Rényi statistics for testing composite hypotheses in general exponential models.

Abstract

We introduce a family of Renyi statistics of orders r is an element of R for testing composite hypotheses in general exponential models, as alternatives to the previously considered generalized likelihood ratio (GLR) statistic and generalized Wald statistic. If appropriately normalized exponential models converge in a specific sense when the sample size (observation window) tends to infinity, and if the hypothesis is regular, then these statistics are shown to be chi(2)-distributed under the hypothesis. The corresponding Renyi tests are shown to be consistent. The exact sizes and powers of asymptotically alpha-size Renyi, GLR and generalized Wald tests are evaluated for a concrete hypothesis about a bivariate Levy process and moderate observation windows. In this concrete situation the exact sizes of the Renyi test of the order r = 2 practically coincide with those of the GLR and generalized Wald tests but the exact powers of the Renyi test are on average somewhat better.