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In this article, we exploit the adaptive properties of wavelets to develop some procedures for testing the equality of nonlinear and nonparametric mean response curves which are assumed by an experimenter to be the underlying functions generating several groups of data with possibly hetereoscedastic errors. The essential feature of the techniques is the transformation of the problem from the domain of the input variable to the wavelet domain through an orthogonal discrete wavelet transformation or a multiresolution expansion. We shall see that this greatly simplifies the testing problem into either a wavelet thresholding problem or a linear wavelet regression problem. The size and power performances of the tests are reported and compared to some existing methods. The tests are also applied to data on dose response curves for vascular relaxation in the absence or presence of a nitric oxide inhibitor.

We discuss the construction of designs for estimation of nonparametric regression models with autocorrelated errors when the mean response is to be approximated by a finite order linear combination of dilated and translated versions of the Daubechies wavelet bases with four vanishing moments. We assume that the parameters of the resulting model will be estimated by weighted least squares (WLS) or by generalized least squares (GLS). The bias induced by the unused components of the wavelet bases, in the linear approximation, then inflates the natural variation of the WLS and GLS estimates. We therefore construct our designs by minimizing the maximum value of the average mean squared error (AMSE). Such designs are said to be robust in the minimax sense. Our illustrative examples are constructed by using the simulated annealing algorithm to select an optimal $n$-point design, which are integers, from a grid of possible values of the explanatory or design variable $x$. We found that the integer-valued designs we constructed based on GLS estimation, have smaller minimum loss when compared to the designs for WLS or ordinary least squares (OLS) estimation, except when the correlation parameter $\rho$ approaches 1.