Robust designs for wavelet approximations of nonlinear models

We consider the construction of designs for the general nonlinear model. Using multiresolution analysis in wavelet theory, the classical nonlinear design problem is transformed into a robust design problem for 'approximately linear' models with orthonormal wavelet basis on the design space S as regressors.; The minimax approach is used to construct designs which are robust against small departures from the finite wavelet representation of the general nonlinear model. We find that the D-optimal design obtained by Herzberg and Traves (1994) is also G-, Q- and A-optimal (in the classical sense) if the Haar wavelet basis is used in the approximation. We provide a proof which we feel is simpler than that of Herzberg and Traves (1994). On the other hand, if the multiwavelets with N = 2 is used, the design which chooses more points in a neighbourhood of the midpoint of the design space and a few at the extremes is shown to be Q- and D-optimal in the simplest case.; Using the nonparametric local averaging procedure with positive weights, we construct optimal weights and designs under the restriction of unbiasedness. We show that under this constraint, the ordinary least squares method is optimal in estimating the parameters of the Haar regression model. In other words, the optimal weight and design obtained were each uniform. For the general N = 2 multiwavelet regression model, we show that the optimal weight and design density are concave and convex paraboloids respectively in each of the 2{dollar}sp{lcub}(m+1){rcub}{dollar} intervals of the design space S = (0,1) with maximum points at the midpoints and minimum points at the endpoints of each interval. We also show that the design is symmetric about the midpoint of the design space.; Strategies for implementing the designs are discussed. The question of how well these wavelets approximate nonlinear models is also considered using specific examples.