Estimation of complex population parameters under the randomized response theory.

*(English)*Zbl 1366.62023
Chaudhuri, Arijit (ed.) et al., Data gathering, analysis and protection of privacy through randomized response techniques: qualitative and quantitative human traits. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-63570-9/hbk; 978-0-444-63571-6/ebook). Handbook of Statistics 34, 119-131 (2016).

Summary: 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 B. G. Greenberg et al. [“Application of randomized response technique in obtaining quantitative data”, J. Am. Stat. Assoc. 66, No. 334, 243–250 (1971; doi:10.2307/2283916)], 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 J. C. Deville [“Variance estimation for complex statistics and estimators: linearization and residual techniques”, Surv. Methodol. 25, No. 2, 193–203 (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.

For the entire collection see [Zbl 1349.62001].

For the entire collection see [Zbl 1349.62001].

##### MSC:

62D05 | Sampling theory, sample surveys |

94A62 | Authentication, digital signatures and secret sharing |

62P20 | Applications of statistics to economics |