^{1}

^{*}

^{1}

In this study, we propose a two stage randomized response model. Improved unbiased estimators of the mean number of persons possessing a rare sensitive attribute under two different situations are proposed. The proposed estimators are evaluated using a relative efficiency comparison. It is shown that our estimators are efficient as compared to existing estimators when the parameter of rare unrelated attribute is known and in unknown case, depending on the probability of selecting a question.

The collection of data through direct questioning on rare sensitive issues such as extramarital affairs, family disturbances and declaring religious affiliation in extremism condition is far-reaching issue. Warner [

Later on, different modifications have been made to improve the methodology for collection of information. Some of them are Lee et al. [

Land et al. [

In this study, we propose improved estimators for the mean and its variance of the number of persons possessing a rare sensitive attribute based on stratified sampling by using Poisson distribution. The estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The proposed estimators are evaluated using a relative efficiency comparing the variances of the estimators reported in Lee et al. [

Consider the population of size N individuals which is divided into L subpopulations (strata) of sizes

(i) “I possessrare sensitive attribute A”

(ii) “Go to randomization device R_{h}_{2}”

with respective probabilities

The randomization device

(i) “I possess rare sensitive attribute A”

(ii) “I possess rare unrelated attribute Y”

with probabilities

By this randomized device, the probability of a yes response in stratum h is given by

where

Let

where

where

The variance of the estimator

where

Thus, the variance expression of the estimator

THEOREM 1.

Proof. From (3), we have

THEOREM 2. The unbiased estimator for

Proof.

Now, we consider the proportional and optimal allocations of the total sample size n into different strata. The method of proportional allocation is used to define sample sizes in each stratum depending on each stratum size. Since the sample size in each stratum is defined as

However, the optimal allocation is a technique to define sample size to minimize variance for a given cost or to minimize the cost for a specified variance. The

riable. In stratified sampling, let cost function is defined as

So the minimum variance of the estimator for the specified cost C under the optimum allocation of sample size is given by

In this section, the estimators for the mean number of rare sensitive attribute are proposed under the assumptions that the sizes of stratum are known; however, ^{th} stratum,

(i) “I possess a sensitive group A”

(ii) “Go to randomization device R_{h}_{2}”

The statements occur with respective probabilities

The two statements of the randomization device

(i) “I possess a sensitive attribute A”

(ii) “I possess unrelated attribute Y”

represented with respective probabilities

The probabilities of the yes responses for the first and second use of pair of randomization devices are respectively given by

and

where ^{th} stratum. We have

Following the expression given in Equations (12) and (13), we have the sample means for both set of responses as

and

By solving (15) and (16), we get estimators of

where

Puttinng (12), (13) and (14) in (19) we get

where

The stratified estimators of

THEOREM 3.

Proof.

Putting the values of

THEOREM 4. The variance of

where

Proof. Since

On putting (20) in (24) we have the theorem.

Corollary 1: An unbiased estimator for the variance of rare sensitive attribute is given by

It can be proved easily.

THEOREM 5.

Proof. From (18), we have

Corollary 2: An unbiased estimator for

where

Now under proportional allocation of sample size, the variance of

However, in optimum allocation, the sample size in stratum h is

and the variance of

Lee et al. [

where

For comparison of the proposed estimator with

Large samples are required to estimate the means of rare sensitive attribute. So we consider a large hypothetical population, in order to study the relative efficiency, setting

Let

From Equation (29) it evident that the relative efficiency of proposed estimator is free from the sample size n. We set the design probabilities as

Let

The relative efficiency of proposed estimator is free from the sample size n. For the analysis, the design probabilities are fixed as_{12} = 0.2, 0.3, 0.4, 0.5 and

estimator outer perform than

W_{1} = 0.4 | W_{1} = 0.6 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

P_{12} | λ_{1Y} | λ_{1A} | P_{11} = 0.6 | 0.7 | 0.8 | 0.9 | P_{11} = 0.6 | 0.7 | 0.8 | 0.9 | |

0.3 | 0.5 | 1.5 | 1.7346 | 1.5829 | 1.4758 | 1.3966 | 1.5630 | 1.4264 | 1.3299 | 1.2585 | |

1.5 | 1.5 | 1.9238 | 1.7016 | 1.5439 | 1.4266 | 1.7336 | 1.5334 | 1.3912 | 1.2855 | ||

1.5 | 0.5 | 2.2198 | 1.9173 | 1.6887 | 1.5016 | 2.0003 | 1.7277 | 1.5217 | 1.3531 | ||

0.4 | 0.5 | 1.5 | 1.8713 | 1.6667 | 1.5228 | 1.4169 | 1.6863 | 1.5018 | 1.3723 | 1.2768 | |

1.5 | 1.5 | 2.1435 | 1.8333 | 1.6166 | 1.4574 | 1.9316 | 1.6520 | 1.4567 | 1.3133 | ||

