.36
							490	400	12.23	9.39	7.48	6.12	5.12
							540	440	11.49	8.86	7.09	5.83	4.89
					360	320	340	250	13.85	10.56	8.36	6.81	5.67
							390	300	12.96	9.92	7.88	6.44	5.38
							440	350	12.20	9.38	7.48	6.13	5.14
							490	400	11.56	8.93	7.15	5.88	4.94
							540	440	10.96	8.50	6.84	5.64	4.76
0.10	0.85	0.05	4	1	260	220	340	250	5.96	5.46	5.02	4.64	4.30
							390	300	5.50	5.05	4.65	4.30	3.99
							440	350	5.14	4.73	4.37	4.05	3.76
							490	400	4.87	4.49	4.15	3.85	3.59
							540	440	4.62	4.27	3.96	3.68	3.43
					310	270	340	250	5.55	5.11	4.71	4.37	4.06
							390	300	5.20	4.79	4.43	4.11	3.82
							440	350	4.93	4.55	4.21	3.91	3.64
							490	400	4.71	4.35	4.03	3.75	3.50
							540	440	4.51	4.17	3.87	3.60	3.37
					360	320	340	250	5.21	4.81	4.45	4.14	3.86
							390	300	4.95	4.57	4.24	3.94	3.67
							440	350	4.73	4.38	4.06	3.78	3.53
							490	400	4.56	4.22	3.92	3.66	3.42
							540	440	4.39	4.07	3.79	3.53	3.31

Table 4.

