Estimation for Two Sensitive Variables Using Randomization Response Model Under Stratified Random Sampling

Abstract

When direct survey are about sensitive characteristics such as addiction to drugs, alcoholism, proneness to tax invasion and sexual violence, nonresponse bias and response bias become serious problems because people oftentimes do not wish to give true information. In this study, when the population is composed of strata such as gender, region, age group, we consider the simple model and crossed model by applying stratified random sampling which can estimate not only the domain population proportion but also the whole population proportion for two sensitive attributes such as drug use and sexual violence in the same time. In addition, when the size of each population stratum is unknown in stratified random sampling, we propose the simple model and crossed model by using stratified double sampling method. In each proposed survey design, the sample allocation of each stratum is dealt with in consideration of proportional allocation and optimal one. We compare the efficiency between the simple model and the crossed model according to the proposed stratified random sampling design.

1. Introduction

The surveys on sensitive attributes as drug use, tax evasion or criminal violence, respondents may avoid responding or give false answers because of the risk of exposing their personal confidentiality, which can lead to non-response bias or response bias. To overcome this problem, the randomized response model (RRM) was first proposed by [1] that could obtain sensitive information while protecting the identity or confidentiality of the respondent through an indirect response using a randomization device and an unrelated question randomized response model was suggested by [2] that improved the Warner’s related question model. Since then many researchers have suggested various randomized response models to improve the estimation quality [3].

For the stratified random sampling design of RRM, a stratified unrelated question randomized response model was suggested by [4] to estimate the whole and domain population proportion, a stratified Warner model was published by [5] and it have developed a discrete quantitative randomized response model using stratified random sampling [6].

Especially, a survey for the two sensitive attributes was suggested by [7] to estimate the proportion of two sensitive characteristics at the same time by having the respondents selected by the simple random sampling with replacement using two randomization devices. They provided two shuffled decks of cards to each respondent in the sample and each deck was comprised two sorts of cards that indicate whether or not the respondent possesses a particular sensitive attributes.

In Deck-I, they structured the first randomization device as “Do you have a sensitive attribute A?” and “Do you have a nonsensitive attribute $A^c$?”, and the second randomization device as “Do you have a sensitive attribute B?” and “Do you have a nonsensitive attribute $B^c$?”, and proposed a simple model that estimates the proportion of the population with sensitive attribute A, sensitive attribute B, and both sensitive attributes A and B simultaneously. In Deck-II, they structured the first randomization device as “Do you have a sensitive attribute A?” and “Do you have a nonsensitive attribute $B^c$?”, and the second randomization device as “Do you have a sensitive attribute B?” and “Do you have a nonsensitive attribute $A^c$?”, and proposed a crossed model that estimates the proportion of the population with sensitive attribute A, sensitive attribute B, and both sensitive attributes A and B at the same time.

In this study, when the population is composed of strata, we extend the simple model and crossed model by applying stratified random sampling which can estimate not only the stratum population proportion but also the whole population proportion for two sensitive attributes. When the size of each stratum is unknown in stratified random sampling, we propose the simple model and crossed model by using stratified double sampling method at the same time. In each proposed model, the sample allocation of each stratum is dealt with in consideration of proportional allocation and optimal one. We are to compare the efficiency between the simple model and the crossed model according to the proposed stratified random sampling method.

In Section 2 we consider the stratified estimation with simple model, and then we consider the crossed model with stratified sampling design in Section 3. In Section 4, we show the efficiency comparison between the stratified random sampling and the stratified double sampling by simple model and crossed model. In Section 5, we make a concluding remarks.

2. Stratified Estimation for Two Sensitive Attributes with the Simple Model

2.1. A Simple Model with the Stratified Random Sampling

In this section, we consider that the application of stratified sampling design to a simple model for estimating two sensitive attributes when the sizes of each stratum are known. We assume a population with size N composed of mutually exclusive strata, each with $N_h$ a known size. From each stratum $N_h(h=1, 2, \cdots , L)$, a sample of size $n_h$ is drawn using simple random sampling with replacement (SRSWR). The ith respondent from each stratum h responds to a survey selected by two randomization devices as following

Table 1. Randomization Device I of the simple model in stratum h.

	Question	Selection Probability
Question1	Do you have sensitive attribute $A_h$?	$P_h$
Question2	Do you have non-sensitive attribute $A_h^c$?	$1-P_h$

and

Table 2. Randomization Device II of the simple model in stratum h.

	Question	Selection Probability
Question1	Do you have sensitive attribute $B_h$?	$T_h$
Question2	Do you have non-sensitive attribute $B_h^c$?	$1-T_h$

, answering either “yes” or “no” by the following devices, with selection probabilities $P_h\ne 0.5$ and $T_h\ne 0.5$.

Therefore, the probability that the respondent in stratum $h(=1, 2, \cdots , L)$ answers (yes, yes) is given by

$$\begin{array}{cc}{\theta }_{h11}=\hfill & (2P_h-1)(2T_h-1){\pi }_{hAB}+(2P_h-1)(1-T_h){\pi }_{hA}\hfill \\ \hfill & +(1-P_h)(2T_h-1){\pi }_{hB}+(1-P_h)(1-T_h).\hfill \end{array}$$

(1)

And the probability that the respondent in stratum h answers (yes, no) is given by

$$\begin{array}{cc}{\theta }_{h10}=\hfill & -(2P_h-1)(2T_h-1){\pi }_{hAB}+(2P_h-1)T_h{\pi }_{hA}\hfill \\ \hfill & -(1-P_h)(2T_h-1){\pi }_{hB}+(1-P_h)T_h.\hfill \end{array}$$

(2)

The probability that the respondent in stratum h answers (no, yes) is given by

$$\begin{array}{cc}{\theta }_{h01}=\hfill & -(2P_h-1)(2T_h-1){\pi }_{hAB}-(2P_h-1)(1-T_h){\pi }_{hA}\hfill \\ \hfill & +P_h(2T_h-1){\pi }_{hB}+P_h(1-T_h).\hfill \end{array}$$

(3)

Finally, the probability that the respondent in stratum h answers (no, no) is given by

$$\begin{array}{cc}{\theta }_{h00}=\hfill & (2P_h-1)(2T_h-1){\pi }_{hAB}-(2P_h-1)T_h{\pi }_{hA}\hfill \\ \hfill & -P_h(2T_h-1){\pi }_{hB}+P_h{T}_h,\hfill \end{array}$$

(4)

where ${\pi }_{hA}$ and ${\pi }_{hB}$ are a population proportion of sensitive attribute A and sensitive attribute B and ${\pi }_{hAB}$ is a population proportion of sensitive attribute A and B, and ${\sum }_{i=0}^1{\sum }_{j=0}^1{\theta }_{hij}=1$ for a stratum h, respectively.

Let the number of person who answered (yes, yes) out of $n_h$ respondents sampled from h stratum is $n_{h11}$, and answered (yes, no) is $n_{h10}$, and answered (no, yes) is $n_{h01}$, and answered (no, no) is $n_{h00}$, then the sample proportions are

$${\widehat{\theta }}_{h11}={\displaystyle \frac{n_{h11}}{n_h}},{\widehat{\theta }}_{h10}={\displaystyle \frac{n_{h10}}{n_h}},{\widehat{\theta }}_{h01}={\displaystyle \frac{n_{h01}}{n_h}},{\widehat{\theta }}_{h00}={\displaystyle \frac{n_{h00}}{n_h}},$$

respectively.

We can define the minimum distance function between the population proportion ${\theta }_{hij}$ and observed sample proportion ${\widehat{\theta }}_{hij}$ for stratum h, as follows

$$D_h={\displaystyle \frac{1}{2}}\sum _{i=0}^1\sum _{j=0}^1{\left({\theta }_{hij}-{\widehat{\theta }}_{hij}\right)}^2.$$

Setting for minimizing the distance function,

$${\displaystyle \frac{\partial D_h}{\partial {\pi }_{hA}}}=0,{\displaystyle \frac{\partial D_h}{\partial {\pi }_{hB}}}=0,{\displaystyle \frac{\partial D_h}{\partial {\pi }_{hAB}}}=0,$$

then using the method of moment we can find the estimators of population proportion ${\pi }_{hA}$, ${\pi }_{hB}$, and ${\pi }_{hAB}$ for stratum h as following

$$\begin{array}{cc}\hfill {\widehat{\pi }}_{hA}& ={\displaystyle \frac{{\widehat{\theta }}_{h11}+{\widehat{\theta }}_{h10}-{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2P_h-1)}{2(2P_h-1)}},\hfill \\ \hfill {\widehat{\pi }}_{hB}& ={\displaystyle \frac{{\widehat{\theta }}_{h11}-{\widehat{\theta }}_{h10}+{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2T_h-1)}{2(2T_h-1)}},\hfill \\ \hfill {\widehat{\pi }}_{hAB}& ={\displaystyle \frac{(P_h+T_h){\widehat{\theta }}_{h11}+(T_h-P_h){\widehat{\theta }}_{h10}-(P_h-T_h){\widehat{\theta }}_{h01}}{2(2P_h-1)(2T_h-1)}}\hfill \\ \hfill & +{\displaystyle \frac{(2-P_h-T_h){\widehat{\theta }}_{h00}-T_h(1-P_h)-P_h(1-T_h)}{2(2P_h-1)(2T_h-1)}}.\hfill \end{array}$$

And the population proportions of sensitive attribute are respectively,

$$\begin{array}{cc}\hfill & {\pi }_A={\displaystyle {\displaystyle \sum _{h=1}^L}}{W}_h{\pi }_{hA},\hfill \\ \hfill & {\pi }_B={\displaystyle {\displaystyle \sum _{h=1}^L}}{W}_h{\pi }_{hB},\hfill \\ \hfill & {\pi }_{AB}={\displaystyle {\displaystyle \sum _{h=1}^L}}{W}_h{\pi }_{hAB},\hfill \end{array}$$

where $W_h=N_h/N$.

Theorem 1. 

The stratified estimators ${\widehat{\pi }}_A$, ${\widehat{\pi }}_B$ and ${\widehat{\pi }}_{AB}$ are an unbiased estimator for the population proportions of sensitive attribute, ${\pi }_A$, ${\pi }_B$, and ${\pi }_{AB}$, respectively.

$$\begin{array}{c}{\widehat{\pi }}_A={\displaystyle \sum _{h=1}^L}{W}_h\left({\displaystyle \frac{{\widehat{\theta }}_{h11}+{\widehat{\theta }}_{h10}-{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2P_h-1)}{2(2P_h-1)}}\right),\hfill \end{array}$$

(5)

$$\begin{array}{c}{\widehat{\pi }}_B={\displaystyle \sum _{h=1}^L}{W}_h\left({\displaystyle \frac{{\widehat{\theta }}_{h11}-{\widehat{\theta }}_{h10}+{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2T_h-1)}{2(2T_h-1)}}\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}{\widehat{\pi }}_{AB}=\hfill & {\displaystyle \sum _{h=1}^L}{W}_h\left({\displaystyle \frac{(P_h+T_h){\widehat{\theta }}_{h11}+(T_h-P_h){\widehat{\theta }}_{h10}-(P_h-T_h){\widehat{\theta }}_{h01}}{2(2P_h-1)(2T_h-1)}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(2-P_h-T_h){\widehat{\theta }}_{h00}-T_h(1-P_h)-P_h(1-T_h)}{2(2P_h-1)(2T_h-1)}}\right).\hfill \end{array}$$

(7)

Proof. 

