Adaptive detection of functions of large number of variables

Yu. I. Ingster ; I. A. Suslina

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 15, Tome 368 (2009), pp. 156-170

Résumé

A major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality”: the rates of accuracy in estimation and separation rates in detection problems behave poorly when the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or Hölder balls. In the paper [9], it was considered functional classes of a new type, first introduced by Sloan and Wożniakowski in [14]. These classes are the balls $\mathcal F_{\sigma,s}$ in the weighed tensor product spaces that are characterized by two parameters: $\sigma>0$ is a “smoothness” parameter, and $s>0$ determines the weight sequence which characterizes “importance” of the variables. In particular, it was shown in [9] that under the white Gaussian noise model, the log-asymptotics of separation rates in detection are similar to those for one-variable functions of the smoothness $\sigma^*=\min(s,\sigma)$ independently of the original problem dimensions; thus the curse of dimensionality is “lifted.” However the test procedure depends on parameters $(\sigma,s)$ which are unknown typically. In this paper, we propose a common test procedure that does not depend on parameters $(\sigma,s)$ and provides the same log-asymptotics of separation rates uniformly over any compact set of parameters $(\sigma,s)$. Also we give independent simple proof of the log-asymptotics of separation rates in the problem. Bibl. – 16 titles.