Multivariate orthogonal series estimates for random design regression

In this paper a new multivariate regression estimate is introduced. It is based on ideas derived in the context of wavelet estimates and is constructed by hard thresholding of estimates of coefficients of a series expansion of the regression function. Multivariate functions constructed analogously to the classical Haar wavelets are used for the series expansion. These functions are orthogonal in L2(μn)L2(μn), where μnμn denotes the empirical design measure. The construction can be considered as designing adapted Haar wavelets.Bounds on the expected L2L2 error of the estimate are presented, which imply that the estimate is able to adapt to local changes in the smoothness of the regression function and to the distribution of the design. This is also illustrated by simulations.