Relative efficiency of the proposed estimator ${\widehat{\mathit{\mu }}}_{\mathit{Y}\mathit{2}}$ with respect to Perri [5] estimator ${\widehat{\mathit{\mu }}}_{\mathit{P}}$ when ${\mathit{\theta }}_{\mathit{1}}=\mathit{4}$ and ${\mathit{\theta }}_{\mathit{2}}=\mathit{1}$.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\sigma }}_{\mathrm{Y}1}$	${\mathrm{\sigma }}_{\mathrm{Y}2}$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	$\mathrm{P}$
									0.1	0.3	0.5	0.7	0.9
0.20	0.7	0.10	4	1	260	220	340	250	0.98	1.01	1.11	1.83	1589.20
							390	300	0.98	1.02	1.12	1.89	1728.00
							440	350	0.98	1.02	1.13	1.95	1854.80
							490	400	0.98	1.02	1.14	2.00	1968.00
							540	440	0.98	1.02	1.15	2.05	2064.70
					310	270	340	250	0.98	1.01	1.10	1.74	1446.10
							390	300	0.98	1.01	1.11	1.80	1568.00
							440	350	0.98	1.01	1.12	1.86	1684.80
							490	400	0.98	1.01	1.13	1.91	1793.90
							540	440	0.98	1.01	1.13	1.96	1890.00
					360	320	340	250	0.99	1.01	1.09	1.68	1343.80
							390	300	0.99	1.01	1.10	1.73	1448.40
							440	350	0.99	1.01	1.11	1.78	1552.40
							490	400	0.99	1.01	1.12	1.83	1653.10
							540	440	0.99	1.01	1.12	1.88	1744.10
0.15	0.75	0.10	4	1	260	220	340	250	0.99	1.02	1.12	1.84	1605.30
							390	300	0.99	1.02	1.13	1.91	1746.40
							440	350	0.99	1.02	1.14	1.97	1875.50
							490	400	0.99	1.02	1.15	2.02	1990.80
							540	440	0.99	1.02	1.16	2.07	2089.90
					310	270	340	250	0.99	1.01	1.11	1.76	1459.00
							390	300	0.99	1.01	1.12	1.82	1582.7
							440	350	0.99	1.02	1.12	1.87	1701.50
							490	400	0.99	1.02	1.13	1.92	1812.50
							540	440	0.99	1.02	1.14	1.98	1910.80
					360	320	340	250	0.99	1.01	1.10	1.70	1345.50
							390	300	0.99	1.01	1.11	1.75	1460.70
							440	350	0.99	1.10	1.12	1.80	1566.40
							490	400	0.99	1.01	1.12	1.85	1668.60
							540	440	0.99	1.02	1.13	1.90	1761.50
0.10	0.80	0.10	4	1	260	220	340	250	0.99	1.02	1.13	1.86	1617.30
							390	300	0.99	1.02	1.14	1.92	1760.20
							440	350	0.99	1.02	1.15	1.98	1890.80
							490	400	0.99	1.03	1.15	2.04	2007.60
							540	440	0.99	1.03	1.16	2.09	2108.50
					310	270	340	250	0.99	1.02	1.12	1.77	1468.60
							390	300	0.99	1.02	1.12	1.83	1593.70
							440	350	0.99	1.02	1.13	1.88	1713.90
							490	400	0.99	1.02	1.14	1.94	1826.20
							540	440	0.99	1.02	1.15	1.99	1926.10
					360	320	340	250	0.99	1.02	1.10	1.70	1362.50
							390	300	0.99	1.02	1.11	1.76	1469.80
							440	350	0.99	1.02	1.12	1.81	1576.60
							490	400	0.99	1.02	1.13	1.86	1680.10
							540	440	0.99	1.02	1.13	1.91	1774.40
0.05	0.85	0.10	4	1	260	220	340	250	1.00	1.03	1.14	1.87	1626.40
							390	300	1.00	1.03	1.15	1.93	1770.50
							440	350	1.00	1.03	1.15	1.99	1902.30
							490	400	1.00	1.03	1.16	2.05	2020.20
							540	440	1.00	1.03	1.17	2.10	2122.40
					310	270	340	250	1.00	1.02	1.12	1.78	1475.70
							390	300	1.00	1.02	1.13	1.84	1601.90
							440	350	1.00	1.02	1.14	1.89	1723.00
							490	400	1.00	1.03	1.14	1.95	1836.40
							540	440	1.00	1.03	1.15	2.00	1937.50
					360	320	340	250	1.00	1.02	1.11	1.71	1368.40
							390	300	1.00	1.02	1.12	1.76	1476.60
							440	350	1.00	1.02	1.12	1.82	1584.20
							490	400	1.00	1.02	1.13	1.87	1688.50
							540	440	1.00	1.02	1.14	1.92	1783.90
0.10	0.85	0.05	4	1	260	220	340	250	0.99	1.02	1.13	1.86	1625.60
							390	300	0.99	1.02	1.14	1.93	1769.60
							440	350	0.99	1.03	1.15	1.99	1901.30
							490	400	0.99	1.03	1.16	2.04	2019.10
							540	440	0.99	1.03	1.17	2.10	2121.20
					310	270	340	250	0.99	1.02	1.12	1.77	1475.10
							390	300	0.99	1.02	1.13	1.83	1601.20
							440	350	0.99	1.02	1.13	1.89	1722.20
							490	400	0.99	1.02	1.14	1.94	1835.50
							540	440	0.99	1.03	1.15	2.00	1936.50
					360	320	340	250	0.99	1.02	1.11	1.71	1367.90
							390	300	0.99	1.02	1.11	1.76	1476.00
							440	350	0.99	1.02	1.12	1.81	1583.50
							490	400	0.99	1.02	1.13	1.86	1687.80
							540	440	0.99	1.02	1.14	1.91	1783.10

Table 5.

