Extremal properties of likelihood-ratio quantizers

M hypotheses and a random variable Y with a different probability distribution under each hypothesis are considered. A quantizer is applied to form a quantized random variable gamma (Y). The extreme points of the set of possible probability distributions of gamma (Y), as gamma ranges over all quantizers, is characterized. Optimality properties of likelihood-ratio quantizers are established for a very broad class of quantization problems, including problems involving the maximization of an Ali-Silvey (1966) distance measure and the Neyman-Pearson variant of the decentralized detection problem.>