Because $E\left({\widehat{\theta }}_{h11}\right)={\theta }_{h11}$, $E\left({\widehat{\theta }}_{h10}\right)={\theta }_{h10}$, $E\left({\widehat{\theta }}_{h01}\right)={\theta }_{h01}$, $E\left({\widehat{\theta }}_{h00}\right)={\theta }_{h00}$, then

$$\begin{array}{cc}\hfill E\left({\widehat{\pi }}_A\right)& =E\left[{\displaystyle \sum _{h=1}^L}{W}_h{\displaystyle \frac{{\widehat{\theta }}_{h11}+{\widehat{\theta }}_{h10}-{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2P_h-1)}{2(2P_h-1)}}\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{W}_h{\displaystyle \frac{E\left({\widehat{\theta }}_{h11}\right)+E\left({\widehat{\theta }}_{h10}\right)-E\left({\widehat{\theta }}_{h01}\right)-E\left({\widehat{\theta }}_{h00}\right)+(2P_h-1)}{2(2P_h-1)}}\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{W}_h{\displaystyle \frac{{\theta }_{h11}+{\theta }_{h10}-{\theta }_{h01}-{\theta }_{h00}+(2P_h-1)}{2(2P_h-1)}}\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hA}\hfill \\ \hfill & ={\pi }_A.\hfill \end{array}$$

In the same way, it can prove that $E\left({\widehat{\pi }}_B\right)={\pi }_B$, $E\left({\widehat{\pi }}_{AB}\right)={\pi }_{AB}$ respectively. □

Theorem 2. 

The variances of stratified estimators ${\widehat{\pi }}_A$, ${\widehat{\pi }}_B$ and ${\widehat{\pi }}_{AB}$ are given by

$$\begin{array}{c}V\left({\widehat{\pi }}_A\right)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right],\hfill \end{array}$$

(8)

$$\begin{array}{c}V\left({\widehat{\pi }}_B\right)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right],\hfill \end{array}$$

(9)

$$\begin{array}{cc}V\left({\widehat{\pi }}_{AB}\right)=\hfill & {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right].\hfill \end{array}$$

(10)

Proof. 

$$\begin{array}{cc}V\left({\widehat{\pi }}_A\right)\hfill & ={\displaystyle \sum _{h=1}^L}{W}_h^{2}V\left({\widehat{\pi }}_{hA}\right)\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{W}_h^{2}V\left[{\displaystyle \frac{{\widehat{\theta }}_{h11}+{\widehat{\theta }}_{h10}-{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2P_h-1)}{2(2P_h-1)}}\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{4{(2P_h-1)}^2}}\left[V\left({\widehat{\theta }}_{h11}\right)+V\left({\widehat{\theta }}_{h10}\right)+2Cov({\widehat{\theta }}_{h11},{\widehat{\theta }}_{h10})\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{4{(2P_h-1)}^2}}\left[{\displaystyle \frac{{\theta }_{h11}(1-{\theta }_{h11})}{n_h}}+{\displaystyle \frac{{\theta }_{h10}(1-{\theta }_{h10})}{n_h}}-2{\displaystyle \frac{{\theta }_{h11}{\theta }_{h10}}{n_h}}\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right],\hfill \end{array}$$

$$\begin{array}{cc}V\left({\widehat{\pi }}_B\right)\hfill & ={\displaystyle \sum _{h=1}^L}{W}_h^{2}V\left({\widehat{\pi }}_{hB}\right)\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{W}_h^{2}V\left[{\displaystyle \frac{{\widehat{\theta }}_{h11}-{\widehat{\theta }}_{h10}+{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2T_h-1)}{2(2T_h-1)}}\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{4{(2T_h-1)}^2}}\left[V\left({\widehat{\theta }}_{h11}\right)+V\left({\widehat{\theta }}_{h01}\right)+2Cov({\widehat{\theta }}_{h11},{\widehat{\theta }}_{h01})\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{4{(2T_h-1)}^2}}\left[{\displaystyle \frac{{\theta }_{h11}(1-{\theta }_{h11})}{n_h}}+{\displaystyle \frac{{\theta }_{h01}(1-{\theta }_{h01})}{n_h}}-2{\displaystyle \frac{{\theta }_{h11}{\theta }_{h01}}{n_h}}\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right],\hfill \end{array}$$

and

$$\begin{array}{cc}V\left({\widehat{\pi }}_{AB}\right)\hfill & ={\displaystyle \sum _{h=1}^L}{W}_h^{2}V\left({\widehat{\pi }}_{hAB}\right)\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{W}_h^{2}V\left[{\displaystyle \frac{(P_h+T_h){\widehat{\theta }}_{h11}+(T_h-P_h){\widehat{\theta }}_{h10}-(P_h-T_h){\widehat{\theta }}_{h01}}{2(2P_h-1)(2T_h-1)}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(2-P_h-T_h){\widehat{\theta }}_{h00}-T_h(1-P_h)-P_h(1-T_h)}{2(2P_h-1)(2T_h-1)}}\right]\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right].\hfill \end{array}$$

□

The covariances of stratified estimator ${\widehat{\pi }}_A$, ${\widehat{\pi }}_B$ and ${\widehat{\pi }}_{AB}$ are given by

$$\begin{array}{c}Cov({\widehat{\pi }}_A,{\widehat{\pi }}_B)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left({\pi }_{hAB}-{\pi }_{hA}{\pi }_{hB}\right),\hfill \end{array}$$

(11)

$$\begin{array}{c}Cov({\widehat{\pi }}_{AB},{\widehat{\pi }}_A)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h){\pi }_{hB}}{{(2P_h-1)}^2}}\right],\hfill \end{array}$$

(12)

$$\begin{array}{c}Cov({\widehat{\pi }}_{AB},{\widehat{\pi }}_B)={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h){\pi }_{hA}}{{(2T_h-1)}^2}}\right].\hfill \end{array}$$

(13)

Next, we consider the sample alloction problem for the stratified sampling design, which are the proportional allocation and the optimum allocation. In the proportional allocation the size of straum h is $n_h=n(N_h/N)$ then the variances of the stratified estimator are given by

$$\begin{array}{c}{V}_P\left({\widehat{\pi }}_A\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h^2\left[{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right],\hfill \end{array}$$

(14)

$$\begin{array}{c}{V}_P\left({\widehat{\pi }}_B\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h^2\left[{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right],\hfill \end{array}$$

(15)

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_{AB}\right)=\hfill & {\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h^2\left[{\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right].\hfill \end{array}$$

(16)

In the optimum allocation, for a fixed cost, the stratum size $n_h$ can be determined to minimize the variance $V\left({\widehat{\pi }}_A\right)$, $V\left({\widehat{\pi }}_B\right)$, $V\left({\widehat{\pi }}_{AB}\right)$ for each estimator. Let the cost function can be defined as follows;

$$C=c_0+\sum _{h=1}^L{n}_h{c}_h,$$

(17)

where $c_0$ be a fixed cost and $c_h$ be a survey cost for unit in stratum h.

Using the Cauchy-Schwartz inequality, we can find the optimum sample size $n_h$ which minimize the variance $V\left({\widehat{\pi }}_A\right)$, $V\left({\widehat{\pi }}_B\right)$, $V\left({\widehat{\pi }}_{AB}\right)$ for a stratum h as following,

$$n_h^A=n\times {\displaystyle \frac{\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hA}(1-{\pi }_{hA})+\frac{P_h(1-P_h)}{{(2P_h-1)}^2}\right)}^{1/2}}{{\displaystyle \sum _{h=1}^L}\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hA}(1-{\pi }_{hA})+\frac{P_h(1-P_h)}{{(2P_h-1)}^2}\right)}^{1/2}}},$$

(18)

$$n_h^B=n\times {\displaystyle \frac{\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hB}(1-{\pi }_{hB})+\frac{T_h(1-T_h)}{{(2T_h-1)}^2}\right)}^{1/2}}{{\displaystyle \sum _{h=1}^L}\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hB}(1-{\pi }_{hB})+\frac{T_h(1-T_h)}{{(2T_h-1)}^2}\right)}^{1/2}}},$$

(19)

$$n_h^{AB}=n\times {\displaystyle \frac{\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+\frac{\begin{array}{cc}\hfill & {(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}\hfill \\ \hfill & +P_h(1-P_h){(2T_h-1)}^2\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)\hfill \end{array}}{{(2P_h-1)}^2{(2T_h-1)}^2}\right)}^{1/2}}{{\displaystyle \sum _{h=1}^L}\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+\frac{\begin{array}{cc}\hfill & {(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}\hfill \\ \hfill & +P_h(1-P_h){(2T_h-1)}^2\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)\hfill \end{array}}{{(2P_h-1)}^2{(2T_h-1)}^2}\right)}^{1/2}}}.$$

(20)

Thus, we can find that the variances due to the optimum allocation are given by

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_A\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}{\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right)}^{1/2}\hfill \\ & \times {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{\sqrt{c_h}}}{\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right)}^{1/2},\hfill \end{array}$$

(21)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_B\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}{\left({\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right)}^{1/2}\hfill \\ & \times {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{\sqrt{c_h}}}{\left({\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right)}^{1/2},\hfill \end{array}$$

(22)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_{AB}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{\begin{array}{cc}\hfill & {(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}\hfill \\ \hfill & +P_h(1-P_h){(2T_h-1)}^2\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)\hfill \end{array}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right)}^{1/2}\hfill \\ \hfill & \times {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{\sqrt{c_h}}}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{\begin{array}{cc}\hfill & {(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}\hfill \\ \hfill & +P_h(1-P_h){(2T_h-1)}^2\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)\hfill \end{array}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right)}^{1/2}.\hfill \end{array}$$

(23)

2.2. A Simple Model with the Stratified Double Sampling

In Section 2.1, the problem of applying the stratified random sampling method to the simple model was discussed, assuming that the sizes of each stratum in the population are precisely known. However, when information about the stratification variables is not available, it is necessary to initially take a large sample to understand the information about the stratification variables. Stratified double sampling is a method after stratifying, a subsample is selected to obtain information about sensitive attributes.

In this section, we consider that the stratified double sampling is applied to the simple model in estimating the proportion of a sensitive attribute in sensitive surveys. Also, we deal the problem of allocating samples to each stratum.

When the sizes of each stratum are unknown, in the first stage, respondents are directly questioned about surveys that match the stratification criteria to classify the strata. In the second stage, the simple model is applied to sensitive questions. The survey involves directly questioning $n^{\prime }$ respondents, who were randomly and uniformly selected in the first stage from a population of size N, which is composed of L strata.

• Question: Do you belong to the h stratum?

In the first stage, let’s classify the sample into h stratum, and denote the number of individuals in each stratum as $n_h^{\prime }$. Then, $W_h$ and $w_h$ can be expressed as follows.

•   $W_h=N_h/N$, $(h=1, 2, \cdots , L)$: the population proportion of stratum h,
•   $w_h=n_h^{\prime }/n^{\prime }$, $(h=1, 2, \cdots , L)$: the sample proportion of stratum h,

where $w_h$ is the unbiased estimator of $W_h$.

In the second stage, the respondents od size $n_h(n={\sum }_h^L{n}_h)$ by simple random sample with replacement selected from the first-stage sample of $n_h^{\prime }$ individuals respond using the randomization device of the simple model of [7].

