## The Annals of Statistics

### Preferred Point Geometry and Statistical Manifolds

#### Abstract

A new mathematical object called a preferred point geometry is introduced in order to (a) provide a natural geometric framework in which to do statistical inference and (b) reflect the distinction between homogeneous aspects (e.g., any point $\theta$ may be the true parameter) and preferred point ones (e.g., when $\theta_0$ is the true parameter). Although preferred point geometry is applicable generally in statistics, we focus here on its relationship to statistical manifolds, in particular to Amari's expected geometry. A symmetry condition characterises when a preferred point geometry both subsumes a statistical manifold and, simultaneously, generalises it to arbitrary order. There are corresponding links with Barndorff-Nielsen's strings. The rather unnatural mixing of metric and nonmetric connections in statistical manifolds is avoided since all connections used are shown to be metric. An interpretation of duality of statistical manifolds is given in terms of the relation between the score vector and the maximum likelihood estimate.

#### Article information

Source
Ann. Statist., Volume 21, Number 3 (1993), 1197-1224.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176349258

Digital Object Identifier
doi:10.1214/aos/1176349258

Mathematical Reviews number (MathSciNet)
MR1241265

Zentralblatt MATH identifier
0798.62009

JSTOR