## The Annals of Statistics

### The Geometry of Exponential Families

#### Abstract

There are two important spaces connected with every multivariate exponential family, the natural parameter space and the expectation parameter space. We describe some geometric results relating the two. (In the simplest case, that of a normal translation family, the two spaces coincide and the geometry is the familiar Euclidean one.) Maximum likelihood estimation, within one-parameter curved subfamilies of the multivariate family, has two simple and useful geometric interpretations. The geometry also relates to the Fisherian question: to what extent can the Fisher information be replaced by $-\partial^2/\partial\theta^2\lbrack\log f_\theta(x)\rbrack\mid_{\theta=\hat{\theta}}$ in the variance bound for $\hat{\theta}$, the maximum likelihood estimator?

#### Article information

Source
Ann. Statist., Volume 6, Number 2 (1978), 362-376.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344130

Digital Object Identifier
doi:10.1214/aos/1176344130

Mathematical Reviews number (MathSciNet)
MR471152

Zentralblatt MATH identifier
0436.62027

JSTOR

Subjects
Primary: 62F10: Point estimation

#### Citation

Efron, Bradley. The Geometry of Exponential Families. Ann. Statist. 6 (1978), no. 2, 362--376. doi:10.1214/aos/1176344130. https://projecteuclid.org/euclid.aos/1176344130