Abstract
The conditional large deviations theorem of Jing and Robinson (1994) is extended in the following sense. Consider a random sample of pairs of random vectors and the sample means of each of the pairs. For p ⩾ 1, the probability that first falls outside a certain p-dimensional convex set given that the second is fixed is shown to decrease with the sample size at an exponential rate which depends on the Kullback-Leibler distance between two distributions in an associated exponential familiy of distributions. Examples are given which include a method of computing the Bahadur exact slope for tests of certain composite hypotheses in exponential families.
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