Asymptotics of guarded weights of evidence

A test at a significance level a yields a decision which can be quantified by assigning the value 1 if the null hypothesis is rejected and 0 otherwise. The theory of guarded weights of evidence for the alternative (Blyth and Staudte, 1995. Statist. Probab. Lett. 23, 45–52) generalizes the usual Neyman–Pearson theory by considering L₂ norm risk. The result is a weight of evidence for the alternative which preserves the bound a on the risk of making a type I error and which takes values between 0 and 1. In this paper we establish, under quite general conditions, the asymptotic convergence of guarded weights of evidence. Both fixed and local alternatives are considered. These results are then applied to yield a consistency theorem, a version of the central limit theorem for guarded weights of evidence and an asymptotic result for guarded weights of evidence based on the F-statistic which in turn is applied to one-way ANOVA.