Among the respondents sampled from stratum h, $n_h$, the number of person who answered (yes, yes) is $n_{h11}^{\prime }$, who answered (yes, no) is $n_{h10}^{\prime }$, and who answered (no, yes) is $n_{h01}^{\prime }$, and who answered (no, no) is $n_{h00}^{\prime }$, then the sample proportions ${\widehat{\theta }}_{h11}$, ${\widehat{\theta }}_{h10}$, ${\widehat{\theta }}_{h01}$ and ${\widehat{\theta }}_{h00}$ are

$$\begin{array}{c}{\widehat{\theta }}_{h11}={\displaystyle \frac{n_{h11}^{\prime }}{n_h}},{\widehat{\theta }}_{h10}={\displaystyle \frac{n_{h10}^{\prime }}{n_h}},{\widehat{\theta }}_{h01}={\displaystyle \frac{n_{h01}^{\prime }}{n_h}},{\widehat{\theta }}_{h00}={\displaystyle \frac{n_{h00}^{\prime }}{n_h}}.\hfill \end{array}$$

The estimators for the population proportion of sensitive attribute ${\pi }_{hA}$, ${\pi }_{hB}$, and ${\pi }_{hAB}$ ara given by

$$\begin{array}{c}{\widehat{\pi }}_{hA}^d={\displaystyle \frac{{\widehat{\theta }}_{h11}+{\widehat{\theta }}_{h10}-{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2P_h-1)}{2(2P_h-1)}},\hfill \end{array}$$

$$\begin{array}{c}{\widehat{\pi }}_{hB}^d={\displaystyle \frac{{\widehat{\theta }}_{h11}-{\widehat{\theta }}_{h10}+{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2T_h-1)}{2(2T_h-1)}},\hfill \end{array}$$

$$\begin{array}{cc}{\widehat{\pi }}_{hAB}^d\hfill & ={\displaystyle \frac{(P_h+T_h){\widehat{\theta }}_{h11}+(T_h-P_h){\widehat{\theta }}_{h10}-(P_h-T_h){\widehat{\theta }}_{h01}}{2(2P_h-1)(2T_h-1)}}\hfill \\ \hfill & +{\displaystyle \frac{(2-P_h-T_h){\widehat{\theta }}_{h00}-T_h(1-P_h)-P_h(1-T_h)}{2(2P_h-1)(2T_h-1)}},\hfill \end{array}$$

respectively.

Theorem 3. 

If the estiamtors ${\widehat{\pi }}_{hA}^d$, ${\widehat{\pi }}_{hB}^d$, and ${\widehat{\pi }}_{hAB}^d$ are unbiased estimator for ${\pi }_{hA}$, ${\pi }_{hB}$ and ${\pi }_{hAB}$ in stratum h, then the stratified double estimators ${\widehat{\pi }}_A^d$, ${\widehat{\pi }}_B^d$ and ${\widehat{\pi }}_{AB}^d$ are unbiased for ${\pi }_A$, ${\pi }_B$ and ${\pi }_{AB}$, respectively

$$\begin{array}{c}{\widehat{\pi }}_A^d={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^d={\displaystyle \sum _{h=1}^L}{w}_h{\displaystyle \frac{{\widehat{\theta }}_{h11}+{\widehat{\theta }}_{h10}-{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2P_h-1)}{2(2P_h-1)}},\hfill \end{array}$$

(24)

$$\begin{array}{c}{\widehat{\pi }}_B^d={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hB}^d={\displaystyle \sum _{h=1}^L}{w}_h{\displaystyle \frac{{\widehat{\theta }}_{h11}-{\widehat{\theta }}_{h10}+{\widehat{\theta }}_{h01}-{\widehat{\theta }}_{h00}+(2T_h-1)}{2(2T_h-1)}},\hfill \end{array}$$

(25)

$$\begin{array}{cc}{\widehat{\pi }}_{AB}^d={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hAB}^d=\hfill & {\displaystyle \sum _{h=1}^L}{w}_h\left[{\displaystyle \frac{(P_h+T_h){\widehat{\theta }}_{h11}+(T_h-P_h){\widehat{\theta }}_{h10}-(P_h-T_h){\widehat{\theta }}_{h01}}{2(2P_h-1)(2T_h-1)}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(2-P_h-T_h){\widehat{\theta }}_{h00}-T_h(1-P_h)-P_h(1-T_h)}{2(2P_h-1)(2T_h-1)}}\right].\hfill \end{array}$$

(26)

Proof. 

Since ${\widehat{\pi }}_{hA}^d$, ${\widehat{\pi }}_{hB}^d$ and ${\widehat{\pi }}_{hAB}^d$ are unbiased estimator of ${\pi }_{hA}$, ${\pi }_{hB}$ and ${\pi }_{hAB}$, respectively, we get

$$\begin{array}{c}E\left({\widehat{\pi }}_A^d\right)=E_1\left[E_2\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^d|w_h\right)\right]=E_1\left({\displaystyle \sum _{h=1}^L}{w}_h{\pi }_{hA}\right)={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hA}={\pi }_A,\hfill \end{array}$$

$$\begin{array}{c}E\left({\widehat{\pi }}_B^d\right)=E_1\left[E_2\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hB}^d|w_h\right)\right]=E_1\left({\displaystyle \sum _{h=1}^L}{w}_h{\pi }_{hB}\right)={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hB}={\pi }_B,\hfill \end{array}$$

$$\begin{array}{c}E\left({\widehat{\pi }}_{AB}^d\right)=E_1\left[E_2\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hAB}^d|w_h\right)\right]=E_1\left({\displaystyle \sum _{h=1}^L}{w}_h{\pi }_{hAB}\right)={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hAB}={\pi }_{AB}.\hfill \end{array}$$

□

Theorem 4. 

If the respondents are selected with simple random sampling from different strata, the variance of the stratified double estimator ${\widehat{\pi }}_A^d$, ${\widehat{\pi }}_B^d$, and ${\widehat{\pi }}_{AB}^d$ are given as following, respectively,

$$\begin{array}{cc}V\left({\widehat{\pi }}_A^d\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right)+{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right],\hfill \end{array}$$

(27)

$$\begin{array}{cc}V\left({\widehat{\pi }}_B^d\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right)+{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right],\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}V\left({\widehat{\pi }}_{AB}^d\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hAB}(1-{\pi }_{hAB})\right.\right.\hfill \\ \hfill & +{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}+P_h(1-P_h){(2T_h-1)}^2}{{(2P_h-1)}^2{(2T_h-1)}^2}}\hfill \\ \hfill & +\left.\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right)+{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right],\hfill \end{array}$$

(29)

where $0\le v_h(=n_h/n_h^{\prime })\le 1$.

Proof. 

By [8] we can rewrite the estimator ${\widehat{\pi }}_A^d$ as follows

$$\begin{array}{c}{\widehat{\pi }}_A^d={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^d={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{\prime }+{\displaystyle \sum _{h=1}^L}{w}_h({\widehat{\pi }}_{hA}^d-{\widehat{\pi }}_{hA}^{\prime })\hfill \end{array}$$

The variance of the first term on right-hand side of equation is

$$\begin{array}{c}V\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{\prime }\right)={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{w}_h\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right)+{\displaystyle \sum _{h=1}^L}{w}_h{({\pi }_{hA}-{\pi }_A)}^2\right],\hfill \end{array}$$

and the variance of the second term on right-hand side of equation is

$$\begin{array}{c}{E}_1\left[V_2\left({\displaystyle \sum _{h=1}^L}{w}_h({\widehat{\pi }}_{hA}^d-{\widehat{\pi }}_{hA}^{\prime })\right)\right]=E_1\left[{\displaystyle \sum _{h=1}^L}\left({\displaystyle \frac{1}{n_h}}-{\displaystyle \frac{1}{n_h^{\prime }}}\right)w_h^2\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right)\right],\hfill \end{array}$$

Since $n_h=v_h{n}_h^{\prime }=v_h{w}_h{n}^{\prime }$, then it becomes

$$\begin{array}{c}=E_1\left[{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{w_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right)\right]\hfill \end{array}$$

$$\begin{array}{c}={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right].\hfill \end{array}$$

It can make (27) from these equation, (28) and (29) of the variance ${\widehat{\pi }}_B^d$, and ${\widehat{\pi }}_{AB}^d$ in the same way. □

Next, we consider the proportional and the optimum allocation as methods of allocating overall sample of n to each stratum of $n_h^{\prime }$ and check variance for each case. If the first samples $n^{\prime }$ and $n_h^{\prime }$ are used instead of the population size N and $N_h$, the sample size of stratum h in proportional distribution is $n_h=n(n_h^{\prime }/n^{\prime })$, so the variances of ${\widehat{\pi }}_A^d$, ${\widehat{\pi }}_B^d$ and ${\widehat{\pi }}_{AB}^d$ are as follows respectively.

$$\begin{array}{c}{V}_P\left({\widehat{\pi }}_A^d\right)={\displaystyle \frac{1}{n^{\prime }}}{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2+{\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}\right],\hfill \end{array}$$

(30)

$$\begin{array}{c}{V}_P\left({\widehat{\pi }}_B^d\right)={\displaystyle \frac{1}{n^{\prime }}}{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2+{\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{T_h(1-T_h)}{{(2T_h-1)}^2}}\right],\hfill \end{array}$$

(31)

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_{AB}^d\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}\right].\hfill \end{array}$$

(32)

We can find the optimum values of $n^{\prime }$ and $v_h$ to minimize $V\left({\widehat{\pi }}_A^d\right)$, $V\left({\widehat{\pi }}_B^d\right)$ and $V\left({\widehat{\pi }}_{AB}^d\right)$ for a specified cost. The cost function is

$$\begin{array}{c}C=c^{\prime }{n}^{\prime }+{\displaystyle \sum _{h=1}^L}{c}_h{n}_h,\hfill \end{array}$$

(33)

where $c^{\prime }$ is a classification cost for unit and $c_h$ is the cost per unit in stratum h.

Since $n_h$ is a random variable, we should minimize the expected cost $E\left(C\right)$ to obtain the optimum values of $n^{\prime }$ and $v_h$.

$$\begin{array}{cc}E\left(C\right)=C^{*}\hfill & =c^{\prime }{n}^{\prime }+{\displaystyle \sum _{h=1}^L}{c}_{h}E\left(n_h\right)\hfill \\ \hfill & =c^{\prime }{n}^{\prime }+n^{\prime }{\displaystyle \sum _{h=1}^L}{c}_h{v}_h{W}_h.\hfill \end{array}$$

(34)

The optimum values of $v_h$ that minimize the products of the expected cost (34) and variances (27)–(29) are obtained using Cauchy-Schwartz inequality respectively.

$$\begin{array}{c}{v}_h^A={\left({\displaystyle \frac{c^{\prime }}{c_h}}{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{{\pi }_{hA}(1-{\pi }_{hA})+\frac{P_h(1-P_h)}{{(2P_h-1)}^2}}{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2}}\right)}^{1/2},\hfill \end{array}$$

(35)

$$\begin{array}{c}{v}_h^B={\left({\displaystyle \frac{c^{\prime }}{c_h}}{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{{\pi }_{hB}(1-{\pi }_{hB})+\frac{T_h(1-T_h)}{{(2T_h-1)}^2}}{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2}}\right)}^{1/2},\hfill \end{array}$$

(36)

$$\begin{array}{cc}{v}_h^{AB}\hfill & =\left({\displaystyle \frac{c^{\prime }}{c_h}}{\pi }_{hAB}(1-{\pi }_{hAB})\right.\hfill \\ \hfill & +{\left.{\displaystyle \frac{{\pi }_{hAB}(1-{\pi }_{hAB})+\frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}+P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}}\right)}^{1/2}.\hfill \end{array}$$

(37)

The optimum values of $n_A^{\prime }$, $n_B^{\prime }$ and $n_{AB}^{\prime }$ can be obtained as follows by substituting the values of $v_h^A$, $v_h^B$ and $v_h^{AB}$ into the expected cost (34).

