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Statistical morphisms and related invariance properties. (English) Zbl 0782.62013

Summary: 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 L. Le Cam’s theory of experiments [see “Asymptotic methods in statistical decision theory.” (1986; Zbl 0605.62002)]. 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 neighborhood 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.

MSC:

62B15 Theory of statistical experiments
62F05 Asymptotic properties of parametric tests
53B99 Local differential geometry

Citations:

Zbl 0605.62002
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References:

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