zbMATH — the first resource for mathematics

Asymptotic minimaxity of wavelet estimators with sampled data. (English) Zbl 1065.62518
Summary: The authors [see, e.g., Ann. Stat. 26, No. 3, 879–921 (1998; Zbl 0935.62041)] studied a setting where data were obtained in the continuum white noise model and showed that scalar nonlinearities applied to wavelet coefficients gave estimators which were asymptotically minimax over Besov balls. They claimed that this implied similar asymptotic minimaxity results in the sampled-data model. In this paper we carefully develop and fully prove this implication.
Our results are based on a careful definition of an empirical wavelet transform and precise bounds on the discrepancy between empirical wavelet coefficiets and the theoretical wavelet coefficients.

62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems