On estimation and detection of a function from tensor product spaces

Yu. I. Ingster   ; I. A. Suslina

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 12, Tome 351 (2007), pp. 180-218

Résumé

We observe an unknown $d$-variables function $f(t)$, $ t\in[0,1]^d$ in the white Gaussian noise of a level $\varepsilon>0$. We suppose that $f\in\mathcal{F}$, where $\mathcal{F}$ is a ball in Hilbert space $\mathcal{L}^d\subset L_2([0,1]^d)$ of tensor product structure. Under minimax setup, we consider two problems: to estimate $f$ (for quadratic losses) and to detect $f$, i.e., to test the null hypothesis $H_0:f=0$ against alternatives $H_1: f\in\mathcal{F}_d$, $\|f\|_2\ge r_\varepsilon$. We are interesting in the case $d=d_\varepsilon\to\infty$. We study sharp, rate and log-asymptotics (as $\varepsilon\to 0$, $d\to\infty$) in the problems. In particular, we show that log-asymptotics are different essentially for $d\ll\log\varepsilon^{-1}$ and for $d\gg\log\varepsilon^{-1}$.

Bibliographie

[1] “On minimax filtring over ellipsoids”, Math. Methods Statist., E. N. Belitser, B. V. Levit, 4, 1995, 259–273 | MR | Zbl

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

[3] G. M. Fikhtengolts, Kurs differentsialnogo i integralnogo ischisleniya, t. III, Fizmatgiz, M., 1963

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

[5] 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

[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 | MR | Zbl | MR | Zbl

[7] Yu. I. Ingster, I. A. Suslina, Nonparametric Goodness-of-Fit Testing under Gaussian Model, 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 (2005), 299–331 | MR

[9] Yu. I. Ingster, I. A. Suslina, “Otsenivanie i proverka gipotez dlya funktsii beskonechnogo chisla peremennykh”, Zap. nauchn. semin. POMI, 328 (2005), 91–113 | MR | Zbl

[10] Yu. I. Ingster, I. A. Suslina, “Estimation and detection of high-variable function from Sloan–Woźniakowski space”, Math. Methods Statist., 16:4 (2007), 318–353 | DOI | MR | Zbl

[11] A. Karol', A. Nazarov, Ya. Nikitin, “Small balls probabilities for Gaussian random fields and tensor products of compact operators”, Trans. AMS, 360:3 (2006), 1443–1474 | DOI | MR

[12] B. Levit, N. Stepanova, “Effisient estimation of multivatiate analytic functions in cube-like domains”, Math. Methods Statist., 13 (2004), 253–281 | MR | Zbl

[13] M. A. Lifshits, E. V. Tulyakova, “Curse of dimensionality in approximation of random fields”, Probab. Math. Stat., 26 (2006), 97–112 | MR | Zbl

[14] Y. Lin, “Tensor product space ANOVA model”, Ann. Stat., 28 (2000), 734–755 | DOI | MR | Zbl

[15] A. Papageorgiou, G. W. Wasilkovski, “On the average complexity of multivariate problems”, J. Complexity, 6 (1990), 1–23 | DOI | MR | Zbl

[16] V. V. Petrov, Summy nezavisimykh sluchainykh velichin, Nauka, M., 1972 | MR

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

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

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