Geodesic
Minimax risk for quadratically convex sets
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 15, Tome 368 (2009), pp. 181-189 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider the problem of estimating the vector $\theta=(\theta_1,\theta_2,\dots)\in\Theta\subset l_2$ on the observations $y_i=\theta_i+\sigma_i\mathbf x_i$, $ i=1,2,\dots$, where $\mathbf x_i$ are i.i.d. $\mathcal N(0,1)$, the parametric set $\Theta$ is compact, orthosymmetric, convex and quadratically convex. We show that in that case the minimax risk is not very different from $\sup\mathfrak R_L(\Pi)$, where $\mathfrak R_L(\Pi)$ is the minimax linear risk in the same problem with the parametric set $\Pi$ and $\sup$ is taken over all the hyperrectangles $\Pi\subset\Theta$. Donoho, Liu, and McGibbon (1990) have obtained this result for the case of equal $\sigma_i$, $i=1,2,\dots$. Bibl. – 4 titles. 
@article{ZNSL_2009_368_a13,
     author = {S. V. Reshetov},
     title = {Minimax risk for quadratically convex sets},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {181--189},
     year = {2009},
     volume = {368},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_368_a13/}
}
TY  - JOUR
AU  - S. V. Reshetov
TI  - Minimax risk for quadratically convex sets
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2009
SP  - 181
EP  - 189
VL  - 368
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2009_368_a13/
LA  - ru
ID  - ZNSL_2009_368_a13
ER  - 
%0 Journal Article
%A S. V. Reshetov
%T Minimax risk for quadratically convex sets
%J Zapiski Nauchnykh Seminarov POMI
%D 2009
%P 181-189
%V 368
%U http://geodesic.mathdoc.fr/item/ZNSL_2009_368_a13/
%G ru
%F ZNSL_2009_368_a13
S. V. Reshetov. Minimax risk for quadratically convex sets. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 15, Tome 368 (2009), pp. 181-189. http://geodesic.mathdoc.fr/item/ZNSL_2009_368_a13/

[1] N. N. Chentsov,, “Otsenka neizvestnoi plotnosti raspredeleniya po nablyudeniyam”, DAN SSSR, 147 (1962), 45–48 | Zbl

[2] M. S. Pinsker, “Optimalnaya filtratsiya kvadratichno-integriruemykh signalov na fone gaussovskogo shuma”, Problemy peredachi informatsii, 16:2 (1980), 52–68 | MR | Zbl

[3] I. A. Ibragimov, R. Z. Has'minskii, “Nonparametric estimation of the value of a linear functional in Gaussian white noise”, Theory Probab. Appl., 29 (1984), 1–32 | MR | Zbl

[4] D. L. Donoho, R. C. Liu, B. MacGibbon, “Minimax risk over hyperrectangles, and implications”, Ann. Statist., 18:3 (1990), 1416–1437 | DOI | MR | Zbl