On the Optimality of Sufficient Statistics-Based Quantizers

Let X be a random variable taking values in a set X, and let {P-?; ? ? Q} be a family of distributions indexed by the parameter vector ? taking values in a set Q. A quantized random variable y(X) is obtained by employing a quantizer y : X? {1, . . . , K}. It is shown that any extreme point of the set of all possible probability distributions of y(X) can be achieved by a deterministic quantizer that decides based only on the sufficient statistics. Using this fact, optimality properties of deterministic sufficient statistics-based quantizers are established for the problem of parameter estimation. It is proven that there always exists an optimal partitioning of sufficient statistics into K convex polytopes which maximizes the trace of the Fisher information matrix when {P-?; ? E Q} belongs to the exponential family. Furthermore, the optimality of likelihood ratio statistic for simple hypothesis testing follows as a consequence of this result, thereby demonstrating a link between parameter estimation and hypothesis testing.