Study of the effect of randomized response techniques on the Jarque and Bera test: real survey application

Expressions for the coefficients of skewness and kurtosis of the estimator of the population proportion of a sensitive characteristic under Warner’s model are developed in this paper. It is a well-established fact that the variance of Warner’s estimator exceeds that of the direct questioning method. In contrast, the coefficients of skewness and kurtosis are shown to decrease. Simulated values of these coefficients are compared with their corresponding theoretical values. The effect of Warner’s randomized response technique on the adjusted Jarque–Bera (AJB) test of normality is examined in comparison with estimates based on direct questioning. The likelihood ratio test and the binomial test are also included in the analysis. The findings reveal the surprising result that an estimate of proportion obtained through direct questioning may not follow a normal distribution, whereas the corresponding estimate derived from a randomized response technique may exhibit approximate normality. Assumptions of normality commonly adopted by researchers for randomized response estimators of proportion are also discussed. The results are further supported by an application to real data.