## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 33, Number 3 (1962), 986-1001.

### 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.

#### Article information

**Source**

Ann. Math. Statist. Volume 33, Number 3 (1962), 986-1001.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177704466

**Digital Object Identifier**

doi:10.1214/aoms/1177704466

**Mathematical Reviews number (MathSciNet)**

MR141184

**Zentralblatt MATH identifier**

0109.37605

**JSTOR**

links.jstor.org

#### Citation

Ferguson, Thomas S. Location and Scale Parameters in Exponential Families of Distributions. Ann. Math. Statist. 33 (1962), no. 3, 986--1001. doi:10.1214/aoms/1177704466. https://projecteuclid.org/euclid.aoms/1177704466.

#### Corrections

- See Correction: Thomas S. Ferguson. Correction Notes: Correction to "Location and Scale Parameters in Exponential Families of Distributions". Ann. Math. Statist., Volume 34, Number 4 (1963), 1603--1603.Project Euclid: euclid.aoms/1177703896