Relative efficiency of the proposed estimator ${\widehat{\mathit{\mu }}}_{\mathit{Y}\mathit{1}}$ with respect to Ahmed et al. [14] estimator ${\widehat{\mathit{\mu }}}_{\mathit{AY}\mathit{1}}$ when ${\mathit{\theta }}_{\mathit{1}}=\mathit{4}$ and ${\mathit{\theta }}_{\mathit{2}}=\mathit{1}$.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\sigma }}_{\mathrm{Y}1}$	${\mathrm{\sigma }}_{\mathrm{Y}2}$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	$\mathrm{P}$
									0.1	0.3	0.5	0.7	0.9
0.20	0.7	0.10	4	1	260	220	340	250	0.81	1.17	2.06	6.14	5899.30
							390	300	0.81	1.17	2.08	6.24	6099.10
							440	350	0.81	1.17	2.09	6.33	6289.70
							490	400	0.81	1.17	2.10	6.42	6466.80
							540	440	0.81	1.17	2.11	6.45	6491.60
					310	270	340	250	0.82	1.16	2.04	6.07	5933.10
							390	300	0.81	1.16	2.05	6.17	6103.60
							440	350	0.81	1.16	2.06	6.26	6272.90
							490	400	0.81	1.17	2.08	6.34	6435.10
							540	440	0.81	1.17	2.09	6.38	6461.80
					360	320	340	250	0.82	1.16	2.01	6.01	5971.40
							390	300	0.82	1.16	2.03	6.10	6115.70
							440	350	0.82	1.16	2.04	6.19	6264.40
							490	400	0.82	1.16	2.06	6.27	6411.20
							540	440	0.81	1.16	2.07	6.31	6438.30
0.15	0.75	0.10	4	1	260	220	340	250	0.89	1.29	2.29	6.85	6603.20
							390	300	0.89	1.29	2.31	6.96	6833.40
							440	350	0.89	1.30	2.32	7.07	7052.40
							490	400	0.89	1.30	2.33	7.17	7255.60
							540	440	0.89	1.30	2.35	7.21	7289.80
					310	270	340	250	0.90	1.28	2.25	6.74	6619.70
							390	300	0.90	1.28	2.27	6.86	6818.50
							440	350	0.90	1.29	2.29	6.97	7015.00
							490	400	0.90	1.29	2.31	7.07	7202.90
							540	440	0.89	1.29	2.32	7.12	7240.80
					360	320	340	250	0.90	1.27	2.22	6.65	6643.30
							390	300	0.90	1.28	2.24	6.76	6813.50
							440	350	0.90	1.28	2.26	6.87	6987.80
							490	400	0.90	1.28	2.28	6.97	7159.30
							540	440	0.90	1.29	2.29	7.03	7198.80
0.10	0.80	0.10	4	1	260	220	340	250	0.98	1.42	2.52	7.56	7317.70
							390	300	0.98	1.42	2.54	7.70	7579.10
							440	350	0.98	1.42	2.56	7.82	7827.10
							490	400	0.98	1.43	2.57	7.93	8056.60
							540	440	0.98	1.43	2.59	7.99	8102.10
					310	270	340	250	0.98	1.40	2.47	7.42	7310.90
							390	300	0.98	1.41	2.50	7.56	7539.20
							440	350	0.98	1.41	2.52	7.68	7763.90
							490	400	0.98	1.41	2.54	7.80	7978.10
							540	440	0.98	1.42	2.55	7.87	8029.50
					360	320	340	250	0.98	1.39	2.43	7.30	7314.40
							390	300	0.98	1.39	2.45	7.43	7512.00
							440	350	0.98	1.40	2.48	7.56	7713.20
							490	400	0.98	1.40	2.50	7.67	7910.40
							540	440	0.98	1.41	2.52	7.75	7964.70
0.05	0.85	0.10	4	1	260	220	340	250	1.06	1.54	2.75	8.28	8040.70
							390	300	1.06	1.55	2.78	8.44	8333.90
							440	350	1.06	1.55	2.80	8.58	8611.10
							490	400	1.06	1.56	2.82	8.70	8866.90
							540	440	1.06	1.56	2.83	8.77	8925.60
					310	270	340	250	1.06	1.52	2.69	8.10	8004.10
							390	300	1.06	1.53	2.72	8.26	8263.00
							440	350	1.06	1.53	2.74	8.40	8516.70
							490	400	1.06	1.54	2.77	8.53	8757.70
							540	440	1.06	1.55	2.79	8.62	8824.70
					360	320	340	250	1.06	1.50	2.63	7.94	7982.30
							390	300	1.06	1.51	2.66	8.09	8208.60
							440	350	1.06	1.51	2.69	8.24	8437.60
							490	400	1.06	1.52	2.72	8.37	8661.30
							540	440	1.06	1.53	2.74	8.47	8732.70
0.10	0.85	0.05	4	1	260	220	340	250	0.99	1.45	2.64	8.07	7968.00
							390	300	0.99	1.46	2.66	8.21	8258.10
							440	350	0.99	1.46	2.68	8.35	8532.40
							490	400	0.99	1.46	2.69	8.47	8785.50
							540	440	0.99	1.47	2.71	8.53	8842.80
					310	270	340	250	0.99	1.44	2.58	7.89	7934.80
							390	300	0.99	1.44	2.61	8.05	8190.50
							440	350	0.99	1.45	2.63	8.19	8441.30
							490	400	0.99	1.45	2.65	8.31	8679.70
							540	440	0.99	1.46	2.67	8.39	8745.00
					360	320	340	250	0.99	1.42	2.53	7.75	7915.80
							390	300	0.99	1.43	2.56	7.89	8139.10
							440	350	0.99	1.43	2.59	8.03	8365.30
							490	400	0.99	1.44	2.61	8.16	8586.30
							540	440	0.99	1.45	2.63	8.25	8655.90

Table 6.

