The power of the optimal asymptotic tests of composite statistical hypotheses

The easily computable asymptotic power of the locally asymptotically optimal test of a composite hypothesis, known as the optimal C(alpha) test, is obtained through a "double" passage to the limit: the number n of observations is indefinitely increased while the conventional measure xi of the error in the hypothesis tested tends to zero so that xi(n)n((1/2)) --> tau not equal 0. Contrary to this, practical problems require information on power, say beta(xi,n), for a fixed xi and for a fixed n. The present paper gives the upper and the lower bounds for beta(xi,n). These bounds can be used to estimate the rate of convergence of beta(xi,n) to unity as n --> infinity. The results obtained can be extended to test criteria other than those labeled C(alpha). The study revealed a difference between situations in which the C(alpha) test criterion is used to test a simple or a composite hypothesis. This difference affects the rate of convergence of the actual probability of type I error to the preassigned level alpha. In the case of a simple hypothesis, the rate is of the order of n(-(1/2)). In the case of a composite hypothesis, the best that it was possible to show is that the rate of convergence cannot be slower than that of the order of n(-(1/2)) ln n.