Large Deviations Theory in Exponential Families

Abstract

We consider repeated independent sampling from one member of an exponential family of probability distributions. The probability that the sample mean of $n$ such observations falls into some set $S_n$ is, by definition, a "large deviations", "small deviations", or "medium deviations" problem depending on the location of the set $S_n$ relative to the expectation of the distribution. We present a theorem which allows the accurate approximation of all such probabilities under a wide variety of circumstances. These approximations are shown to yield simple and numerically accurate expressions for the small sample power functions of hypothesis tests in the exponential family. Various large sample properties of exponential families are presented, many of which are seen to be extensions and refinements of familiar large deviations results. The method employed is to replace the given exponential family by a suitably modified normal translation family, which is shown to approximate the original family uniformly well over any bounded subset of the parameter space. The simple and tractable nature of normal translation families then provides our results.