The easily computable asymptotic power of the locally asymptotically optimal test of a composite hypothesis, known as the optimal C(α) test, is obtained through a “double” passage to the limit: the number n of observations is indefinitely increased while the conventional measure ξ of the error in the hypothesis tested tends to zero so that ξnn½ → τ ≠ 0. Contrary to this, practical problems require information on power, say β(ξ,n), for a fixed ξ and for a fixed n. 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 as n → ∞. The results obtained can be extended to test criteria other than those labeled C(α). The study revealed a difference between situations in which the C(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 of n-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 of n-½ ln n.
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