$$\begin{array}{c}{n}_A^{\prime }={\displaystyle \frac{C^{*}}{c^{\prime }+{\displaystyle \sum _{h=1}^L}{c}_h{W}_h{\left(\frac{c^{\prime }}{c_h}{\pi }_{hA}(1-{\pi }_{hA})+\frac{{\pi }_{hA}(1-{\pi }_{hA})+\frac{P_h(1-P_h)}{{(2P_h-1)}^2}}{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2}\right)}^{1/2}}}\hfill \end{array}$$

(38)

$$\begin{array}{c}{n}_B^{\prime }={\displaystyle \frac{C^{*}}{c^{\prime }+{\displaystyle \sum _{h=1}^L}{c}_h{W}_h{\left(\frac{c^{\prime }}{c_h}{\pi }_{hB}(1-{\pi }_{hB})+\frac{{\pi }_{hB}(1-{\pi }_{hB})+\frac{P_h(1-P_h)}{{(2P_h-1)}^2}}{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2}\right)}^{1/2}}}\hfill \end{array}$$

(39)

$$\begin{array}{c}{n}_{AB}^{\prime }={\displaystyle \frac{C^{*}}{c^{\prime }+{\displaystyle \sum _{h=1}^L}{c}_h{W}_h{\left[\frac{c^{\prime }}{c_h}{\pi }_{hAB}(1-{\pi }_{hAB})+\frac{\frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}+P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}\right]}^{1/2}}}.\hfill \end{array}$$

(40)

The minimum variances of ${\widehat{\pi }}_A$, ${\widehat{\pi }}_B$ and ${\widehat{\pi }}_{AB}$ are obtained by substituting the optimum values of $n_A^{\prime }$, $n_B^{\prime }$ and $n_{AB}^{\prime }$ and $v_h^A$, $v_h^B$ and $v_h^{AB}$ into (27), (28) and (29) respectively.

$$\begin{array}{c}{V}_{min}\left({\widehat{\pi }}_A^d\right)={\displaystyle \frac{1}{C^{*}}}{\left[\sqrt{c^{\prime }}\sqrt{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2}+{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}\sqrt{{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}}\right]}^2,\hfill \end{array}$$

(41)

$$\begin{array}{c}{V}_{min}\left({\widehat{\pi }}_B^d\right)={\displaystyle \frac{1}{C^{*}}}{\left[\sqrt{c^{\prime }}\sqrt{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2}+{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}\sqrt{{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{P_h(1-P_h)}{{(2P_h-1)}^2}}}\right]}^2,\hfill \end{array}$$

(42)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_{AB}^d\right)\hfill & ={\displaystyle \frac{1}{C^{*}}}\left[\sqrt{c^{\prime }}\sqrt{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}\right.\hfill \\ \hfill & +{\left.{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}\sqrt{{\displaystyle \frac{{(2P_h-1)}^2{T}_h(1-T_h){\pi }_{hA}+P_h(1-P_h){(2T_h-1)}^2+P_h{T}_h(1-P_h)(1-T_h)}{{(2P_h-1)}^2{(2T_h-1)}^2}}}\right]}^2.\hfill \end{array}$$

(43)

3. Stratified Estimation of Two Sensitive Attributes with the Crossed Model

3.1. Crossed Model with the Stratified Random Sampling

In this section, we will take the problem of applying stratified sampling to crossed model for estimating two sensitive attributes in the case where the size of each stratum for the population is precisely known. Let the population with size N be divided into disjoint L strata with size $N_h(h=1, 2, \cdots , L)$ and assume each stratum size is known. In stratum h, $n_h(n={\sum }_{h=1}^L{n}_h)$ respondents are selected by simple random sampling with replacement. Each respondent is asked to answer “yes” or “no” according to his/her status using the following two randomization devices as following

Table 3. Randomization Device I of crossed model in stratum h.

	Question	Selection Probability
Question1	Do you have a rare sensitive attribute $A_h$?	$P_h$
Question2	Do you have non-sensitive attribute $B_h^c$?	$1-P_h$

and

Table 4. Randomization Device II of crossed model in stratum h.

	Question	Selection Probability
Question1	Do you have sensitive attribute $B_h$?	$T_h$
Question2	Do you have non-sensitive attribute $A_h^c$?	$1-T_h$

without reporting the selected question to the interviewer with selection probability subject to $P_h+T_h=1$.

Hence, the probability of getting the response (yes, yes) from respondents in stratum h is given by

$$\begin{array}{cc}{\theta }_{h11}^{*}\hfill & ={\pi }_{hAB}\{P_h{T}_h+(1-P_h)(1-T_h)\}-{\pi }_{hA}(1-P_h)(1-T_h)\hfill \\ \hfill & -{\pi }_{hB}(1-P_h)(1-T_h)+(1-P_h)(1-T_h).\hfill \end{array}$$

(44)

Similarly, the probability of getting the response (yes, no) from respondents in stratum h is given by

$$\begin{array}{cc}{\theta }_{h10}^{*}\hfill & =-{\pi }_{hAB}\{P_h{T}_h+(1-P_h)(1-T_h)\}-{\pi }_{hA}\{(1-P_h)T_h-1\}\hfill \\ \hfill & -{\pi }_{hB}(1-P_h)T_h+(1-P_h)T_h.\hfill \end{array}$$

(45)

And, the probability of getting the response (no, yes) from respondents in stratum h is given by

$$\begin{array}{cc}{\theta }_{h01}^{*}\hfill & =-{\pi }_{hAB}\{P_h{T}_h+(1-P_h)(1-T_h)\}-{\pi }_{hA}{P}_h(1-T_h)\hfill \\ \hfill & -{\pi }_{hB}\{P_h(1-T_h)T_h-1\}+P_h(1-T_h).\hfill \end{array}$$

(46)

Finally, the probability of getting the response (no, no) from respondents in stratum h is given by

$$\begin{array}{c}{\theta }_{h00}^{*}={\pi }_{hAB}\{P_h{T}_h+(1-P_h)(1-T_h)\}-{\pi }_{hA}{P}_h{T}_h-{\pi }_{hB}{P}_h{T}_h+P_h{T}_h,\hfill \end{array}$$

(47)

where ${\pi }_{hA}$ is the population proportion of respondents in stratum h possessing sensitive attribute A, ${\pi }_{hB}$ is the population proportion of respondents in stratum h possessing sensitive attribute B, ${\pi }_{hAB}$ is the population proportion of respondents in stratum h possessing both sensitive attributes A and B, and that ${\theta }_{h11}^{*}+{\theta }_{h10}^{*}+{\theta }_{h01}^{*}+{\theta }_{h00}^{*}=1$.

If $n_{h11}^{*}$, $n_{h10}^{*}$, $n_{h01}^{*}$ and $n_{h00}^{*}$ are the numbers of respondents responding (yes, yes), (yes, no), (no, yes), and (no, no) respectively from $n_h$ sampled respondents in stratum h, the observed proportions ${\theta }_{h11}^{*}$, ${\theta }_{h10}^{*}$, ${\theta }_{h01}^{*}$ and ${\theta }_{h00}^{*}$ are given by, respectively,

$$\begin{array}{c}{\widehat{\theta }}_{h11}^{*}={\displaystyle \frac{n_{h11}^{*}}{n_h}},{\widehat{\theta }}_{h10}^{*}={\displaystyle \frac{n_{h10}^{*}}{n_h}},{\widehat{\theta }}_{h01}^{*}={\displaystyle \frac{n_{h01}^{*}}{n_h}},{\widehat{\theta }}_{h00}^{*}={\displaystyle \frac{n_{h00}^{*}}{n_h}}.\hfill \end{array}$$

The estimators of ${\pi }_{hA}$, ${\pi }_{hB}$ and ${\pi }_{hAB}$, the population proportions of sensitive attribute for stratum h are given by

$$\begin{array}{c}{\widehat{\pi }}_{hA}^{*}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{(T_h-P_h+1)({\widehat{\theta }}_{h11}^{*}-{\widehat{\theta }}_{h00}^{*})+(P_h+T_h-1)({\widehat{\theta }}_{h10}^{*}-{\widehat{\theta }}_{h01}^{*})}{2(P_h+T_h-1)}},\hfill \end{array}$$

(48)

$$\begin{array}{c}{\widehat{\pi }}_{hB}^{*}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{(P_h-T_h+1)({\widehat{\theta }}_{h11}^{*}-{\widehat{\theta }}_{h00}^{*})+(P_h+T_h-1)({\widehat{\theta }}_{h01}^{*}-{\widehat{\theta }}_{h10}^{*})}{2(P_h+T_h-1)}},\hfill \end{array}$$

(49)

and

$$\begin{array}{c}{\widehat{\pi }}_{hAB}^{*}={\displaystyle \frac{P_h{T}_h{\widehat{\theta }}_{h11}^{*}-(1-P_h)(1-T_h){\widehat{\theta }}_{h00}^{*}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}(P_h+T_h-1)}}.\hfill \end{array}$$

(50)

Thus the stratified estimators of the proportion of sensitive attribute A, B, and both A and B are

$$\begin{array}{c}{\widehat{\pi }}_A^{*}={\displaystyle \sum _{h=1}^L}{W}_h{\widehat{\pi }}_{hA}^{*},\hfill \end{array}$$

$$\begin{array}{c}{\widehat{\pi }}_B^{*}={\displaystyle \sum _{h=1}^L}{W}_h{\widehat{\pi }}_{hB}^{*},\hfill \end{array}$$

$$\begin{array}{c}{\widehat{\pi }}_{AB}^{*}={\displaystyle \sum _{h=1}^L}{W}_h{\widehat{\pi }}_{hAB}^{*},\hfill \end{array}$$

where $W_h=N_h/N$.

The variances of the stratified estimators ${\widehat{\pi }}_A^{*}$, ${\widehat{\pi }}_B^{*}$ and ${\widehat{\pi }}_{AB}^{*}$ are given by

$$\begin{array}{cc}V\left({\widehat{\pi }}_A^{*}\right)\hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hA}(1-{\pi }_{hA})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-P_h)\left[T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(51)

$$\begin{array}{cc}V\left({\widehat{\pi }}_B^{*}\right)\hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hB}(1-{\pi }_{hB})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-T_h)\left[P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(52)

$$\begin{array}{cc}V\left({\widehat{\pi }}_{AB}^{*}\right)\hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hAB})\right.\hfill \\ \hfill & +{\displaystyle \frac{{\pi }_{hAB}\{P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2\}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(53)

And, the covariances of stratum estimators ${\widehat{\pi }}_A^{*}$, ${\widehat{\pi }}_B^{*}$, and ${\widehat{\pi }}_{AB}^{*}$ are given by

$$\begin{array}{cc}Cov({\widehat{\pi }}_A^{*},{\widehat{\pi }}_B^{*})\hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[({\pi }_{hAB}-{\pi }_{hA}{\pi }_{hB})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{2P_h{T}_h(1-P_h)(1-T_h)(1+2{\pi }_{hAB}-{\pi }_{hA}-{\pi }_{hB})}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(54)

$$\begin{array}{cc}Cov({\widehat{\pi }}_{AB}^{*},{\widehat{\pi }}_A^{*})\hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hA})+{\displaystyle \frac{{\pi }_{hAB}{T}_h(1-P_h)(P_h-T_h+1)}{{(P_h+T_h-1)}^2}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)(T_h-P_h+1)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(55)

$$\begin{array}{cc}Cov({\widehat{\pi }}_{AB}^{*},{\widehat{\pi }}_B^{*})\hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h^2}{n_h}}\left[{\pi }_{hAB}(1-{\pi }_{hB})+{\displaystyle \frac{{\pi }_{hAB}{P}_h(1-T_h)(T_h-P_h+1)}{{(P_h+T_h-1)}^2}}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)(P_h-T_h+1)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(56)

