Abstract
In honor of the 100th birth anniversary of Lucien Le Cam (November 18, 1924–April 24, 2000), we work out a version of the van Trees inequality in a Hájek–Le Cam spirit, that is, 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 Cramér–Rao bound, that is, the van Trees inequality is not just a Bayesian analog of the Cramér–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.
Funding Statement
Elisabeth Gassiat was supported by Institut Universitaire de France and by ANR Grants ANR-21-CE23-0035-02 and ANR-23-CE40-0018-02.
Acknowledgments
The authors would like to thank David Pollard for suggesting to study the van Trees inequality under the angle of a Cramér–Rao bound for a location model, and for following and encouraging this work since 2001, when he delivered a series of lectures during the statistics semester at Institut Henri Poincaré, Paris.
Citation
Elisabeth Gassiat. Gilles Stoltz. "The van Trees Inequality in the Spirit of Hájek and Le Cam." Statist. Sci. 39 (4) 644 - 653, November 2024. https://doi.org/10.1214/24-STS941
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