**ABSTRACT**

**Background and Objective:**The crux of this paper is to develop a new a distinct question randomized response model for estimating a rare sensitive attribute using Poisson distribution.

**Methodology:**The equilibrium point of the model was investigated and a new stratified sampling and stratified sampling randomized response model is proposed.

**Results:**It has suggested an unbiased estimator of the mean number of persons possessing the rare sensitive attribute in presence of the unknown proportion of persons possessing a rare unrelated attribute. The addressed issue is resolved by using Lagrange multipliers technique and the optimum allocation is acquired in the form of fuzzy numbers.

**Conclusion:**Anew dexterous stratified randomized response model has been proposed and properties of the proposed randomized response model have been studied. Numerical it has shown that the suggested randomized response model is superior one.

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**How to cite this article**

*Trends in Applied Sciences Research, 14: 1-6.*

**URL:**https://scialert.net/abstract/?doi=tasr.2019.1.6

**INTRODUCTION**

The randomized response (RR) data-gathering device to procure trustworthy data on sensitive issues by protecting privacy of the respondent was first developed by Warner^{1}. Feeling that the co-operation of the respondent might be further enhanced if one of the two questions referred to a non-sensitive, innocuous attribute, say Y, unrelated to the sensitive attribute A. Horvitz *et al*.^{2} proposed an unrelated question randomized response model. Greenberg *et al*.^{3} provided theoretical framework for a rectification to the Warner’s^{1} model envisaged by Horvitz *et al*.^{2}. Numerous randomized response techniques have been developed for reducing non sampling errors in sample surveys, protecting a respondent’s privacy and increasing response rates. The key references of the randomized response model are Singh and Mathur^{4,5}, Singh *et al*.^{6}, Kim and Warde^{7}, Kim and Elam^{8,9} and Singh and Tarray^{10,11}.

Land *et al*.^{12} have considered a different and unique problem where the number of persons possessing a rare sensitive attribute is very small and a huge sample size is required to estimate this number. They proposed a method to estimate the mean of the number of persons possessing a rare sensitive attribute by utilizing the Poisson distribution in survey sampling. Land *et al*.^{12} have discussed two different situations that when the proportion of persons possessing a rare unrelated attribute is known and that when it is unknown. Lee *et al*.^{13} has extended the studies of Land *et al*.^{12} to the stratified sampling.

Singh and Tarray^{14} have further considered the problem of estimating the mean of the number of persons possessing a rare sensitive attribute using the Poisson distribution in the situation where the proportion of persons possessing a rare unrelated attribute is known. Singh and Tarray^{14} have suggested an alternative randomized response model based on Singh *et al*.^{15} model and studied its properties in presence of known population proportion of rare unrelated attributes. The main problem with the use of the method due to Singh and Tarray^{14} is that sometimes the mean value of the rare unrelated attribute remains unknown.

**ESTIMATION OF PROPORTION OF A RARE SENSITIVE ATTRIBUTE WHEN PROPORTION OF A RARE UNRELATED ATTRIBUTE IS UNKNOWN**

Let π_{1} be the true proportion of the rare sensitive attribute A_{1} in the population U. For example, the proportion of AIDS/HIV patients who continue having affairs with strangers, the proportion of persons who have witnessed a murder, the proportion of persons who are told by the doctors that they will not survive long due to a ghastly disease, for more examples, the reader is referred to Land *et al*.^{12}. Consider selecting a large sample of n persons from the population such that as n → ∞ and π_{1}→0 then, lim (n π_{1}) = δ_{1} (finite). Let π_{2} be the true proportion of the population having the rare unrelated attribute A_{2} such that as n → ∞ and π_{2}→0 then, lim (n π_{2}) = δ_{2} (finite and known). For instance, π_{2} might be the proportion of persons who are born exactly at 12:00 o’clock, the proportion of babies born blind, see Land *et al*.^{12}.

In the proposed procedure, each respondent in the sample of n persons, selecting using simple random sampling with replacement (SRSWR) from the given population, is requested to use the deck of cards marked as Deck-I and Deck-II. Each respondent in the sample is requested to use Deck-I consists of three types of cards bearing statements:

• | Do you possess the rare sensitive attribute A_{1}? |

• | Do you possess the rare unrelated attribute A_{2}? |

• | Draw one more card |

with probabilities P_{1}, P_{2} and P_{3}, respectively such that . The respondent is required to draw one card randomly from Deck-I and give answer in term of “Yes” or “No” according to his/her actual status and the statement, (i) or (ii), drawn. However if the statement (iii) is drawn, he/she is required to repeat the above process without replacing that card. If the statement (iii) is drawn in the second phase, he/she is directed to report “No”. If m is the total number of cards in the Deck-I, the probability of a “Yes” answer is given by:

(1) |

Note that both attributes A_{1} and A_{2} are very rare in population. Assuming that, as → ∞ and θ_{1} → 0 such that (finite), thus it is clear that:

(2) |

Next, the respondent is requested to use Deck-II consists of three types of cards bearing statements:

• | Do you possess the rare sensitive attribute A_{1}? |

• | Do you possess the rare unrelated attribute A_{2}? |

• | Draw one more card |

with probabilities T_{1}, T_{2} and T_{3}, respectively such that . The respondent is required to draw one card randomly from Deck-II and give answer in term of “Yes” or “No” according to his/her actual status and the statement, (i) or (ii) drawn. However, if the statement (iii) is drawn, he/she is required to repeat the above process without replacing that card. If the statement (iii) is drawn in the second phase, he/she is instructed to report “No”. If m is the total number of cards in the Deck - II, the probability of a “Yes” answer is given by:

(3) |

As before assuming that as → ∞ and θ_{2} → 0 such that (finite). Thus it is obvious that:

(4) |

By following the procedure as adopted by Singh and Tarray^{14}, we have:

(5) |

and:

(6) |

where, y_{1i} and y_{2i} denotes the observed values in the first and the second response from the ith respondent, respectively. Solving Eq. 5 and 6 for it has established the following theorems.

**Theorem 1:** An unbiased estimator of the parameter δ_{1} for the rare sensitive attribute A_{1} is given by:

(7) |

Where:

and:

**Proof:** Since y_{1i }~ iid Poisson (δ^{*}_{1}) and y_{2i} ~ iid Poisson (δ^{*}_{2}), thus by taking expected value on both sides of the Eq. 7, we have:

which proves the theorem.

**Theorem 2:** The variance of the unbiased estimator of the parameter δ^{*}_{1} is given by:

(8) |

**Proof:** Since y_{1i}~ iid Poisson (δ^{*}_{1}) and y_{2i}~ iid Poisson (δ^{*}_{2}), thus V(y_{1i}) = δ^{*}_{1} and V(y_{2i}) = δ^{*}_{2}. It is to be mentioned that both responses are not independent, thus we have Eq. 9 where:

(9) |

(10) |

(11) |

and

(12) |

Putting Eq. 10-12 in 9 it established the theorem.

**Corollary 1:** An unbiased estimator to estimate the parameter δ_{2} for rare unrelated attribute A_{2} is given by:

(13) |

with the variance:

(14) |

**Proof:** Analogous to the proof of the theorems 1 and 2.

**Corollary 2:** An unbiased estimator of the variance of the estimator _{} is given by:

(15) |

and an unbiased estimator of the variance of the estimator is given by:

(16) |

where,_{} and are, respectively defined in Eq. 7 and 13, respectively.

**RELATIVE EFFICIENCY**

The percentage of relative efficiency of the proposed estimator _{} with respect to the Land *et al*.^{12} estimator δ_{1}^{*} is given by:

(17) |

Where:

See Land *et al*.^{12}, Eq. 14, p. 7.

It is observed from Eq. 17 that the percentage of relative efficiency of the proposed estimator_{} with respect to Land *et al*.^{12} estimator ^{} is free from the sample size n. To see the performance of the proposed estimator _{} relative to Land *et al*.^{12} estimator ^{}, it has computed the values of PRE (_{}, ^{}) using the formula given in Eq. 17 for fixed (m =100) and different parametric values as given in Table 1. The resulting values of PRE (_{},^{}) are shown in Table 1.

Table 1 exhibited that the values of PREs are greater than 100 for all the parametric values considered here. Thus the proposed procedure is better than the Land *et al*.^{12} procedure.

Table 1: | Percentage of relative efficiency of the proposed estimator _{1} with respect to Land et al.^{12} estimator |

For the choice of δ_{1}, δ_{2} as 0.5, 1.50, the percentage of relative efficiency remains considerably larger than the other two cases, which reveals that it is appropriate to use the rare unrelated attribute Y, one with a mean value greater than that of the rare sensitive attribute A without affecting the cooperation of the respondents in using the suggested randomization device. The choice of (P_{i}, T_{i} ), I = 1, 2 should be made in such a way that the respondents should not feel that their privacy is threatened, while the difference (P_{1}T_{2}-P_{2}T_{1}) should be kept large as compared to P_{1}-T_{1}. Finally, our recommendation is to use the suggested estimator _{1} in practice.

**CONCLUSION**

This study advocates the problem where the number of persons possessing a rare sensitive attribute is very small and huge sample size is required to estimate. A more practical situation is discussed, when the proportion of persons possessing a rare unrelated attributes is unknown. Properties of the proposed randomized response model have been studied along with recommendations. Efficiency comparison is worked out to investigate the performance of the suggested procedures. It is interesting to mention that the proposed procedure is superior.

**SIGNIFICANCE STATEMENT**

This study discovers a new Stratified randomized response model and random sampling is generally obtained by dividing the population into non-overlapping groups called strata and selecting a simple random sample from each stratum. An RR technique using a stratified random sampling gives the group characteristics related to each stratum estimator. Also, stratified sample protect a researcher from the possibility of obtaining a poor sample. This study will help the researchers to uncover the critical areas related to randomized response technique. For the future research, researcher can be considering a new theory for randomized response model.

**ACKNOWLEDGMENT**

The authors are grateful to the referees for there valuable suggestions.

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