In proportional allocation for stratified random sampling, the size of sample in stratum h is $n_h=n(N_h/N)$, so the variances of the stratified estimators ${\widehat{\pi }}_A^{*}$, ${\widehat{\pi }}_B^{*}$ and ${\widehat{\pi }}_{AB}^{*}$ are given by

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_A^{*}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hA}(1-{\pi }_{hA})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-P_h)\left[T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(57)

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_B^{*}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hB}(1-{\pi }_{hB})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-T_h)\left[P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(58)

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_{AB}^{*}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hAB}(1-{\pi }_{hAB})\right.\hfill \\ \hfill & +{\displaystyle \frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(59)

We use Cauchy-Schwartz inequality to obtain $n_h$ to minimize $V\left({\widehat{\pi }}_A^{*}\right)$, $V\left({\widehat{\pi }}_B^{*}\right)$ and $V\left({\widehat{\pi }}_{AB}^{*}\right)$ for specified cost function C in (17).

$$\begin{array}{c}{n}_h^A=n{\displaystyle \frac{\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hA}(1-{\pi }_{hA})+\frac{(1-P_h)\left[T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}\right)}^{1/2}}{{\displaystyle \sum _{h=1}^L}\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hA}(1-{\pi }_{hA})+\frac{(1-P_h)\left[T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}\right)}^{1/2}}}\hfill \end{array},$$

(60)

$$\begin{array}{c}{n}_h^B=n{\displaystyle \frac{\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hB}(1-{\pi }_{hB})+\frac{(1-T_h)\left[P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}\right)}^{1/2}}{{\displaystyle \sum _{h=1}^L}\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hB}(1-{\pi }_{hB})+\frac{(1-T_h)\left[P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}\right)}^{1/2}}}\hfill \end{array},$$

(61)

$$\begin{array}{c}{n}_h^{AB}=n{\displaystyle \frac{\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+\frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}\right)}^{1/2}}{{\displaystyle \sum _{h=1}^L}\frac{W_h}{\sqrt{c_h}}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+\frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}\right)}^{1/2}}}\hfill \end{array}.$$

(62)

Hence. the minimum variances are

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_A^{*}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}{\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{(1-P_h)\left[T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right)}^{1/2}\hfill \\ \hfill & \times {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{\sqrt{c_h}}}{\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{(1-P_h)\left[T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right)}^{1/2},\hfill \end{array}$$

(63)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_B^{*}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}{\left({\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{(1-T_h)\left[P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right)}^{1/2}\hfill \\ \hfill & \times {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{\sqrt{c_h}}}{\left({\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{(1-T_h)\left[P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})\right]}{{(P_h+T_h-1)}^2}}\right)}^{1/2},\hfill \end{array}$$

(64)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_{AB}^{*}\right)\hfill & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right)}^{1/2}\hfill \\ \hfill & \times {\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{\sqrt{c_h}}}{\left({\pi }_{hAB}(1-{\pi }_{hAB})+{\displaystyle \frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right)}^{1/2},\hfill \end{array}$$

(65)

3.2. Crossed Model with the Stratified Double Sampling

In the previous section, we addressed the problem of applying stratified sampling to crossed model for estimating two sensitive attributes in the case where the size of each stratum for the population is precisely known.

In this section, we apply stratified double sampling to crossed model, and we are going to deal with the problem of allocating samples to each stratum. The first sample is used to classify stratum by asking direct question and the second sample is used to estimate a sensitive attribute by using crossed model. According to the procedure in Section 2.2, the first sample of $n^{\prime }$ respondents is selected by SRSWR from the population of size N consisted of L strata, and is asked to answer to direct question such as “Are you in stratum h?”. Then we classify the first sample into h stratum. In second sample, $n_h$ respondents are selected by SRSWR from first sample and asked to answer according to the randomization device of crossed model.

Hence, the probability of getting the response (yes, yes) from respondents in stratum h is the same as (44), the probability of getting the response (yes, no) is the same as (45), the probability of getting the response (no, yes) is the same as (46), and the probability of getting the response (no, no) is the same as (47).

If $n_{h11}^{\prime }$, $n_{h10}^{\prime }$, $n_{h01}^{\prime }$ and $n_{h00}^{\prime }$ are the numbers of respondents responding (yes, yes), (yes, no), (no, yes), and (no, no) respectively from $n_h$ sampled respondents in stratum h, the observed proportions ${\widehat{\theta }}_{h11}^{*}$, ${\widehat{\theta }}_{h10}^{*}$, ${\widehat{\theta }}_{h01}^{*}$ and ${\widehat{\theta }}_{h00}^{*}$ are given by

$$\begin{array}{c}{\widehat{\theta }}_{h11}^{*}={\displaystyle \frac{n_{h11}^{\prime }}{n_h}},{\widehat{\theta }}_{h10}^{*}={\displaystyle \frac{n_{h10}^{\prime }}{n_h}},{\widehat{\theta }}_{h01}^{*}={\displaystyle \frac{n_{h01}^{\prime }}{n_h}},{\widehat{\theta }}_{h00}^{*}={\displaystyle \frac{n_{h00}^{\prime }}{n_h}}.\hfill \end{array}$$

The estimators of the population proportion of sensitive attributes in stratum h${\pi }_{hA}$, ${\pi }_{hB}$ and ${\pi }_{hAB}$ are given by

$$\begin{array}{c}{\widehat{\pi }}_{hA}^{*d}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{(T_h-P_h+1)({\widehat{\theta }}_{h11}^{*}-{\widehat{\theta }}_{h00}^{*})+(P_h+T_h-1)({\widehat{\theta }}_{h10}^{*}-{\widehat{\theta }}_{h01}^{*})}{2(P_h+T_h-1)}},\hfill \end{array}$$

$$\begin{array}{c}{\widehat{\pi }}_{hB}^{*d}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{(P_h-T_h+1)({\widehat{\theta }}_{h11}^{*}-{\widehat{\theta }}_{h00}^{*})+(P_h+T_h-1)({\widehat{\theta }}_{h01}^{*}-{\widehat{\theta }}_{h10}^{*})}{2(P_h+T_h-1)}},\hfill \end{array}$$

$$\begin{array}{c}{\widehat{\pi }}_{hAB}^{*d}={\displaystyle \frac{P_h{T}_h{\widehat{\theta }}_{h11}^{*}-(1-P_h)(1-T_h){\widehat{\theta }}_{h00}^{*}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}(P_h+T_h-1)}}.\hfill \end{array}$$

Theorem 5. 

The stratified double estimators ${\widehat{\pi }}_A^{*d}$, ${\widehat{\pi }}_B^{*d}$, and ${\widehat{\pi }}_{AB}^{*d}$ using the crossed model for sensitive attributes A, B and both A and B are unbiased estimators of ${\pi }_A$, ${\pi }_B$, and ${\pi }_{AB}$, respectively,

$$\begin{array}{cc}{\widehat{\pi }}_A^{*d}\hfill & ={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{*d}\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{w}_h\left[{\displaystyle \frac{1}{2}}+{\displaystyle \frac{(T_h-P_h+1)({\widehat{\theta }}_{h11}^{*}-{\widehat{\theta }}_{h00}^{*})+(P_h+T_h-1)({\widehat{\theta }}_{h10}^{*}-{\widehat{\theta }}_{h01}^{*})}{2(P_h+T_h-1)}}\right],\hfill \end{array}$$

(66)

$$\begin{array}{cc}{\widehat{\pi }}_B^{*d}\hfill & ={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hB}^{*d}\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{w}_h\left[{\displaystyle \frac{1}{2}}+{\displaystyle \frac{(P_h-T_h+1)({\widehat{\theta }}_{h11}^{*}-{\widehat{\theta }}_{h00}^{*})+(P_h+T_h-1)({\widehat{\theta }}_{h01}^{*}-{\widehat{\theta }}_{h10}^{*})}{2(P_h+T_1-1)}}\right],\hfill \end{array}$$

(67)

$$\begin{array}{cc}{\widehat{\pi }}_{AB}^{*d}\hfill & ={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hAB}^{*d}\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{w}_h\left[{\displaystyle \frac{P_h{T}_h{\widehat{\theta }}_{h11}^{*}-(1-P_h)(1-T_h){\widehat{\theta }}_{h00}^{*}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}(P_h+T_h-1)}}\right].\hfill \end{array}$$

(68)

Proof. 

Since ${\widehat{\pi }}_{hA}^{*d}$, ${\widehat{\pi }}_{hB}^{*d}$ and ${\widehat{\pi }}_{hAB}^{*d}$ are unbiased estimators of ${\pi }_{hA}$, ${\pi }_{hB}$ and ${\pi }_{hAB}$, respectively, we

$$\begin{array}{c}E\left({\widehat{\pi }}_A^{*d}\right)=E_1\left[E_2\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{*d}|w_h\right)\right]=E_1\left({\displaystyle \sum _{h=1}^L}{w}_h{\pi }_{hA}\right)={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hA}={\pi }_A,\hfill \end{array}$$

$$\begin{array}{c}E\left({\widehat{\pi }}_B^{*d}\right)=E_1\left[E_2\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hB}^{*d}|w_h\right)\right]=E_1\left({\displaystyle \sum _{h=1}^L}{w}_h{\pi }_{hB}\right)={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hB}={\pi }_B,\hfill \end{array}$$

$$\begin{array}{c}E\left({\widehat{\pi }}_{AB}^{*d}\right)=E_1\left[E_2\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hAB}^{*d}|w_h\right)\right]=E_1\left({\displaystyle \sum _{h=1}^L}{w}_h{\pi }_{hAB}\right)={\displaystyle \sum _{h=1}^L}{W}_h{\pi }_{hAB}={\pi }_{AB},\hfill \end{array}$$

□

Theorem 6. 

The variances of ${\widehat{\pi }}_A^{*d}$, ${\widehat{\pi }}_B^{*d}$ and ${\widehat{\pi }}_{AB}^{*d}$ are given by

$$\begin{array}{cc}V\left({\widehat{\pi }}_A^{*d}\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hA}(1-{\pi }_{hA})\right.\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right)\hfill \\ \hfill & +\left.{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hA}(1-{\pi }_{hA})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(69)

$$\begin{array}{cc}V\left({\widehat{\pi }}_B^{*d}\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hB}(1-{\pi }_{hB})\right.\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-T_h)P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right)\hfill \\ \hfill & +\left.{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hB}(1-{\pi }_{hB})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-T_h)P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(70)

$$\begin{array}{cc}V\left({\widehat{\pi }}_{AB}^{*d}\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hAB}(1-{\pi }_{hAB})\right.\right.\hfill \\ \hfill & +{\displaystyle \frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right)\hfill \\ \hfill & +\left.{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hAB}(1-{\pi }_{hAB})\right.\hfill \\ \hfill & +{\displaystyle \frac{{\pi }_{hAB}[P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2]}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\hfill \\ \hfill & +\left.{\displaystyle \frac{P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

(71)

where $0\le v_h(=n_h/n_h^{\prime })\le 1$.

Proof. 

