On a theorem of bahadur on the rate of convergence of test statistics

Abstract

Let xl, x2, * * *, xn be n independent and identically distributed random variables whose distribution depends on a parameter 0, 0 E 0. Let 00 be a subset of 0 and consider the test of the hypothesis that 0 E 00. Ln(X1, , xj) is the level attained by a test statistic T(xl, , * * X x,) in the sense that it is the maximum probability under the hypothesis of obtaining a value as large or larger than Tn where large values of Tn are significant for the hypothesis. Under some assumptions Bahadur [3] showed that where a non-null 0 obtains Ln cannot tend to zero at a rate faster than [p(0)]f where p is a function defined in terms of Kullback-Liebler information numbers. In this paper this result has been shown to be true without any assumptions whatsoever (Theorem 1). Some aspects of the relationship between the rate of convergence of Ln and rate of convergence of the size of the tests are also studied and an equivalence property is shown (Theorem 2).