Regression in random design and warped wavelets

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 $\left\{{\psi }_{\mathrm{jk}}\right(G),j,k\}$ warped with the design. This allows us to employ 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 behaves quite similarly 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.