Wavelet designs for estimating nonparametric curves with heteroscedastic error 3

In this paper, we discuss the problem of constructing designs in order to maximize the accuracy 9 of nonparametric curve estimation in the possible presence of heteroscedastic errors. Our approach is to exploit the 3exibility of wavelet approximations to approximate the unknown response 11 curve by its wavelet expansion thereby eliminating the mathematical di5culty associated with the unknown structure. It is expected that only 6nitely many parameters in the resulting wavelet 13 response can be estimated by weighted least squares. The bias arising from this, compounds the natural variation of the estimates. Robust minimax designs and weights are then constructed 15 to minimize mean-squared-error-based loss functions of the estimates. We 6nd the periodic and symmetric properties of the Euclidean norm of the multiwavelet system useful in eliminating 17 some of the mathematical di5culties involved. These properties lead us to restrict the search for robust minimax designs to a speci6c class of symmetric designs. We also construct minimum 19 variance unbiased designs and weights which minimize the loss functions subject to a side condition of unbiasedness. We discuss an example from the nonparametric literature. 21 c © 2002 Published by Elsevier Science B.V. MSC: primary 62K05; 62G35; secondary 62G07; 41A30 23