Relative efficiency of the proposed estimator ${\widehat{\mathit{\mu }}}_{\mathit{Y}\mathit{2}}$ with respect to Ahmed et al. [14] estimator ${\widehat{\mathit{\mu }}}_{\mathit{AY}\mathit{2}}$ when ${\mathit{\theta }}_{\mathit{1}}=\mathit{4}$ and ${\mathit{\theta }}_{\mathit{2}}=\mathit{1}$.

${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{\theta }}_1$	${\mathrm{\theta }}_2$	${\mathrm{\sigma }}_{\mathrm{Y}1}$	${\mathrm{\sigma }}_{\mathrm{Y}2}$	${\mathrm{\mu }}_{\mathrm{Y}1}$	${\mathrm{\mu }}_{\mathrm{Y}2}$	${\mathrm{RE}}_1$	${\mathrm{RE}}_2$
0.85	0.10	0.05	4	1	260	220	340	250	3.5688	3.5201
							390	300	3.5701	3.5221
							440	350	3.5712	3.5241
							490	400	3.5722	3.5258
							540	440	3.5723	3.5255
					310	270	340	250	3.5691	3.5227
							390	300	3.5702	3.5241
							440	350	3.5712	3.5255
							490	400	3.5721	3.5269
							540	440	3.5722	3.5265
					360	320	340	250	3.5695	3.5251
							390	300	3.5704	3.5260
							440	350	3.5712	3.5270
							490	400	3.5720	3.5281
							540	440	3.5721	3.5276
0.80	0.15	0.05	4	1	260	220	340	250	116.44	105.89
							390	300	117.42	106.52
							440	350	118.22	107.12
							490	400	118.87	107.66
							540	440	119.17	107.59
					310	270	340	250	115.91	106.64
							390	300	116.84	107.08
							440	350	117.63	107.54
							490	400	118.30	107.94
							540	440	118.65	107.87
					360	320	340	250	115.48	107.35
							390	300	116.34	107.64
							440	350	117.10	107.97
							490	400	117.77	108.31
							540	440	118.15	108.18
0.75	0.20	0.05	4	1	260	220	340	250	355.18	328.81
							390	300	363.11	332.28
							440	350	369.70	335.58
							490	400	375.11	338.62
							540	440	378.32	338.29
					310	270	340	250	348.93	332.66
							390	300	356.59	335.15
							440	350	363.24	337.70
							490	400	368.94	340.18
							540	440	372.59	339.68
					360	320	340	250	343.89	336.31
							390	300	351.03	338.03
							440	350	357.49	339.94
							490	400	363.22	341.91
							540	440	367.14	341.23
0.70	0.25	0.10	4	1	260	220	340	250	637.70	649.04
							390	300	661.97	658.47
							440	350	682.56	667.49
							490	400	699.81	675.82
							540	440	711.27	675.11
					310	270	340	250	616.17	658.67
							390	300	639.28	665.62
							440	350	659.84	672.73
							490	400	677.80	679.63
							540	440	690.56	678.47
					360	320	340	250	599.19	667.92
							390	300	620.45	672.86
							440	350	640.14	678.31
							490	400	657.98	683.88
							540	440	671.33	682.30
0.65	0.30	0.05	4	1	260	220	340	250	894.78	1048.40
							390	300	941.37	1067.30
							440	350	981.73	1085.40
							490	400	1016.20	1102.10
							540	440	1040.80	1101.10
					310	270	340	250	851.25	1066.00
							390	300	894.75	1080.20
							440	350	934.31	1094.70
							490	400	969.53	1108.80
							540	440	996.20	1106.90
					360	320	340	250	817.76	1083.00
							390	300	857.09	1093.40
							440	350	894.29	1104.70
							490	400	928.64	1116.30
							540	440	955.85	1113.70
0.60	0.35	0.05	4	1	260	220	340	250	1098.70	1512.50