1.5 | 0.5 | 2.6070 | 2.1568 | 1.8251 | 1.5615 | 2.3492 | 1.9436 | 1.6447 | 1.4071 | ||

0.5 | 0.5 | 1.5 | 2.0097 | 1.7510 | 1.5701 | 1.4372 | 1.8109 | 1.5779 | 1.4148 | 1.2951 | |

1.5 | 1.5 | 2.3751 | 1.9699 | 1.6908 | 1.4885 | 2.2100 402 | 1.7751 | 1.5327 | 1.3413 | ||

1.5 | 0.5 | 3.0537 | 2.4245 | 1.9727 | 1.6238 | 2.7517 | 2.1848 | 1.7776 | 1.4633 | ||

0.6 | 0.5 | 1.5 | 1.6090 | 1.01489 | 1.2107 | 1.0910 | 1.9370 | 1.6545 | 1.4576 | 1.3135 | |

1.5 | 1.5 | 1.9600 | 1.4204 | 1.3225 | 1.1377 | 2.3596 | 1.9026 | 1.5921 | 1.3698 | ||

1.5 | 0.5 | 2.6727 | 1.6326 | 1.5961 | 1.2642 | 3.2177 | 2.4550 | 1.9215 | 1.5219 | ||

0.7 | 0.5 | 1.5 | 1.7147 | 1.4383 | 1.2464 | 1.1063 | 2.0642 | 1.7315 | 1.5005 | 1.3318 | |

1.5 | 1.5 | 2.1511 | 1.6900 | 1.3806 | 1.1616 | 2.5897 | 2.0346 | 1.6621 | 1.3984 | ||

1.5 | 0.5 | 3.1223 | 2.2915 | 1.7258 | 1.3150 | 3.7592 | 2.7587 | 2.0776 | 1.5831 |

P_{11} = P_{21} | P_{12} = P_{22} | T_{11} = T_{21} | T_{12} = T_{22} | λ_{1A} = λ_{2A} | λ_{1Y} = λ_{2Y} | RE (W_{1} = 0.4) | RE (W_{1} = 0.5) |
---|---|---|---|---|---|---|---|

0.6 | 0.6 | 0.3 | 0.2 | 1.5 | 0.5 | 12.5971 | 15.7464 |

1.5 | 1.5 | 16.9517 | 21.1896 | ||||

0.5 | 1.5 | 10.0051 | 12.5064 | ||||

0.3 | 1.5 | 0.5 | 10.3926 | 12.9908 | |||

1.5 | 1.5 | 13.9851 | 17.4814 | ||||

0.5 | 1.5 | 8.2542 | 10.3178 | ||||

0.4 | 1.5 | 0.5 | 8.1881 | 10.2352 | |||

1.5 | 1.5 | 11.0186 | 13.7732 | ||||

0.5 | 1.5 | 6.5033 | 8.1292 | ||||

0.5 | 1.5 | 0.5 | 5.9836 | 7.4795 | |||

1.5 | 1.5 | 8.0520 | 10.0651 | ||||

0.5 | 1.5 | 4.7524 | 5.9405 | ||||

0.6 | 0.6 | 0.4 | 0.2 | 1.5 | 0.5 | 3.1703 | 3.9629 |

1.5 | 1.5 | 4.4483 | 5.5603 | ||||

0.5 | 1.5 | 2.7607 | 2.4509 | ||||

0.3 | 1.5 | 0.5 | 2.5759 | 3.2198 | |||

1.5 | 1.5 | 3.6142 | 4.5178 | ||||

0.5 | 1.5 | 2.2431 | 2.8038 | ||||

0.4 | 1.5 | 0.5 | 1.9814 | 2.4768 | |||

1.5 | 1.5 | 2.7801 | 3.4752 | ||||

0.5 | 1.5 | 1.7254 | 2.1568 | ||||

0.5 | 1.5 | 0.5 | 1.3870 | 1.7338 | |||

1.5 | 1.5 | 1.9461 | 2.4326 | ||||

0.5 | 1.5 | 1.2078 | 1.5098 |

In this study, a two stage randomized response model is proposed with improved estimators for the mean and its variance of the number of persons possessing a rare sensitive attribute based on stratified sampling by using Poisson distribution. It is shown that our proposed method have better efficiencies than the existing randomized response model, when the parameter of rare unrelated attribute is known and in unknown case, depending on the probability of selecting a question. For future work, we can obtain more sensitive information from respondents by using stratified double sampling with the proposed model.

AbdulWakeel,MasoodAnwar, (2016) Improved Estimation of Rare Sensitive Attribute in a Stratified Sampling Using Poisson Distribution. Open Journal of Statistics,06,85-95. doi: 10.4236/ojs.2016.61011