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arXiv · 2402.06431

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

Gassiat Elisabeth·Stoltz Gilles
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Abstract

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.

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Gassiat Elisabeth, Stoltz Gilles. 2024-02-09. The van Trees inequality in the spirit of Hajek and Le Cam. https://arxiv.org/abs/2402.06431

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