caa a22 4500 605524521 CHVBK 20210128100754.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s10958-015-2302-z doi (NATIONALLICENCE)springer-10.1007/s10958-015-2302-z Detection of a Sparse Variable Function [Elektronische Daten] [Yu. Ingster, I. Suslina] We observe an unknown d-variable function f = f(t), t = (t 1, . . . , t d) ∈ [0, 1] d , f ∈ L 2([0, 1] d ), in Gaussian white noise of level ε > 0. We test the null hypothesis H 0 : f = 0 against an alternative H 1. Under the alternative, we assume that the unknown function is separated from zero: f ≥ r ε $$ \left\Vert f\right\Vert \ge {r}_{\varepsilon } $$ for some positive family r ε → ε → 0 0 $$ {r}_{\varepsilon}\underset{\varepsilon \to 0}{\to }0 $$ . Moreover, we assume that the unknown d-variable function f is a function of a smaller number of variables s ("sparse variable” function) that satisfies some regularity constraints. We also consider the problem of adaptation in k = 1, . . . , s. We assume that d = d ε → ∞. The integer s ∈ ℕ is either fixed or s = s ε → ∞, s = o(d). We study minimax error probabilities and obtain minimax separation rates that provide distinguishability in the problems. Then we apply the results obtained in the case of alternatives from Sobolev balls with a deleted L 2-ball. Springer Science+Business Media New York, 2015 Ingster Yu St. Petersburg National Research University of Information Technologies, Mechanics, and Optics, St. Petersburg, Russia aut Suslina I. St. Petersburg National Research University of Information Technologies, Mechanics, and Optics, St. Petersburg, Russia aut Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 206/2(2015-04-01), 181-196 1072-3374 206:2<181 2015 206 10958 https://doi.org/10.1007/s10958-015-2302-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-015-2302-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ingster Yu St. Petersburg National Research University of Information Technologies, Mechanics, and Optics, St. Petersburg, Russia aut NATIONALLICENCE 700 1- Suslina I. St. Petersburg National Research University of Information Technologies, Mechanics, and Optics, St. Petersburg, Russia aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 206/2(2015-04-01), 181-196 1072-3374 206:2<181 2015 206 10958