Rényi statistics for testing composite hypotheses in general exponential models

Abstract

We introduce a family of Rényi statistics of orders r ∈ 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 χ²-distributed under the hypothesis. The corresponding Rényi tests are shown to be consistent. The exact sizes and powers of asymptotically α-size Rényi, GLR and generalized Wald tests are evaluated for a concrete hypothesis about a bivariate Lévy process and moderate observation windows. In this concrete situation the exact sizes of the Rényi test of the order r = 2 practically coincide with those of the GLR and generalized Wald tests but the exact powers of the Rényi test are on average somewhat better.

Keywords:

• Natural exponential models
• Testing composite hypotheses
• Generalized likelihood ratio statistics
• Generalized Wald statistics
• Rényi statistics
• Hypotheses about Lévy processes

We thank two referees for their critical comments and helpful suggestions leading to a substantially rewritten revised version. This paper was supported by the grants BMF2003-00892, BMF2003-04820 and GACR 201/02/1391.