Robust designs for wavelet approximations of regression models

Abstract

We consider the construction of designs for the estimation of a regression function, when it is anticipated that this function is to be approximated by the dominant terms in its wavelet expansion. We consider both the Haar wavelet basis and the multiwavelet system. The experimenter estimates the coefficients of those wavelets included in the approximation, hoping that the omitted terms will be inconsequential. This introduces bias into the least squares estimates, which we propose handling at the design stage by one of two methods: (i) implementing a minimax robust design, which enjoys the property of minimizing the maximum value of an mse-based loss function, with the maximum being taken as the remainder in the wavelet expansion varies over an L2 -neighbourhood; (ii) implementing a minimum variance unbiased (mvu) design which, when employed with weighted least squares and weights derived here, minimizes the variance subject to a side condition of unbiasedness. For the Haar wavelet system we show that the uniform design is both minimax robust and mvu. For multiwavelet approximations we give examples of both minimax robust and mvu designs. Two examples from the nonparametric regression literature are discussed, and designs are presented for each type of wavelet approximation.