Two stage randomized response schemes for estimating a multinomial

Abstract

Warner's (1965) original randomized response technique for dichotomous data was extended by Abul-Ela, et al. (1967) to the case of a multinomial with t ≥ 3 groups, r of which are stigmatizing. A competing approach for estimating a multinomial via the randomized response technique is discussed by Warner (1971). Yet a third approach, for the multinomial is considered by Greenberg, et al. (1969) and is based on using an alternate independent question.

Here we outline some alternative schemes for estimating the t group proportions when r < t −1, using only one sample. Their realizations for any sampled individual, constitute two-stage schemes. The second stage is conditional on the random individual's response in the first stage.

In Section 1 we briefly summarize the three existing methods. Section 2 describes the new procedures and the resulting estimators. In Section 3 we identify the new schemes as special cases of Warner's (1971) general linear randomized response model thereby obtaining alternative estimators based on modified generalized least squares method due to Zellner (1962).

Keywords:

• randomized response
• estimating a multinomial
• two stage schemes
• dichotomous data
• general linear randomized response model
• modified generalized least secures method
• Warner's general linear model