By [8], if we redefine the stratified double estimator ${\widehat{\pi }}_A^{*d}$ of ${\pi }_A$ for sensitive attribute A using crossed model as follow;

$$\begin{array}{c}{\widehat{\pi }}_A^{*d}={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{*d}={\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{{*}^{\prime }}+{\displaystyle \sum _{h=1}^L}{w}_h({\widehat{\pi }}_{hA}^{*d}-{\widehat{\pi }}_{hA}^{{*}^{\prime }})\hfill \end{array}$$

The variance of the first term on the right hand of equation is

$$\begin{array}{cc}V\left({\displaystyle \sum _{h=1}^L}{w}_h{\widehat{\pi }}_{hA}^{{*}^{\prime }}\right)\hfill & ={\displaystyle \frac{1}{n^{\prime }}}\left[{\displaystyle \sum _{h=1}^L}{W}_h\left({\pi }_{hA}(1-{\pi }_{hA})\right.\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right)\hfill \\ \hfill & +\left.{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2\right],\hfill \end{array}$$

and the variance of the second term is

$$\begin{array}{cc}\hfill & E_1\left[V_2\left({\displaystyle \sum _{h=1}^L}{w}_h({\widehat{\pi }}_{hA}^{*d}-{\widehat{\pi }}_{hA}^{{*}^{\prime }})\right)\right]\hfill \\ \hfill & =E_1\left[{\displaystyle \sum _{h=1}^L}\left({\displaystyle \frac{1}{n_h}}-{\displaystyle \frac{1}{n_h^{\prime }}}\right)w_h^2\left({\pi }_{hA}(1-{\pi }_{hA})\right.\right.\hfill \\ \hfill & +\left.\left.{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right)\right].\hfill \end{array}$$

Since $n_h=v_h{n}_h^{\prime }=v_h{w}_h{n}^{\prime }$

$$\begin{array}{cc}\hfill & =E_1\left[{\displaystyle \sum _{h=1}^L}{\displaystyle \frac{w_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left({\pi }_{hA}(1-{\pi }_{hA})\right.\right.\hfill \\ \hfill & +\left.\left.{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right)\right],\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^L}{\displaystyle \frac{W_h}{n^{\prime }}}\left({\displaystyle \frac{1}{v_h}}-1\right)\left[{\pi }_{hA}(1-{\pi }_{hA})\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}\right],\hfill \end{array}$$

We have (69) from those equations. We can obtain (70) and (71) in the same way.

□

Next, we consider the proportional and the optimum allocation as methods of allocating overall sample of to each stratum of n and check variance for each case. If the first samples $n^{\prime }$ and $n_h^{\prime }$ are used instead of the population size N and $N_h$, the sample size of stratum h in proportional distribution is $n_h=n(n_h^{\prime }/n^{\prime })$, so the variances of ${\widehat{\pi }}_A^{*d}$, ${\widehat{\pi }}_B^{*d}$ and ${\widehat{\pi }}_{AB}^{*d}$ are as follows, respectively.

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_A^{*d}\right)\hfill & {\displaystyle =\frac{1}{n^{\prime }}{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2}\hfill \\ \hfill & {\displaystyle +\frac{1}{n}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hA}(1-{\pi }_{hA})\right.}\hfill \\ \hfill & {\displaystyle +\left.\frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}\right],}\hfill \end{array}$$

(72)

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_B^{*d}\right)\hfill & {\displaystyle =\frac{1}{n^{\prime }}{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2}\hfill \\ \hfill & {\displaystyle +\frac{1}{n}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hB}(1-{\pi }_{hB})\right.}\hfill \\ \hfill & {\displaystyle +\left.\frac{(1-T_h)P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}\right],}\hfill \end{array}$$

(73)

and

$$\begin{array}{cc}{V}_P\left({\widehat{\pi }}_{AB}^{*d}\right)\hfill & {\displaystyle =\frac{1}{n^{\prime }}{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}\hfill \\ \hfill & {\displaystyle +\frac{1}{n}{\displaystyle \sum _{h=1}^L}{W}_h\left[{\pi }_{hAB}(1-{\pi }_{hAB})\right.}\hfill \\ \hfill & {\displaystyle +\frac{{\pi }_{hAB}\{P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2-\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2\}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}\hfill \\ \hfill & {\displaystyle +\left.\frac{P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}\right].}\hfill \end{array}$$

(74)

We have to find the optimum values of $n^{\prime }$ and $v_h$ that minimize $V\left({\widehat{\pi }}_A^{*d}\right)$, $V\left({\widehat{\pi }}_B^{*d}\right)$ and $V\left({\widehat{\pi }}_{AB}^{*d}\right)$ for a specified cost. The optimum values of $v_h$ that minimize the products of the expected cost (34) and the variances (72)–(74) are obtained using Cauchy-Schwartz inequality respectively.

$$\begin{array}{c}{v}_h^A={\left({\displaystyle \frac{c^{\prime }}{c_h}}{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{{\pi }_{hA}(1-{\pi }_{hA})+\frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}{{\sum }_{h=1}^L{W}_h{({\pi }_{hA}-{\pi }_A)}^2}}\right)}^{1/2},\hfill \end{array}$$

(75)

$$\begin{array}{c}{v}_h^B={\left({\displaystyle \frac{c^{\prime }}{c_h}}{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{{\pi }_{hB}(1-{\pi }_{hB})+\frac{(1-T_h)P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}{{\sum }_{h=1}^L{W}_h{({\pi }_{hB}-{\pi }_B)}^2}}\right)}^{1/2},\hfill \end{array}$$

(76)

$$\begin{array}{cc}{v}_h^{AB}\hfill & =\left({\displaystyle \frac{c^{\prime }}{c_h}}{\pi }_{hAB}(1-{\pi }_{hAB})\right.\hfill \\ \hfill & +{\left.{\displaystyle \frac{{\pi }_{hAB}(1-{\pi }_{hAB})+\frac{\begin{array}{cc}\hfill & {\pi }_{hAB}\{P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2\hfill \\ \hfill & -\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2\}\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})\hfill \end{array}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}{{\sum }_{h=1}^L{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}}\right)}^{1/2},\hfill \end{array}$$

(77)

The optimum values of $n^{\prime }$. (78)–(80) can be obtained as follows by substituting the values $v_h$ of (75), (76), and (77) into (34).

$$\begin{array}{c}{n}_A^{\prime }={\displaystyle \frac{C^{*}}{c^{\prime }+{\sum }_{h=1}^L{c}_h{W}_h{\left(\frac{c^{\prime }}{c_h}{\pi }_{hA}(1-{\pi }_{hA})+\frac{{\pi }_{hA}(1-{\pi }_{hA})+\frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}{{\sum }_{h=1}^L{W}_h{({\pi }_{hA}-{\pi }_A)}^2}\right)}^{1/2}}},\hfill \end{array}$$

(78)

$$\begin{array}{c}{n}_B^{\prime }={\displaystyle \frac{C^{*}}{c^{\prime }+{\sum }_{h=1}^L{c}_h{W}_h{\left(\frac{c^{\prime }}{c_h}{\pi }_{hB}(1-{\pi }_{hB})+\frac{{\pi }_{hB}(1-{\pi }_{hB})+\frac{(1-T_h)P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}{{\sum }_{h=1}^L{W}_h{({\pi }_{hB}-{\pi }_B)}^2}\right)}^{1/2}}},\hfill \end{array}$$

(79)

$$\begin{array}{c}{n}_{AB}^{\prime }={\displaystyle \frac{C^{*}}{c^{\prime }+{\sum }_{h=1}^L{c}_h{W}_h{\left(\frac{c^{\prime }}{c_h}{\pi }_{hAB}(1-{\pi }_{hAB})+\frac{{\pi }_{hAB}(1-{\pi }_{hAB})+\frac{\begin{array}{cc}\hfill & {\pi }_{hAB}\{P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2\hfill \\ \hfill & -\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2\}\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})\hfill \end{array}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}{{\sum }_{h=1}^L{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}\right)}^{1/2}}}.\hfill \end{array}$$

(80)

The minimum variances of ${\widehat{\pi }}_A^{*d}$, ${\widehat{\pi }}_B^{*d}$ and ${\widehat{\pi }}_{AB}^{*d}$ are obtained by substituting the optimum values of $n_A^{\prime }$, $n_B^{\prime }$ and $n_{AB}^{\prime }$, $v_h^A$, $v_h^B$ and $v_h^{AB}$ into (72), (73) and (74), respectively.

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_A^{*d}\right)\hfill & ={\displaystyle \frac{1}{C^{*}}}\left[\sqrt{c^{\prime }}\sqrt{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hA}-{\pi }_A)}^2}\right.\hfill \\ \hfill & +{\left.{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}\sqrt{{\pi }_{hA}(1-{\pi }_{hA})+{\displaystyle \frac{(1-P_h)T_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}}\right]}^2,\hfill \end{array}$$

(81)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_B^{*d}\right)\hfill & ={\displaystyle \frac{1}{C^{*}}}\left[\sqrt{c^{\prime }}\sqrt{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hB}-{\pi }_B)}^2}\right.\hfill \\ \hfill & +{\left.{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}\sqrt{{\pi }_{hB}(1-{\pi }_{hB})+{\displaystyle \frac{(1-T_h)P_h\{P_h{T}_h+(1-P_h)(1-T_h)\}(1-{\pi }_{hA}-{\pi }_{hB}+2{\pi }_{hAB})}{{(P_h+T_h-1)}^2}}}\right]}^2,\hfill \end{array}$$

(82)

$$\begin{array}{cc}{V}_{min}\left({\widehat{\pi }}_{AB}^{*d}\right)\hfill & ={\displaystyle \frac{1}{C^{*}}}\left[\sqrt{c^{\prime }}\sqrt{{\displaystyle \sum _{h=1}^L}{W}_h{({\pi }_{hAB}-{\pi }_{AB})}^2}\right.\hfill \\ \hfill & +{\left.{\displaystyle \sum _{h=1}^L}{W}_h\sqrt{c_h}\sqrt{{\displaystyle \frac{\begin{array}{cc}\hfill & {\pi }_{hAB}\{P_h^2{T}_h^2{(1-P_h)}^2{(1-T_h)}^2\hfill \\ \hfill & -\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2\}\hfill \\ \hfill & +P_h{T}_h(1-P_h)(1-T_h)(1-{\pi }_{hA}-{\pi }_{hB})\hfill \end{array}}{\{P_h{T}_h+(1-P_h)(1-T_h)\}{(P_h+T_h-1)}^2}}}\right]}^2.\hfill \end{array}$$

(83)

4. Efficiency Comparisons

In this section, we compare the efficiency of the simple model and the crossed model by the stratified sampling method proposed in Section 2 and Section 3, respectively. We will compare the variances such as between (8) and (51), (9) and (52), and (10) and (53). Let $N=\mathrm{10,000} (N_1=7000, N_2=3000)$, $n=1000 (n_1=700, n_2=300)$, $W_1=0.7$, $W_2=0.3$ and we calculate the relative efficiencies (REs) such that

$$\begin{array}{c}\hfill R{E}_1={\displaystyle \frac{V\left({\widehat{\pi }}_A\right)}{V\left({\widehat{\pi }}_A^{*}\right)}},R{E}_2={\displaystyle \frac{V\left({\widehat{\pi }}_B\right)}{V\left({\widehat{\pi }}_B^{*}\right)}},R{E}_3={\displaystyle \frac{V\left({\widehat{\pi }}_{AB}\right)}{V\left({\widehat{\pi }}_{AB}^{*}\right)}},\end{array}$$

We consider changing ${\pi }_{hA}$, ${\pi }_{hB}$ and ${\pi }_{hAB}$ and in cases of selection probability for the first and second randomization device (1) $P_h=T_h$, (2) $P_h>T_h$, and (3) $P_h<T_h$, respectively. The values of over than 1 indicate that the crossed model of [7] by using stratified sampling method is more efficient than the simple model of [7].

In first case $P_h=T_h$,

Table 5. The means and standard deviations of $RE$ for the simple and crossed model when $P_h=T_h$.