							390	300	1168.00	1544.30
							440	350	1229.00	1574.90
							490	400	1281.70	1603.20
							540	440	1321.40	1602.10
					310	270	340	250	1032.30	1539.30
							390	300	1095.80	1563.70
							440	350	1154.50	1588.50
							490	400	1207.60	1612.70
							540	440	1249.60	1610.30
					360	320	340	250	917.00	1565.30
							390	300	981.16	1583.60
							440	350	1042.50	1603.40
							490	400	1099.60	1623.60
							540	440	1146.60	1620.10
0.55	0.40	0.05	4	1	260	220	340	250	1250.10	2029.90
							390	300	1338.80	2077.90
							440	350	1418.00	2124.00
							490	400	1487.20	2167.00
							540	440	1541.10	2166.20
					310	270	340	250	1163.40	2065.80
							390	300	1243.70	2103.40
							440	350	1318.80	2141.50
							490	400	1387.50	2178.50
							540	440	1443.50	2176.20
					360	320	340	250	1099.00	2101.10
							390	300	1169.70	2129.90
							440	350	1238.50	2160.80
							490	400	1303.50	2192.10
							540	440	1358.60	2188.40
0.50	0.45	0.5	4	1	260	220	340	250	1359.40	2591.10
							390	300	1463.50	2658.30
							440	350	1557.20	2723.00
							490	400	1639.70	2783.20
							540	440	1705.70	2783.60
					310	270	340	250	1256.30	2635.20
							390	300	1349.60	2688.70
							440	350	1437.60	2742.80
							490	400	1518.70	2795.20
							540	440	1586.30	2794.00
					360	320	340	250	1180.70	2678.70
							390	300	1262.20	2720.60
							440	350	1341.90	2765.20
							490	400	1418.10	2810.20
							540	440	1483.80	2807.20
0.45	0.50	0.5	4	1	260	220	340	250	1437.80	3188.60
							390	300	1553.50	3277.70
							440	350	1658.20	3363.40
							490	400	1751.00	3443.30
							540	440	1826.70	3445.70
					310	270	340	250	1322.00	3238.70
							390	300	1424.90	3310.70
							440	350	1522.60	3383.30
							490	400	1613.10	3453.60
							540	440	1689.90	3454.50
					360	320	340	250	1237.80	3288.40
							390	300	1327.10	3345.90
							440	350	1415.10	3406.70
							490	400	1499.60	3467.70
							540	440	1573.50	3466.80
0.40	0.55	0.5	4	1	260	220	340	250	1494.20	3816.00
							390	300	1618.50	3929.40
							440	350	1731.40	4038.20
							490	400	1831.80	4139.70
							540	440	1915.00	4145.40
					310	270	340	250	1368.80	3869.00
							390	300	1478.70	3961.90
							440	350	1583.60	4055.30
							490	400	1681.00	4145.50
							540	440	1764.70	4150.00
					360	320	340	250	1278.00	3922.10
							390	300	1373.10	3997.50
							440	350	1467.10	4076.80
							490	400	1557.60	4155.90
							540	440	1637.80	4158.80
0.35	0.60	0.5	4	1	260	220	340	250	1535.20	4468.10
							390	300	1665.70	4607.70
							440	350	1784.70	4741.50
							490	400	1890.70	4866.30
							540	440	1979.50	4876.50
					310	270	340	250	1402.30	4520.00
							390	300	1517.50	4636.00
							440	350	1627.50	4752.20
							490	400	1730.00	4864.20
							540	440	1819.00	4873.80
					360	320	340	250	1306.70	4573.00
							390	300	1406.00	4668.60
							440	350	1504.40	4768.30
							490	400	1599.30	4867.60
							540	440	1684.00	4876.00

