Randomized response surveys of finite populations: A unified approach with quantitative data
The obtained results follow suggestions of \textit{M. E. Thompson} [Discussion of the paper by \textit{A. K. Adhikari}, \textit{A. Chaudhuri} and \textit{K. Vijayan}, Int. Stat. Rev. 52, 115-125 (1984; Zbl 0579.62007)]. A random response mechanism generates variables \(Z_ i\) independently and they are related with the true variable value \(Y_ i\) through a model such that \(E_ R(Z_ i)=a_ iY_ i+b_ i\) and \(E_ R(Z^ 2_ i)=a_ iY^ 2_ i+d_ i\). \(a_ i\), \(b_ i\) and \(d_ i\) are known variables and \(E_ R\) is the expectation with respect to the random response technique used. Two models \((M_ 1,M_ 2)\) are proposed for studying the behaviour of strategies. They fix certain properties of expectations, variances and covariances of the \(Y_ i's\). Classes of estimators are constructed. The non-existence of uniformly minimum variance estimators is proved in Theorem 1 for the class \(C_ 1\). Theorem 2 proves the optimality of a certain strategy in the class \(C_ 2\) under \(M_ 2\) and in \(C_ 1\) if \(M_ 1\) is acceptable. When the linear correlation coefficient is zero \((M_ 3)\) the estimator proposed has always better results and if the \textit{V. P. Godambe} and \textit{M. E. Thompson} permutation model \((M_ 4)\) [Ann. Stat. 1, 1212-1221 (1973; Zbl 0275.62011)] holds the strategy still is optimum in \(C_ 1.\) The paper brings an analysis of the optimality of the estimator when direct response is used. Different unequal probability strategies are studied and the errors formulae are worked out.
scientific article; zbMATH DE number 3996836
scientific article