The obtained results follow suggestions of M. E. Thompson [Discussion of the paper by A. K. Adhikari, A. Chaudhuri and 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 V. P. Godambe and 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.