Table 7.

Relative efficiency of the proposed estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ and ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ with respect to Ahmed et al. [14] estimators ${\widehat{\mathit{\mu }}}_{\mathit{AY}\mathit{1}}$ and ${\widehat{\mathit{\mu }}}_{\mathit{AY}\mathit{2}}$ respectively when ${\mathit{\theta }}_{\mathit{1}}=\mathit{4}$, ${\mathit{\theta }}_{\mathit{2}}=\mathit{1}$ and $\mathit{P}=\mathit{0.9}$.

The behaviour of the estimators in Tables 3–7 indicate that the proposed estimators perform better than Perri [5] and Ahmed et al. [14] estimators in terms of efficiency.

1. When the proposed estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ and ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ are compared with Perri [5] estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{P}}$, the estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ gives lesser efficiency than the estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$. Also, it is clear from Tables 5 and 6 that when the proposed estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ and ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$ are compared with Ahmed et al. [14] estimators ${\widehat{\mathrm{\mu }}}_{\mathrm{AY}1}$ and ${\widehat{\mathrm{\mu }}}_{\mathrm{AY}2}$, the estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ gives lesser efficiency than the estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}2}$.

2. From the simulation results in Tables 5 and 6, it can be interpreted that the values of relative efficiencies are coming out to be near 1 when P = 0.1, this is the cost to be paid for perturbing the data, so that privacy of respondents is protected.

3. Further, it is observed that for the fix values of ${\mathrm{P}}_{\mathrm{i}}$, ${\mathrm{\theta }}_{\mathrm{i}}$, ${\mathrm{\mu }}_{\mathrm{Yi}}$ and ${\mathrm{\sigma }}_{\mathrm{Yi}}$ (i = 1, 2, 3) the value of relative efficiencies are decreasing in Tables 3 and 4 while in Tables 5 and 6 it is increasing as the values of P increase.

The rest of the results can be read out from the given tables.

5. Conclusions

The main objective of this paper is to estimate the population means of two quantitative sensitive variables. It is to be pointed out that the proposed model is more efficient in terms of relative efficiencies and respondent’s privacy protection. Therefore, these results advocate that the proposed technique is appreciatively favourable in obtaining the truthful response from the respondents.

Proof: Given that $\mathrm{E}\left({\mathrm{S}}_1\right)={\mathrm{\theta }}_1$ and $\mathrm{E}\left({\mathrm{S}}_2\right)={\mathrm{\theta }}_2$. Following Singh [9], I define

$${\mathrm{\gamma }}_{\mathrm{rs}}=\mathrm{E}{\left[{\mathrm{S}}_1‐{\mathrm{\theta }}_1\right]}^{\mathrm{r}}{\left[{\mathrm{S}}_2‐{\mathrm{\theta }}_2\right]}^{\mathrm{s}}$$E34

Then due to independence of the scramble variables, I have

$$\mathrm{E}\left({\mathrm{S}}_1^2\right)={\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2$$E35

$$\mathrm{E}\left({\mathrm{S}}_1^3\right)={\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3$$E36

$$\mathrm{E}\left({\mathrm{S}}_1^4\right)={\mathrm{\gamma }}_{40}+4{\mathrm{\gamma }}_{30}{\mathrm{\theta }}_1+6{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1^2+{\mathrm{\theta }}_1^4$$E37

$$\mathrm{E}\left({\mathrm{S}}_2^2\right)={\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2$$E38

$$\mathrm{E}\left({\mathrm{S}}_2^3\right)={\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3$$E39

$$\mathrm{E}\left({\mathrm{S}}_2^4\right)={\mathrm{\gamma }}_{04}+4{\mathrm{\gamma }}_{03}{\mathrm{\theta }}_2+6{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2^2+{\mathrm{\theta }}_2^4$$E40

$$\mathrm{E}\left({\mathrm{S}}_1{\mathrm{S}}_2\right)={\mathrm{\theta }}_1{\mathrm{\theta }}_2$$E41

$$\mathrm{E}\left({\mathrm{S}}_1^2{\mathrm{S}}_2^2\right)=\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)$$E42

$$\mathrm{E}\left({\mathrm{S}}_1^3{\mathrm{S}}_2\right)=\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right){\mathrm{\theta }}_2$$E43

and

$$\mathrm{E}\left({\mathrm{S}}_1{\mathrm{S}}_2^3\right)={\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)$$E44

$$\mathrm{V}\left({\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}\right)=\frac{\begin{array}{l}{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]}^2{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{V}\left({\mathrm{Z}}_{1\mathrm{i}}\right)+{\mathrm{\theta }}_2^2{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{V}\left({\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)\\ ‐2{\mathrm{\theta }}_2\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\sum }_{\mathrm{i}=1}^{\mathrm{n}}cov\left({\mathrm{Z}}_{1\mathrm{i}}{\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)\end{array}}{{\mathrm{n}}^2{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}^2}$$

$$=\frac{\begin{array}{l}{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]}^2{\mathrm{\sigma }}_{{\mathrm{Z}}_1}^2+{\mathrm{\theta }}_2^2{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2\\ ‐2{\mathrm{\theta }}_2\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]{\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2^{\prime }}\end{array}}{\mathrm{n}{\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\gamma }}_{02}‐\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2{\mathrm{\gamma }}_{20}\right]}^2}$$E45

