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. All it requires is the orthonormal expansion using the Karhunen-Loève [KL] theorem, which is employed to compute weighted averages of the data. The test statistics are ratios of quadratic forms of these averages, and hence turn out to be scale-invariant by construction: The scaling parameter cancels asymptotically without having to be estimated. Limiting distributions are obtained. Critical values and asymptotic local power functions can be calculated by means of standard numerical means. The method to construct such ratio tests is illustrated with three limiting normal cases (Brownian motion, Brownian bridge, and Ornstein-Uhlenbeck process), which are applicable in many empirical situations. The ratio tests are directed against local alternatives and turn out to be almost as powerful as optimal competitors.