Geodesic
Detection of a sparse-variable function
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 103-126 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We observe an unknown $d$-variable function $f=f(t)$, $t=(t_1,\dots,t_d)\in[0,\,1]^d,$ $f\in L_2([0,\,1]^d)$ in Gaussian white noise of level $\varepsilon>0$. We test the null hypothesis $H_0\colon f=0$ against the alternative $H_1$. Under the alternative, we suppose that unknown function is bounded away from zero: $$ \|f\|\ge r_\varepsilon,$$ for some positive family $\underset{\varepsilon\to0}{r_\varepsilon\to0}$. Moreover, we assume that unknown $d$-variable $f$ is a function of a smaller number of variables $s$ (“sparse variable” function), and this function satisfies some regularity constraints. We also consider the problem of adaptation in $k=1,\dots,s$. We assume that $d=d_\varepsilon\to\infty$. The integer $s\in\mathbb N$ could be fixed or $s=s_\varepsilon\to\infty$, $s=o(d)$. We study the minimax error probabilities and obtain the minimax separation rates that provide distinguishability in the problems. Then, we apply the results obtained for the case of the alternatives from the Sobolev balls with the remote $L_2$-ball. 
@article{ZNSL_2013_420_a5,
     author = {Yu. I. Ingster and I. A. Suslina},
     title = {Detection of a~sparse-variable function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {103--126},
     year = {2013},
     volume = {420},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a5/}
}
TY  - JOUR
AU  - Yu. I. Ingster
AU  - I. A. Suslina
TI  - Detection of a sparse-variable function
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 103
EP  - 126
VL  - 420
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a5/
LA  - ru
ID  - ZNSL_2013_420_a5
ER  - 
%0 Journal Article
%A Yu. I. Ingster
%A I. A. Suslina
%T Detection of a sparse-variable function
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 103-126
%V 420
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a5/
%G ru
%F ZNSL_2013_420_a5
Yu. I. Ingster; I. A. Suslina. Detection of a sparse-variable function. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 103-126. http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a5/

[1] C. Butucea, Yu. I. Ingster, “Detection of a sparse submatrix of a high-dimensional noisy matrix”, Bernoulli, 19:5B (2013), 2153–2779

[2] G. Gayraud, Yu. I. Ingster, “Detection of sparse additive variable functions”, Electronic J. Statist., 6 (2012), 1409–1448 | DOI | MR | Zbl

[3] Yu. I. Ingster, I. A. Suslina, Nonparametric Goodness-of-Fit Testing under Gaussian Model, Lect. Notes Statist., 169, 2003 | DOI | MR | Zbl

[4] Yu. I. Ingster, O. Lepski, “On multichannel signal detection”, Math. Methods Statist., 12 (2003), 247–275 | MR

[5] Yu. I. Ingster, I. A. Suslina, “On estimation and detection of smooth function of many variables”, Math. Methods Statist., 14 (2005), 299–331 | MR

[6] Yu. I. Ingster, I. A. Suslina, “Otsenivanie i proverka gipotez dlya funktsii iz tenzornykh proizvedenii prostranstv”, Zap. nauchn. semin. POMI, 351, 2007, 180–218 | MR