Bayesian Estimation Using Warner's Randomized Response Model through Simple and Mixture Prior Distributions

Abstract

Bayesian estimation of population proportion of a sensitive characteristic is proposed by adopting a simple beta distribution and a mixture of Beta distributions as quantification of prior information using simple random sampling with replacement. In the sequel application of the stratified random sampling is also studied in Bayesian scenario. It is assumed that data are collected through Warner (1965) randomized response technique. To study the performance of Bayesian estimators we have used Mean Squared Error (MSE) and/or Relative Efficiency (RE) as performance criterion. Further, comparison of the suggested estimator is made with Kim et al. (2006) stratified estimator and usual maximum likelihood estimator in case of stratified random sampling. It is observed that unlike the moment and maximum likelihood methods, proposed Bayesian estimation method is free of the problems of having an estimate of population proportion outside the interval (0, 1) and large variance when the sample proportion of yes responses is very low or very high.

Keywords:

• Bayesian estimation
• Mixture prior information
• Population proportion
• Randomized response technique
• Stratified random sampling
• Sensitive attributes

Mathematics Subject Classification:

• 62D05

Acknowledgments

The authors are grateful to the editor, the associate editor and the referees for their recommendations and proposals which helped in improving substantially the earlier version of this article. The authors are indebted to Quaid-i-Azam University, Islamabad, Pakistan, for providing excellent research facilities. The author Muhammad Riaz is also grateful to King Fahad University of Petroleum and Minerals, Dhahran, Saudi Arabia, for ensuring an outstanding research environment and facilities.