Location and Scale Parameters in Exponential Families of Distributions
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  and Lindley . 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.
Permanent link to this document: http://projecteuclid.org/euclid.aoms/1177704466
Digital Object Identifier: doi:10.1214/aoms/1177704466
Mathematical Reviews number (MathSciNet): MR141184
Zentralblatt MATH identifier: 0109.37605