We use a Bayesian version of the Cramér-Rao lower bound due to van Trees to give an elementary proof that the limiting distribution of any regular estimator cannot have a variance less than the classical information bound, under minimal regularity conditions. We also show how minimax convergence rates can be derived in various non- and semi-parametric problems from the van Trees inequality. Finally we develop multivariate versions of the inequality and give applications.
@article{1186078362,
author = {Gill, Richard D. and Levit, Boris Y.},
title = {Applications of the van Trees inequality: a Bayesian Cram\'er-Rao bound},
journal = {Bernoulli},
volume = {1},
number = {3},
year = {1995},
pages = { 59-79},
language = {en},
url = {http://dml.mathdoc.fr/item/1186078362}
}
Gill, Richard D.; Levit, Boris Y. Applications of the van Trees inequality: a Bayesian Cramér-Rao bound. Bernoulli, Tome 1 (1995) no. 3, pp. 59-79. http://gdmltest.u-ga.fr/item/1186078362/