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Pré-Publication, Document De Travail Année : 2024

The van Trees inequality in the spirit of Hajek and Le Cam

Résumé

We work out a version of the van Trees inequality in a Hajek--Le Cam spirit, i.e., under minimal assumptions that, in particular, involve no direct pointwise regularity assumptions on densities but rather almost-everywhere differentiability in quadratic mean of the model. Surprisingly, it suffices that the latter differentiability holds along canonical directions -- not along all directions. Also, we identify a (slightly stronger) version of the van Trees inequality as a very instance of a Cramer--Rao bound, i.e., the van Trees inequality is not just a Bayesian analog of the Cramer--Rao bound. We provide, as an illustration, an elementary proof of the local asymptotic minimax theorem for quadratic loss functions, again assuming differentiability in quadratic mean only along canonical directions.

Dates et versions

hal-04452222 , version 1 (12-02-2024)

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Élisabeth Gassiat, Gilles Stoltz. The van Trees inequality in the spirit of Hajek and Le Cam. 2024. ⟨hal-04452222⟩
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