On preferred point geometry in statistics

Critchley, Frank; Marriott, Paul and Salmon, Mark (2002). On preferred point geometry in statistics. Journal of Statistical Planning and Inference, 102(2) pp. 229–245.

DOI: https://doi.org/10.1016/S0378-3758(01)00115-X

Abstract

A brief synopsis of progress in differential geometry in statistics is followed by a note of some points of tension in the developing relationship between these disciplines. The preferred point nature of much of statistics is described and suggests the adoption of a corresponding geometry which reduces these tensions. Applications of preferred point geometry in statistics are then reviewed. These include extensions of statistical manifolds, a statistical interpretation of duality in Amari's expected geometry, and removal of the apparent incompatibility between (Kullback–Leibler) divergence and geodesic distance. Equivalences between a number of new expected preferred point geometries are established and a new characterisation of total flatness shown. A preferred point geometry of influence analysis is briefly indicated. Technical details are kept to a minimum throughout to improve accessibility.

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