Some New Statistics for Testing Hypotheses in Parametric Models

The paper deals with simple and composite hypotheses in statistical models with i.i.d. observations and with arbitrary families dominated byσ-finite measures and parametrized by vector-valued variables. It introducesφ-divergence testing statistics as alternatives to the classical ones: the generalized likelihood ratio and the statistics of Wald and Rao. It is shown that, under the assumptions of standard type about hypotheses and model densities, the results about asymptotic distribution of the classical statistics established so far for the counting and Lebesgue dominating measures (discrete and continuous models) remain true also in the general case. Further, these results are extended to theφ-divergence statistics with smooth convex functionsφ. The choice ofφ-divergence statistics optimal from the point of view of power is discussed and illustrated by several examples.