On estimation and detection of infinite-variable function

Yu. I. Ingster ; I. A. Suslina

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 91-113

Résumé

We observe an unknown infinite-variable function $f=f(t)$, $t=(t_1,\ldots,t_n,\ldots)\in[0,1]^\infty$, in the white Gaussian noise of a level $\varepsilon>0$. We suppose that, in each variable, there exist 1-periodical $\sigma$-smooth extensions of functions $f(t)$ on $\mathbb R^\infty$. Taking a quantity $\sigma>0$ and a positive sequence $\mathbf a=\{a_k\}$, we consider the set $\mathcal F_{\sigma,\mathbf a}$ that consists of functions $f$ such that $\sum_{k=1}^\infty a_k^2\|\partial^\sigma f/\partial t_k^{\sigma}\|_2^2\le 1$. We consider the cases $a_k=k^\alpha$ and $a_k=\exp(\lambda k)$, $\alpha>0$, $\lambda>0$. We want to estimate a function $f\in\mathcal F_{\sigma,\mathbf a}$ or to test the null hypothesis $H_0$: $f=0$ against alternatives $f\in\mathcal F_{\sigma,\mathbf a}(r_\varepsilon)$ where the set $\mathcal F_{\sigma,\mathbf a}(r)$ consists of functions of $f\in \mathcal F_{\sigma,\mathbf a}$ such that $\|f\|_2\ge r$. In the estimation problem, we obtain the asymptotics (as $\varepsilon\to 0$) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates $r_\varepsilon^*$ that provide distiguishability in the problems.

Bibliographie

[1] M. S. Ermakov, “Minimaksnoe obnaruzhenie signala v gaussovskom belom shume”, Teoriya veroyatnostei i ee primen., 35:4 (1990), 704–715 | MR

[2] I. A. Ibragimov, R. Z. Khasminskii, “Odna zadacha statisticheskogo otsenivaniya v gaussovskom belom shume”, Doklady AN SSSR, 236:6 (1977), 1300–1302 | MR | Zbl

[3] I. A. Ibragimov, R. Z. Khasminskii, “Some estimation problems on Infinite dimensional Gaussian white noise”, Festschrift for Lusien Le Cam. Research papers in Probability and Statictics, Springer-Verlag, NY, 1997, 275–296 | MR

[4] Yu. I. Ingster, “O minimaksnom neparametricheskom obnaruzhenii signala v gaussovskom belom shume”, Problemy peredachi informatsii, 18:2 (1982), 61–73 | MR | Zbl

[5] Yu. I. Ingster, “Asymptotically minimax testing of nonparametric hypotheses”, Proc. 4th Vilnius Conference on Probab. Theory and Math. Stat., 1, VNU Science Press, 1987, 553–573 | MR

[6] Yu. I. Ingster, “Asymptotically minimax hypothesis testing for nonparametric alternatives, I, II, III”, Math. Methods Statist., 2 (1993), 85–114, 171–189, 249–268 | MR | Zbl

[7] Yu. I. Ingster, I. A. Suslina, Nonparametric Goodness-of-Fit Testing under Gaussian Model, Lectures Notes in Statistics, 169, Springer-Verlag, New York, 2002 | MR

[8] Yu. I. Ingster, I. A. Suslina, “On estimation and detection of smooth function of many variables”, Math. Methods Statist., 14:3 (2005), 299–331 | MR

[9] M. Nusbaum, “Ob optimalnoi filtratsii funktsii mnogikh peremennykh na fone gaussovskogo belogo shuma”, Problemy peredachi informatsii, 19:2 (1983), 23–29 | MR | Zbl

[10] M. S. Pinsker, “Optimalnaya filtratsiya kvadratichno-integriruemykh funktsii v gaussovskom shume”, Problemy peredachi informatsii, 16:2 (1980), 52–68 | MR | Zbl

[11] A. V. Skorokhod, Integrirovanie v gilbertovom prostranstve, Nauka, M., 1975

[12] Ch. Stone, “Additive regression and other nonparametric models”, Ann. Stat., 13:2 (1985), 689–705 | DOI | MR | Zbl