			${\mathit{T}}_{\mathit{h}}$
${\mathit{P}}_{\mathit{h}}={\mathit{T}}_{\mathit{h}}$			0.4			0.6			0.7
${\mathit{\pi }}_{\mathit{hAB}}$	${\mathit{\pi }}_{\mathit{hA}}$	${\mathit{\pi }}_{\mathit{hB}}$	${\mathit{RE}}_{\mathit{1}}$	${\mathit{RE}}_{\mathit{2}}$	${\mathit{RE}}_{\mathit{3}}$	${\mathit{RE}}_{\mathit{1}}$	${\mathit{RE}}_{\mathit{2}}$	${\mathit{RE}}_{\mathit{3}}$	${\mathit{RE}}_{\mathit{1}}$	${\mathit{RE}}_{\mathit{2}}$	${\mathit{RE}}_{\mathit{3}}$
0.1	0.2	0.2	2.411	2.411	9.9	2.411	2.411	9.9	1.96	1.961	3.884
			(0.201)	(0.201)	(0.952)	(0.201)	(0.201)	(0.952)	(0.13)	(0.133)	(0.419)
		0.3	2.638	2.652	10.826	2.638	2.652	10.826	2.109	2.159	4.154
			(0.241)	(0.243)	(1.139)	(0.241)	(0.243)	(1.139)	(0.15)	(0.157)	(0.479)
		0.4	2.911	2.938	11.944	2.911	2.938	11.944	2.282	2.368	4.465
			(0.294)	(0.297)	(1.387)	(0.294)	(0.297)	(1.387)	(0.176)	(0.185)	(0.554)
		0.5	3.25	3.283	13.325	3.25	3.283	13.325	2.486	2.592	4.828
			(0.367)	(0.371)	(1.729)	(0.367)	(0.371)	(1.729)	(0.208)	(0.220)	(0.648)
	0.3	0.2	2.613	2.598	10.93	2.613	2.598	10.93	2.056	2.009	4.264
			(0.235)	(0.234)	(1.149)	(0.235)	(0.234)	(1.149)	(0.14)	(0.140)	(0.492)
		0.3	2.88	2.88	12.06	2.88	2.88	12.06	2.216	2.216	4.584
			(0.286)	(0.286)	(1.400)	(0.286)	(0.286)	(1.400)	(0.162)	(0.165)	(0.569)
		0.4	3.208	3.219	13.454	3.208	3.219	13.454	2.403	2.437	4.956
			(0.356)	(0.357)	(1.745)	(0.356)	(0.357)	(1.745)	(0.19)	(0.196)	(0.665)
		0.5	3.622	3.639	15.22	3.622	3.639	15.22	2.626	2.674	5.395
			(0.455)	(0.457)	(2.24)	(0.455)	(0.457)	(2.240)	(0.227)	(0.234)	(0.789)
0.2	0.2	0.2	2.058	2.058	8.19	2.058	2.058	8.19	1.718	1.719	3.068
			(0.147)	(0.147)	(0.652)	(0.147)	(0.147)	(0.652)	(0.101)	(0.103)	(0.258)
		0.3	2.221	2.233	8.81	2.221	2.233	8.81	1.831	1.875	3.231
			(0.171)	(0.172)	(0.754)	(0.171)	(0.172)	(0.754)	(0.114)	(0.119)	(0.286)
		0.4	2.411	2.433	9.533	2.411	2.433	9.533	1.96	2.034	3.412
			(0.201)	(0.203)	(0.882)	(0.201)	(0.203)	(0.882)	(0.13)	(0.138)	(0.319)
		0.5	2.638	2.664	10.386	2.638	2.664	10.386	2.109	2.198	3.615
			(0.241)	(0.244)	(1.047)	(0.241)	(0.244)	(1.047)	(0.150)	(0.160)	(0.358)
	0.3	0.2	2.205	2.193	8.895	2.205	2.193	8.895	1.797	1.756	3.315
			(0.167)	(0.167)	(0.761)	(0.167)	(0.167)	(0.761)	(0.107)	(0.107)	(0.294)
		0.3	2.392	2.392	9.625	2.392	2.392	9.625	1.918	1.918	3.501
			(0.197)	(0.197)	(0.89)	(0.197)	(0.197)	(0.890)	(0.122)	(0.125)	(0.327)
		0.4	2.613	2.622	10.486	2.613	2.622	10.486	2.056	2.085	3.71
			(0.235)	(0.236)	(1.057)	(0.235)	(0.236)	(1.057)	(0.14)	(0.144)	(0.367)
		0.5	2.88	2.893	11.519	2.88	2.893	11.519	2.216	2.257	3.945
			(0.286)	(0.288)	(1.275)	(0.286)	(0.288)	(1.275)	(0.162)	(0.168)	(0.415)
0.3	0.2	0.2	1.796	1.796	7.001	1.796	1.796	7.001	1.53	1.53	2.565
			(0.112)	(0.112)	(0.477)	(0.112)	(0.112)	(0.477)	(0.08)	(0.082)	(0.179)
		0.3	1.918	1.929	7.448	1.918	1.929	7.448	1.619	1.657	2.676
			(0.127)	(0.128)	(0.54)	(0.127)	(0.128)	(0.540)	(0.089)	(0.094)	(0.195)
		0.4	2.058	2.077	7.957	2.058	2.077	7.957	1.718	1.783	2.798
			(0.147)	(0.148)	(0.615)	(0.147)	(0.148)	(0.615)	(0.101)	(0.107)	(0.213)
		0.5	2.221	2.243	8.54	2.221	2.243	8.54	1.831	1.909	2.931
			(0.171)	(0.173)	(0.708)	(0.171)	(0.173)	(0.708)	(0.114)	(0.121)	(0.234)
	0.3	0.2	1.908	1.897	7.52	1.908	1.897	7.52	1.596	1.559	2.745
			(0.125)	(0.125)	(0.545)	(0.125)	(0.125)	(0.545)	(0.085)	(0.085)	(0.200)
		0.3	2.046	2.046	8.033	2.046	2.046	8.033	1.69	1.69	2.87
			(0.144)	(0.144)	(0.621)	(0.144)	(0.144)	(0.621)	(0.095)	(0.097)	(0.218)
		0.4	2.205	2.213	8.623	2.205	2.213	8.623	1.797	1.822	3.007
			(0.167)	(0.168)	(0.715)	(0.167)	(0.168)	(0.715)	(0.107)	(0.111)	(0.239)
		0.5	2.392	2.403	9.306	2.392	2.403	9.306	1.918	1.953	3.157
			(0.197)	(0.198)	(0.832)	(0.197)	(0.198)	(0.832)	(0.122)	(0.127)	(0.264)

(Note: The first and second row values in each cell represent the mean and standard deviation of RE).

shows that the means of relative efficiency are symmetrically on the left and right based on $P_h=T_h=0.5$. For example, when $P_h=T_h=0.4$, ${\pi }_{hAB}=0.1$, ${\pi }_{hA}=0.2$, ${\pi }_{hB}=0.1$ then the mean value of $R{E}_1$ is 2.411 in A, $R{E}_2$ is 2.411 in B, and $R{E}_3$ is 9.9 in AB, which is the same as when $P_h=T_h=0.6$, indicating that the efficiency of the cross model is higher than that of the simple model. As shown in Table 5, in all possible cases where $P_h=T_h$, the crossed model is more efficient than the simple model. In second case $P_h>T_h$,

Table 6. The means and standard deviations of $RE$ for the simple and crossed model when $P_h>T_h$.

			${\mathit{T}}_{\mathit{h}}$
${\mathit{P}}_{\mathit{h}}=\mathbf{0}.\mathbf{7}$			0.2			0.4			0.6
${\mathit{\pi }}_{\mathit{hAB}}$	${\mathit{\pi }}_{\mathit{hA}}$	${\mathit{\pi }}_{\mathit{hB}}$	${\mathit{RE}}_{\mathbf{1}}$	${\mathit{RE}}_{\mathbf{2}}$	${\mathit{RE}}_{\mathbf{3}}$	${\mathit{RE}}_{\mathbf{1}}$	${\mathit{RE}}_{\mathbf{2}}$	${\mathit{RE}}_{\mathbf{3}}$	${\mathit{RE}}_{\mathbf{1}}$	${\mathit{RE}}_{\mathbf{2}}$	${\mathit{RE}}_{\mathbf{3}}$
0.1	0.2	0.2	0.776	0.038	0.115	0.339	0.414	0.578	1.484	4.226	6.066
			(0.065)	(0.004)	(0.01)	(0.03)	(0.037)	(0.059)	(0.108)	(0.327)	(0.649)
		0.3	0.846	0.044	0.126	0.372	0.458	0.636	1.606	4.624	6.569
			(0.077)	(0.005)	(0.013)	(0.036)	(0.046)	(0.07)	(0.127)	(0.39)	(0.757)
		0.4	0.932	0.05	0.14	0.412	0.511	0.706	1.75	5.088	7.164
			(0.093)	(0.006)	(0.016)	(0.045)	(0.057)	(0.086)	(0.15)	(0.471)	(0.897)
		0.5	1.036	0.057	0.157	0.462	0.576	0.794	1.922	5.64	7.879
			(0.115)	(0.008)	(0.02)	(0.056)	(0.072)	(0.109)	(0.181)	(0.578)	(1.081)
	0.3	0.2	0.849	0.042	0.128	0.377	0.454	0.661	1.584	4.48	6.825
			(0.075)	(0.004)	(0.013)	(0.037)	(0.045)	(0.073)	(0.12)	(0.368)	(0.785)
		0.3	0.933	0.049	0.142	0.418	0.507	0.734	1.72	4.929	7.443
			(0.091)	(0.006)	(0.016)	(0.045)	(0.056)	(0.089)	(0.142)	(0.443)	(0.929)
		0.4	1.035	0.057	0.159	0.468	0.573	0.825	1.881	5.46	8.186
			(0.112)	(0.008)	(0.02)	(0.056)	(0.072)	(0.112)	(0.169)	(0.542)	(1.12)
		0.5	1.163	0.066	0.181	0.532	0.657	0.943	2.076	6.101	9.098
			(0.141)	(0.01)	(0.026)	(0.073)	(0.094	(0.146)	(0.206)	(0.678)	(1.38)
0.2	0.2	0.2	0.665	0.032	0.098	0.288	0.35	0.487	1.289	3.637	4.902
			(0.048)	(0.003)	(0.007)	(0.022)	(0.027)	(0.042)	(0.082)	(0.242)	(0.429)
		0.3	0.716	0.037	0.106	0.311	0.381	0.527	1.38	3.932	5.222
			(0.055)	(0.003)	(0.009)	(0.026)	(0.032)	(0.049)	(0.094)	(0.282)	(0.484)
		0.4	0.776	0.041	0.115	0.339	0.417	0.574	1.484	4.264	5.587
			(0.065)	(0.004)	(0.01)	(0.03)	(0.038)	(0.058)	(0.108)	(0.33)	(0.551)
		0.5	0.846	0.046	0.126	0.372	0.46	0.63	1.606	4.645	6.008
			(0.077)	(0.005)	(0.012)	(0.036)	(0.046)	(0.069)	(0.127)	(0.391)	(0.634)
	0.3	0.2	0.72	0.035	0.107	0.316	0.378	0.547	1.368	3.824	5.425
			(0.054)	(0.003)	(0.009)	(0.026)	(0.031)	(0.051)	(0.09)	(0.268)	(0.501)
		0.3	0.78	0.04	0.117	0.344	0.415	0.596	1.468	4.15	5.804
			(0.064)	(0.004)	(0.01)	(0.031)	(0.037)	(0.06)	(0.104)	(0.314)	(0.571)
		0.4	0.849	0.045	0.128	0.377	0.458	0.655	1.584	4.521	6.242
			(0.075)	(0.005)	(0.013)	(0.037)	(0.046)	(0.071)	(0.12)	(0.371)	(0.657)
		0.5	0.933	0.051	0.142	0.418	0.51	0.726	1.72	4.951	6.751
			(0.091)	(0.006)	(0.015)	(0.045)	(0.056)	(0.087)	(0.142)	(0.445)	(0.765)
0.3	0.2	0.2	0.581	0.028	0.086	0.25	0.303	0.421	1.14	3.193	4.137
			(0.037)	(0.002)	(0.006)	(0.017)	(0.02)	(0.032)	(0.065)	(0.187)	(0.31)
		0.3	0.62	0.031	0.091	0.268	0.327	0.45	1.21	3.42	4.361
			(0.042)	(0.002)	(0.006)	(0.019)	(0.023)	(0.036)	(0.073)	(0.213)	(0.343)
		0.4	0.665	0.035	0.098	0.288	0.353	0.484	1.289	3.67	4.612
			(0.048)	(0.003)	(0.007)	(0.022)	(0.027)	(0.042)	(0.082)	(0.245)	(0.381)
		0.5	0.716	0.038	0.106	0.311	0.383	0.523	1.38	3.949	4.893
			(0.055)	(0.003)	(0.009)	(0.026)	(0.032)	(0.048)	(0.094)	(0.283)	(0.426)
	0.3	0.2	0.626	0.03	0.093	0.272	0.324	0.468	1.204	3.336	4.53
			(0.041)	(0.002)	(0.006)	(0.019)	(0.023)	(0.038)	(0.07)	(0.204)	(0.355)
		0.3	0.67	0.034	0.099	0.293	0.351	0.503	1.281	3.584	4.79
			(0.047)	(0.003)	(0.007)	(0.022)	(0.027)	(0.043)	(0.079)	(0.234)	(0.394)
		0.4	0.72	0.038	0.107	0.316	0.382	0.543	1.368	3.859	5.082
			(0.054)	(0.003)	(0.009)	(0.026)	(0.032)	(0.05)	(0.09)	(0.27)	(0.441)
		0.5	0.78	0.042	0.117	0.344	0.417	0.591	1.468	4.168	5.413
			(0.064)	(0.004)	(0.01)	(0.031)	(0.038)	(0.058)	(0.104)	(0.315)	(0.497)

