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STATISTICAL MORPHISMS AND RELATED

INVARIANCE PROPERTIES

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DOMINIQUE B. PICARD

*Université Paris VII, U.F.R. de Mathematiques, Tour 45-55, 5eme etage. 2,*

Place Jussieu 75251 Paris, France

CNRS Unites associées 1321 et 1323 Statistique des Processus en

Milieu Médical Université Paris V, 45 Rue des Saints Pères, 75005 Paris, France
(Received December 12, 1988; revised September 25, 1990)

**Abstract.**
Our aim is to investigate a way to characterize the
elements of a statistical manifold (the metric and the family of
connections) using invariance properties suggested by Le Cam's theory of
experiments. We distinguish the case where the statistical manifold is
flat. Then, there naturally exists an entropy and it is proven that
experiment invariance is equivalent to entropy invariance. If the
statistical manifold is not flat, we introduce a notion of local
invariance of selected order associated to the asymptotic (on *n*
observations, *n* tending to infinity) expansion of the power of the
Neymann Pearson test in a contiguous neighborough of some point. This
invariance provides a substantial number of morphisms. This was not
always true for the entropy invariance: particularly, the case of
Gaussian experiments is investigated where it can be proven that entropy
invariance does not characterize a metric or a family of connections.

*Key words and phrases*:
Statistical manifold, Amari connections,
comparison of experiments, likelihood expansions, asymptotic properties of
tests.

**Source**
( TeX ,
DVI ,
PS )