Regression in random design and warped wavelets
Open Access
- 1 December 2004
- journal article
- Published by Bernoulli Society for Mathematical Statistics and Probability in Bernoulli
- Vol. 10 (6), 1053-1105
- https://doi.org/10.3150/bj/1106314850
Abstract
We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis f jk(G);j; kg warped with the design. This allows to perform a very stable and computable thresholding algorithm. We investigate the properties of this new basis. In particular, we prove that if the design has a property of Muckenhoupt type, this new basis has a behavior quite similar to a regular wavelet basis. This enables us to prove that the associated thresholding procedure achieves rates of convergence which have been proved to be minimax in the uniform design case.Keywords
This publication has 25 references indexed in Scilit:
- Smooth Design-Adapted Wavelets for Nonparametric Stochastic RegressionJournal of the American Statistical Association, 2004
- Estimating deformations of stationary processesThe Annals of Statistics, 2003
- Extending the Scope of Wavelet Regression Methods by Coefficient-Dependent ThresholdingJournal of the American Statistical Association, 2000
- Wavelet shrinkage for nonequispaced samplesThe Annals of Statistics, 1998
- Nonlinear approximationActa Numerica, 1998
- Random design wavelet curve smoothingStatistics & Probability Letters, 1997
- Interpolation methods for nonlinear wavelet regression with irregularly spaced designThe Annals of Statistics, 1997
- Wavelets for period analysis of unevenly sampled time seriesThe Astronomical Journal, 1996
- Neo-Classical Minimax Problems, Thresholding and Adaptive Function EstimationBernoulli, 1996
- Weighted Norm Inequalities for the Hardy Maximal FunctionTransactions of the American Mathematical Society, 1972