(Note: The first and second row values in each cell represent the mean and standard deviation of RE).

shows the relative efficiency of the cross model compared to the simple model when $P_h$ is relatively larger than $T_h$. When $P_h$ and $T_h$ have similar value, i.e., $P_h=0.7$, $T_h=0.6$, the efficiency of the crossed model is high, and the other case of the difference between $P_h$ and $T_h$ is relatively large, $P_h=0.7$, $T_h=0.2$, the relative efficiency of the crossed model is low. In third case $P_h<T_h$,

Table 7. The means and standard deviations of $RE$ for the simple and crossed model when $P_h<T_h$.

			${\mathit{T}}_{\mathit{h}}$
${\mathit{P}}_{\mathit{h}}=\mathbf{0}.\mathbf{2}$			0.4			0.6			0.7
${\mathit{\pi }}_{\mathit{hAB}}$	${\mathit{\pi }}_{\mathit{hA}}$	${\mathit{\pi }}_{\mathit{hB}}$	${\mathit{RE}}_{\mathbf{1}}$	${\mathit{RE}}_{\mathbf{2}}$	${\mathit{RE}}_{\mathbf{3}}$	${\mathit{RE}}_{\mathbf{1}}$	${\mathit{RE}}_{\mathbf{2}}$	${\mathit{RE}}_{\mathbf{3}}$	${\mathit{RE}}_{\mathbf{1}}$	${\mathit{RE}}_{\mathbf{2}}$	${\mathit{RE}}_{\mathbf{3}}$
0.1	0.2	0.2	0.601	12.388	5.393	0.148	7.263	1.101	0.038	0.776	0.079
			(0.047)	(0.652)	(0.682)	(0.014)	(0.491)	(0.131)	(0.004)	(0.065)	(0.008)
		0.3	0.65	13.241	5.752	0.162	7.878	1.2	0.042	0.866	0.087
			(0.055)	(0.74)	(0.768)	(0.017)	(0.574)	(0.153)	(0.004)	(0.079)	(0.01)
		0.4	0.709	14.178	6.162	0.179	8.578	1.319	0.046	0.967	0.096
			(0.065)	(0.845)	(0.873)	(0.02)	(0.678)	(0.182)	(0.006)	(0.097)	(0.012)
		0.5	0.779	15.215	6.637	0.201	9.389	1.464	0.052	1.08	0.108
			(0.078)	(0.971)	(1.003)	(0.025)	(0.812)	(0.22)	(0.007)	(0.12)	(0.015)
	0.3	0.2	0.663	12.25	6.266	0.169	7.499	1.307	0.044	0.83	0.092
			(0.054)	(0.637)	(0.825)	(0.017)	(0.523)	(0.164)	(0.005)	(0.074)	(0.01)
		0.3	0.72	13.085	6.713	0.187	8.154	1.436	0.049	0.933	0.102
			(0.063)	(0.722)	(0.939)	(0.021)	(0.615)	(0.195)	(0.006)	(0.091)	(0.013)
		0.4	0.788	13.999	7.231	0.21	8.906	1.594	0.055	1.049	0.115
			(0.075)	(0.824)	(1.08)	(0.026)	(0.731)	(0.237)	(0.007)	(0.114)	(0.016)
		0.5	0.871	15.009	7.836	0.239	9.784	1.792	0.063	1.184	0.131
			(0.091)	(0.945)	(1.259)	(0.034)	(0.882)	(0.295)	(0.01)	(0.144)	(0.02)
0.2	0.2	0.2	0.521	11.079	4.207	0.125	6.341	0.909	0.032	0.665	0.068
			(0.036)	(0.522)	(0.432)	(0.01)	(0.374)	(0.093)	(0.003)	(0.048)	(0.006)
		0.3	0.558	11.763	4.419	0.136	6.809	0.975	0.035	0.733	0.074
			(0.041)	(0.585)	(0.471)	(0.012)	(0.429)	(0.105)	(0.003)	(0.057)	(0.007)
		0.4	0.601	12.501	4.653	0.148	7.329	1.05	0.038	0.805	0.08
			(0.047)	(0.658)	(0.517)	(0.014)	(0.495)	(0.12)	(0.004)	(0.067)	(0.008)
		0.5	0.65	13.301	4.914	0.162	7.913	1.139	0.042	0.882	0.088
			(0.055)	(0.743)	(0.57)	(0.017)	(0.577)	(0.138)	(0.004)	(0.08)	(0.01)
	0.3	0.2	0.572	10.968	4.81	0.142	6.52	1.061	0.037	0.704	0.078
			(0.041)	(0.512)	(0.503)	(0.012)	(0.396)	(0.112)	(0.003)	(0.054)	(0.007)
		0.3	0.614	11.639	5.066	0.155	7.014	1.143	0.04	0.78	0.085
			(0.046)	(0.573)	(0.552)	(0.014)	(0.455)	(0.127)	(0.004)	(0.064)	(0.009)
		0.4	0.663	12.361	5.349	0.169	7.567	1.239	0.044	0.861	0.093
			(0.054)	(0.643)	(0.61)	(0.017)	(0.528)	(0.147)	(0.005)	(0.077)	(0.01)
		0.5	0.72	13.144	5.667	0.187	8.191	1.353	0.049	0.95	0.103
			(0.063)	(0.726)	(0.678)	(0.021)	(0.618)	(0.173)	(0.006)	(0.093)	(0.012)
0.3	0.2	0.2	0.461	10.02	3.49	0.109	5.627	0.777	0.028	0.581	0.06
			(0.029)	(0.428)	(0.31)	(0.008)	(0.295)	(0.071)	(0.002)	(0.037)	(0.005)
		0.3	0.489	10.582	3.632	0.117	5.997	0.823	0.03	0.635	0.064
			(0.032)	(0.474)	(0.332)	(0.009)	(0.333)	(0.078)	(0.002)	(0.043)	(0.005)
		0.4	0.521	11.179	3.788	0.125	6.399	0.876	0.032	0.69	0.069
			(0.036)	(0.527)	(0.357)	(0.01)	(0.378)	(0.087)	(0.003)	(0.05)	(0.006)
		0.5	0.558	11.816	3.957	0.136	6.84	0.936	0.035	0.746	0.074
			(0.041)	(0.587)	(0.385)	(0.012)	(0.431)	(0.097)	(0.003)	(0.058)	(0.007)
	0.3	0.2	0.503	9.929	3.952	0.122	5.767	0.896	0.031	0.611	0.068
			(0.032)	(0.42)	(0.353)	(0.009)	(0.31)	(0.083)	(0.002)	(0.041)	(0.006)
		0.3	0.535	10.482	4.121	0.131	6.155	0.953	0.034	0.67	0.073
			(0.036)	(0.465)	(0.38)	(0.011)	(0.351)	(0.092)	(0.003)	(0.048)	(0.006)
		0.4	0.572	11.068	4.305	0.142	6.579	1.018	0.037	0.73	0.079
			(0.041)	(0.516)	(0.41)	(0.012)	(0.399)	(0.103)	(0.003)	(0.056)	(0.007)
		0.5	0.614	11.692	4.506	0.155	7.046	1.093	0.04	0.794	0.086
			(0.046)	(0.575)	(0.445)	(0.014)	(0.457)	(0.117)	(0.004)	(0.065)	(0.008)

(Note: The first and second row values in each cell represent the mean and standard deviation of RE).

shows the relative efficiency of the cross model compared to the simple model when $P_h$ is relatively smaller than $T_h$. When $P_h$ and $T_h$ have similar value, i.e., $P_h=0.2$, $T_h=0.4$, the efficiency of the crossed model is high, and the other case of the difference between $P_h$ and $T_h$ is relatively large, $P_h=0.2$, $T_h=0.7$, the relative efficiency of the crossed model is low.

From the numerical study, we can find that the proposed crossed model with stratified random sampling is more efficient than the simple model when the selection probability of randomization device I, $P_h$, is similar with that of randomization device II, $T_h$, that is, the difference between $P_h$ and $T_h$ is small.

5. Concluding Remarks

When the population is composed of strata as gender, region, or age group, we consider the simple model and crossed model by applying stratified random sampling which can estimate not only the stratum population proportion but also the whole population proportion for two sensitive attributes such as criminal violence and tax evasion at the same time. In addition, when the size of each stratum is unknown in stratified random sampling, we propose the simple model and crossed model by using stratified double sampling method. In each proposed model, the sample allocation of each stratum is dealt with in consideration of proportional allocation and optimal one. We compare the efficiency between the simple model and the crossed model according to the proposed stratified random sampling method. As a result, it was found that the crossed model by the stratified sampling method was more efficient than the simple model by the stratified sampling method when the difference between $P_h$ and $T_h$ is small. In particular, the bigger the difference between ${\pi }_{hA}$, ${\pi }_{hB}$, and ${\pi }_{hAB}$ and the smaller difference between $P_h$ and $T_h$, then the crossed model is more efficient than the simple model under stratified random sampling.

When trying to estimate the proportion of domestic violence and drug use experience in a social survey at the same time, a randomized response model that considers two sensitive variables can be used. At this time, the stratified random sampling design proposed in this paper can be applied to secure the representativeness of the sample, and has the advantage of enabling estimation for the entire group and estimation by demographic characteristics.

Finally based on this study, we will consider the two sensitive variables by applying the probability proportional to size design as larger sampling strategy.