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September, 1962 Location and Scale Parameters in Exponential Families of Distributions
Thomas S. Ferguson
Ann. Math. Statist. 33(3): 986-1001 (September, 1962). DOI: 10.1214/aoms/1177704466

Abstract

Location and scale parameters, on the one hand, and distributions admitting sufficient statistics for the parameters, on the other, have played a large role in the development of modern statistics. This paper deals with the problem of finding those distributions involved in the intersection of these two domains. In Sections 2 through 4 the preliminary definitions and lemmas are given. The main results found in Theorems 1 through 4 may be considered as a strengthening of the results of Dynkin [3] and Lindley [8]. Theorem 1 discovers the only possible forms assumed by the density of an exponential family of distributions having a location parameter. These forms were discovered by Dynkin under the superfluous assumptions that a density with respect to Lebesgue measure exist and have piecewise continuous derivatives of order one. Theorem 2 consists of the specialization of Theorem 1 to one-parameter exponential families of distributions. The resulting distributions, as found by Lindley, are either (1), the distributions of $(1/\gamma) \log X$, where $X$ has a gamma distribution and $\gamma \neq 0$, or (2), corresponding to the case $\gamma = 0$, normal distributions. In Theorem 3, the result analogous to Theorem 2 for scale parameters is stated. In Theorem 4, those $k$-parameter exponential families of distributions which contain both location and scale parameters are found. If the parameters of a two-parameter exponential family of distributions may be taken to be location and scale parameters, then the distributions must be normal. The final section contains a discussion of the family of distributions obtained from the distributions of Theorem 2 and their limits as $\gamma \rightarrow \pm \infty$. These limits are "non-regular" location parameter distributions admitting a complete sufficient statistic. This family of distributions is a main class of distributions to which Basu's theorem (on statistics independent of a complete sufficient statistic) applies. Furthermore, this family is seen to provide a natural setting in which to prove certain characterization theorems which have been proved separately for the normal and gamma distributions. Concluding the section is a theorem which, essentially, characterizes the gamma distribution by the maximum likelihood estimate of its scale parameter.

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Thomas S. Ferguson. "Location and Scale Parameters in Exponential Families of Distributions." Ann. Math. Statist. 33 (3) 986 - 1001, September, 1962. https://doi.org/10.1214/aoms/1177704466

Information

Published: September, 1962
First available in Project Euclid: 27 April 2007

zbMATH: 0109.37605
MathSciNet: MR141184
Digital Object Identifier: 10.1214/aoms/1177704466

Rights: Copyright © 1962 Institute of Mathematical Statistics

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Vol.33 • No. 3 • September, 1962
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