An Efficient Survey Technique for Estimating the Proportion and Sensitivity Attributes in a Dichotomous Finite Populatio
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RESEARCH ARTICLE
An Efficient Survey Technique for Estimating the Proportion and Sensitivity Attributes in a Dichotomous Finite Population Amod Kumar1 • G. N. Singh1 • Gajendra K. Vishwakarma1
Received: 20 December 2016 / Revised: 22 October 2018 / Accepted: 19 December 2018 The National Academy of Sciences, India 2019
Abstract In this paper, a simple survey technique is applied to estimate the population proportion p of a sensitive trait, in addition to T, the probability that a respondent truthfully states that he or she bears a sensitive character when questioned directly and examined its properties. It has been found that the suggested model is efficient. Numerical illustrations are presented to support the theoretical results. Keywords Randomized response Direct response Estimation of proportion Privacy and sensitive attributes Mathematics Subject Classification 62D05
1 Introduction Refusals to respond or intentionally misleading replies are known to be two of the main sources of non-sampling bias in sample surveys of human populations. It is difficult to collect reliable data from interviewees and hard to raise the quality of responses when the survey topic is sensitive in nature. To obtain more reliable information in interviews, Warner [1] developed a pioneering technique to estimate the population proportion of individuals who possess & Amod Kumar [email protected] G. N. Singh [email protected] Gajendra K. Vishwakarma [email protected] 1
Department of Applied Mathematics, Indian Institute of Technology (ISM) Dhanbad, Dhanbad 826004, India
sensitive characteristic from a survey data. When every person in a population either belongs to sensitive group A or Ac , the respondents selected in a sample requested to select a question ‘‘Do you belong to A?’’ or ‘‘Do you belong to Ac ?’’ with probabilities P and ð1 PÞ, respectively, without revealing to the interviewer which of the alternative questions has been chosen. With the help of a randomized device, the respondent replies only ‘‘Yes’’ or ‘‘No’’ answers in a random sample of n respondents. Assuming truthful reporting, the probability of ‘‘Yes’’ answer hw , is given by hw ¼ Pp þ ð1 PÞð1 pÞ
ð1Þ
Warner [1] showed that the maximum likelihood estimate is unbiased, and its value is given by h^w ð1 PÞ ; P 6¼ 0:5 ð2Þ p^w ¼ ð2P 1Þ where h^w is the observed proportion of ‘‘Yes’’ answer in the sample. Since h^w follows the binomial distribution with parameters n and hw , the variance of the estimator p^w is given by V ðp^w Þ ¼
pð1 pÞ Pð 1 PÞ þ n nð2P 1Þ2
ð3Þ
It is clear from Eq. (3) that the variance of the Warner [1] estimator has two components. The first part is the usual binomial variance associated with a direct question and completely truthful replies by all respondents and other one is the additional term or cost one pays for the uncertainty associated with the randomized response. Following the work of Warner, many modifications were proposed [2–12]. Some other works on randomized response technique (RRT) in rece
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