Open Access
October, 1968 Large Deviations Theory in Exponential Families
Bradley Efron, Donald Traux
Ann. Math. Statist. 39(5): 1402-1424 (October, 1968). DOI: 10.1214/aoms/1177698121


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.


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Bradley Efron. Donald Traux. "Large Deviations Theory in Exponential Families." Ann. Math. Statist. 39 (5) 1402 - 1424, October, 1968.


Published: October, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0196.20502
MathSciNet: MR234552
Digital Object Identifier: 10.1214/aoms/1177698121

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 5 • October, 1968
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