Location and Scale Parameters in Exponential Families of Distributions

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.