where, the variance ${\mathrm{\sigma }}_{{\mathrm{Z}}_1}^2$ is given by:

$${\mathrm{\sigma }}_{{\mathrm{Z}}_1}^2=\mathrm{E}\left({\mathrm{Z}}_{1\mathrm{i}}^2\right)‐{\left[\mathrm{E}\left({\mathrm{Z}}_{1\mathrm{i}}\right)\right]}^2$$

$$=\mathrm{E}{\left({\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}\right)}^2‐{\left[\mathrm{E}\left({\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}\right)\right]}^2$$

$$=\mathrm{E}\left({\mathrm{S}}_1^2{\mathrm{Y}}_{1\mathrm{i}}^2+{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}^2+2{\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}{\mathrm{Y}}_{2\mathrm{i}}\right)‐{\left[{\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right]}^2$$

$$=\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)$$

$$+2{\mathrm{\theta }}_1{\mathrm{\theta }}_2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)‐{\left[{\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right]}^2$$

$$={\mathrm{\gamma }}_{20}\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)+{\mathrm{\gamma }}_{02}\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)+{\mathrm{\theta }}_1^2{\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\theta }}_2^2{\mathrm{\sigma }}_{\mathrm{Y}2}^2+2{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}$$E46

The variance ${\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2$ is given by:

$${\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2=\mathrm{E}\left({{\mathrm{Z}}^{\prime }}_{2\mathrm{i}}^2\right)‐{\left[\mathrm{E}\left({\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)\right]}^2$$

$$\begin{array}{l}=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\mathrm{E}{\left({\mathrm{S}}_1^2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\right)}^2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\mathrm{E}{\left({\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}\right)}^2\\ ‐{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]}^2\end{array}$$

$$\begin{array}{l}=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\mathrm{E}\left({\mathrm{S}}_1^4{\mathrm{Y}}_{1\mathrm{i}}^2+{\mathrm{S}}_1^2{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}^2+2{\mathrm{S}}_1^3{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}{\mathrm{Y}}_{2\mathrm{i}}\right)\\ +\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\mathrm{E}\left({\mathrm{S}}_1^2{\mathrm{S}}_2^2{\mathrm{Y}}_{1\mathrm{i}}^2+{\mathrm{S}}_2^4{\mathrm{Y}}_{2\mathrm{i}}^2+2{\mathrm{S}}_1{\mathrm{S}}_2^3{\mathrm{Y}}_{1\mathrm{i}}{\mathrm{Y}}_{2\mathrm{i}}\right)\\ ‐{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]}^2\end{array}$$

$$\begin{array}{l}=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left[\left({\mathrm{\gamma }}_{40}+4{\mathrm{\gamma }}_{30}{\mathrm{\theta }}_1+6{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1^2+{\mathrm{\theta }}_1^4\right)\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)+\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\right.\\ \left.+2\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right){\mathrm{\theta }}_2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\right]+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left[\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)\right.\\ +\left({\mathrm{\gamma }}_{04}+4{\mathrm{\gamma }}_{03}{\mathrm{\theta }}_2+6{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2^2+{\mathrm{\theta }}_2^4\right)\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)+2{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\\ ‐{\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]}^2\end{array}$$

After simplification, this gives

$$\begin{array}{l}{\mathrm{\sigma }}_{{\mathrm{Z}}_2^{\prime }}^2=\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{40}+4{\mathrm{\gamma }}_{30}{\mathrm{\theta }}_1+6{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1^2+{\mathrm{\theta }}_1^4\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ +\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\left[\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{04}+4{\mathrm{\gamma }}_{03}{\mathrm{\theta }}_2+6{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2^2+{\mathrm{\theta }}_2^4\right)+\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ +2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\right]\\ ‐\left[\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right\}{\mathrm{\mu }}_{\mathrm{Y}1}\right.\\ {\left.+\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right\}{\mathrm{\mu }}_{\mathrm{Y}2}\right]}^2\end{array}$$E47

and the covariance ${\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2^{\prime }}$ between ${\mathrm{Z}}_{1\mathrm{i}}$ and ${\mathrm{Z}}_{2\mathrm{i}}^{\prime }$ is given by:

