Semiparametric testing of statistical functionals revisited

Abstract

Along the lines of Janssen's and Pfanzagl's work the testing theory for statistical functionals is further developed for non-parametric one-sample problems. Efficient tests for the one-sided and two-sided problems are derived for nonparametric statistical functionals. The asymptotic power function is calculated under implicit alternatives and hypotheses, which are given by the functional itself, for the one- sided and two-sided cases. Under mild regularity assumptions it is shown that these tests are asymptotic most powerful. The combination of the modern theory of Le Cam and approximation in limit experiments provide a deep insight into the upper bounds for asymptotic power functions of tests for the one-sided and two-sided problems. As example tests based on the von Mises functional are treated in nonparametric context.