Minimax Estimation of the Mixing Proportion of Two Known Distributions

Abstract

Abstract Let X 1 …, X n, be a sample from P ϑ = ϑP 1 + (1 − ϑ)P 0, with P 0 and P 1 known. Several estimators for ϑ have been proposed in the literature. The estimator proposed by Boes (1966) is minimax unbiased. In this article the unbiasedness is dropped and minimax estimators are derived with respect to the squared error loss function within the class of all moment-type estimators . For larger values of n, the minimax estimator coincides with a shrunken version of the Boes estimator. To give an example, let P ϑ have density fϑ(x) = ϑ(1 + 1.2x) + (1 − ϑ)(1 − .8x)(−.5 ≦ x ≦ .5). The Boes estimator is , with maximal risk equal to 3/n. For n ≧ 132 the minimax estimator is given by , with . Its maximal risk equals . For smaller values of n the minimax estimator is a bit harder to compute. Table 1 shows that for n = 20 the minimax estimator reduces the maximal risk from .150 to .088. In the last section minimax estimators are constructed with respect to the more general loss function L(ϑ, a) = w(ϑ)(ϑ − a)2. As an application it is shown how this estimator can be used in empirical Bayes classification.