The van Trees inequality in the spirit of Hajek and Le Cam
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910324301496320 |
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| author | Elisabeth, Gassiat Gilles, Stoltz |
| author_facet | Elisabeth, Gassiat Gilles, Stoltz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06431 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The van Trees inequality in the spirit of Hajek and Le Cam Elisabeth, Gassiat Gilles, Stoltz Statistics Theory 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. |
| title | The van Trees inequality in the spirit of Hajek and Le Cam |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2402.06431 |