Mallows has conjectured that among distributions which are Gaussian but for occasional contamination by additive noise, the one having least Fisher information has (two-sided) geometric contamination. A very similar problem arises in estimation of a nonnegative vector parameter in Gaussian white noise when it is known also that most [i.e., $(1 - \varepsilon)$] components are zero. We provide a partial asymptotic expansion of the minimax risk as $\varepsilon \rightarrow 0$. While the conjecture seems unlikely to be exactly true for finite $\varepsilon$, we verify it asymptotically up to the accuracy of the expansion. Numerical work suggests the expansion is accurate for $\varepsilon$ as large as 0.05. The best $l_1$-estimation rule is first- but not second-order minimax. The results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases.
Publié le : 1994-03-14
Classification:
Fisher information,
minimax decision theory,
least favorable prior,
nearly black object,
robustness,
white noise model,
62C20,
62C10,
62G05
@article{1176325368,
author = {Johnstone, Iain M.},
title = {On Minimax Estimation of a Sparse Normal Mean Vector},
journal = {Ann. Statist.},
volume = {22},
number = {1},
year = {1994},
pages = { 271-289},
language = {en},
url = {http://dml.mathdoc.fr/item/1176325368}
}
Johnstone, Iain M. On Minimax Estimation of a Sparse Normal Mean Vector. Ann. Statist., Tome 22 (1994) no. 1, pp. 271-289. http://gdmltest.u-ga.fr/item/1176325368/