$${\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2^{\prime }}=cov\left({\mathrm{Z}}_{1\mathrm{i}},{\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)=\mathrm{E}\left({\mathrm{Z}}_{1\mathrm{i}}{\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)‐\mathrm{E}\left({\mathrm{Z}}_{1\mathrm{i}}\right)\mathrm{E}\left({\mathrm{Z}}_{2\mathrm{i}}^{\prime }\right)$$

$$\begin{array}{l}=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\mathrm{E}\left\{\left({\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\right)\left({\mathrm{S}}_1^2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\right)\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\mathrm{E}\left\{\left({\mathrm{S}}_1{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2{\mathrm{Y}}_{2\mathrm{i}}\right)\left({\mathrm{S}}_1{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}+{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}\right)\right\}\\ ‐\left({\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right){\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]}^2\end{array}$$

$$\begin{array}{l}=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\mathrm{E}\left({\mathrm{S}}_1^3{\mathrm{Y}}_{1\mathrm{i}}^2+2{\mathrm{S}}_1^2{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}{\mathrm{Y}}_{2\mathrm{i}}+{\mathrm{S}}_1{\mathrm{S}}_2^2{\mathrm{Y}}_{2\mathrm{i}}^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\mathrm{E}\left({\mathrm{S}}_1^2{\mathrm{S}}_2{\mathrm{Y}}_{1\mathrm{i}}^2+2{\mathrm{S}}_1{\mathrm{S}}_2^2{\mathrm{Y}}_{1\mathrm{i}}{\mathrm{Y}}_{2\mathrm{i}}+{\mathrm{S}}_2^3{\mathrm{Y}}_{2\mathrm{i}}^2\right)\\ ‐\left({\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right){\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]}^2\end{array}$$

$$\begin{array}{l}=\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left[\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right)\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)+2\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\theta }}_2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)+{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\right]\\ +\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left[\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\theta }}_2\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)+2{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)+\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\right]\\ ‐\left({\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right){\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left\{\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right){\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}2}\right\}+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left\{{\mathrm{\theta }}_1{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}1}+\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right){\mathrm{\mu }}_{\mathrm{Y}2}\right\}\right]}^2\end{array}$$

After some algebra, I get

$$\begin{array}{l}{\mathrm{\sigma }}_{{\mathrm{Z}}_1{\mathrm{Z}}_2^{\prime }}=\left({\mathrm{\sigma }}_{\mathrm{Y}1}^2+{\mathrm{\mu }}_{\mathrm{Y}1}^2\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{30}+3{\mathrm{\gamma }}_{20}{\mathrm{\theta }}_1+{\mathrm{\theta }}_1^3\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)\right]\\ +\left({\mathrm{\sigma }}_{\mathrm{Y}2}^2+{\mathrm{\mu }}_{\mathrm{Y}2}^2\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{03}+3{\mathrm{\gamma }}_{02}{\mathrm{\theta }}_2+{\mathrm{\theta }}_2^3\right)\right]\\ +2\left({\mathrm{\sigma }}_{\mathrm{Y}1}{\mathrm{\sigma }}_{\mathrm{Y}2}+{\mathrm{\mu }}_{\mathrm{Y}1}{\mathrm{\mu }}_{\mathrm{Y}2}\right)\left[\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_2\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right]\\ ‐\left({\mathrm{\theta }}_1{\mathrm{\mu }}_{\mathrm{Y}1}+{\mathrm{\theta }}_2{\mathrm{\mu }}_{\mathrm{Y}12}\right)\left[\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right)\left({\mathrm{\gamma }}_{20}+{\mathrm{\theta }}_1^2\right)+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}{\mathrm{\theta }}_1{\mathrm{\theta }}_2\right\}{\mathrm{\mu }}_{\mathrm{Y}1}\right.\\ \left.+\left\{\left({\mathrm{P}}_1+{\mathrm{P}}_3\mathrm{P}\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2+\left\{{\mathrm{P}}_2+{\mathrm{P}}_3\left(1‐\mathrm{P}\right)\right\}\left({\mathrm{\gamma }}_{02}+{\mathrm{\theta }}_2^2\right)\right\}{\mathrm{\mu }}_{\mathrm{Y}2}\right]\end{array}$$E48

Finally, substituting the expressions given in Eqs. (13), (14) and (15) in Eq. (12), I get the variance of the estimator ${\widehat{\mathrm{\mu }}}_{\mathrm{Y}1}$ as given in Eq. (24).