Ratio tests under limiting normality

We propose a class of ratio tests that is applicable whenever a cumulation (of transformed) data is asymptotically normal upon appropriate normalization. The Karhunen–Loève theorem is employed to compute weighted averages. The test statistics are ratios of quadratic forms of these averages and hence scale-invariant, also called self-normalizing: The scaling parameter cancels asymptotically. Limiting distributions are obtained. Critical values and asymptotic local power functions can be calculated by standard numerical means. The ratio tests are directed against local alternatives and turn out to be almost as powerful as optimal competitors, without being plagued by nuisance parameters at the same time. Also in finite samples they perform well relative to self-normalizing competitors.