Estimation of Complex Population Parameters Under the Randomized Response Theory

When a sensitive quantitative variable is under study and the Randomized Response Theory is adopted, a great deal of literature has been devoted to the estimation of the population mean (or total) or - at most - simple functions of population totals. However, in many real surveys the main interest might rely on the estimation of a complex parameter, usually a nonlinear combination of population totals. Hence, in order to face with this problem, we suppose to collect data by means of the well-known unrelated question method proposed by Greenberg et al. (1971), and under the design-based framework, we propose to handle such a complex parameter as a population functional by suitably extending the linearization approach proposed by Deville (1999). The considered strategy permits to obtain parameter estimation by means of the substitution method based on the empirical functional, and to achieve the corresponding variance estimator. Some selected illustrative examples are provided mostly concerning the estimation of two inequality indices, namely the Gini concentration index and the Atkinson index, widely discussed in the social and economic literature.