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 optimalC (α) test, is obtained through a “double” passage to the limit: the numbern of observations is indefinitely increased while the conventional measure ξ of the error in the hypothesis tested tends to zero so that ξn n ½ → τ ≠ 0. Contrary to this, practical problems require information on power, say β(ξ,n ), for a fixed ξ and for a fixedn . The present paper gives the upper and the lower bounds for β(ξ,n ). These bounds can be used to estimate the rate of convergence of β(ξ,n ) to unity asn → ∞. The results obtained can be extended to test criteria other than those labeledC (α). The study revealed a difference between situations in which theC (α) 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 α. In the case of a simple hypothesis, the rate is of the order ofn -½ . 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 ofn -½ lnn .