## The Annals of Statistics

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

### Preferred Point Geometry and Statistical Manifolds

Frank Critchley, Paul Marriott, and Mark Salmon

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 53B99: None of the above, but in this section

Secondary: 62F05: Asymptotic properties of tests 62F12: Asymptotic properties of estimators

**Keywords**

Preferred point geometry statistical manifolds Amari's expected geometry

#### Citation

Critchley, Frank; Marriott, Paul; Salmon, Mark. Preferred Point Geometry and Statistical Manifolds. Ann. Statist. 21 (1993), no. 3, 1197--1224. doi:10.1214/aos/1176349258. https://projecteuclid.org/euclid.aos/1176349258