Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series A, Pt. 3, pp. 328--366

NONPARAMETRIC STOCHASTIC REGRESSION WITH DESIGN-ADAPTED WAVELETS

By

V. DELOUILLE, Université catholique de Louvain, Belgium
J. FRANKE, Universität Kaiserslautern, Germany

and

R. VON SACHS, Université catholique de Louvain, Belgium

SUMMARY. We present a new approach of nonparametric regression with wavelets if the design is stochastic. In contrast to existing approaches we use a new construction of a design-adapted wavelet basis which is constructed given the random regressors. To exemplify the potential of our new methodology we treat the case of using orthogonal design-adapted Haar wavelets for regression with (non-Gaussian) i.i.d.~errors. We derive results on the near-optimal rate of convergence of the minimax L2-risk of non-linear threshold estimators over a certain function class which parallel those of the classical case of fixed equidistant design. We indicate generalisations in various directions and cover parts of those by empirical investigations in our simulation examples.

AMS (1991) subject classification. 62G08.

Key words and phrases. Gaussian approximation, minimax L2-rate, stochastic design, Unbalanced Haar wavelets, wavelet thresholding.

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