ON NON-EQUALLY SPACED WAVELET REGRESSION

MARIANNA PENSKY^{1} AND BRANI VIDAKOVIC^{2}

1Department of Mathematics, University of Central Florida, Orlando, FL 32816, U.S.A.

2Institute of Statistics and Decision Sciences, Duke University, Box 90251, Durham, NC 27708-0251, U.S.A.

(Received December 14, 1998; revised December 13, 1999)

Abstract. Wavelet-based regression analysis is widely used mostly for equallyspaced designs. For such designs wavelets are superior to other traditional orthonormal bases because of their versatility and ability to parsimoniously describe irregular functions. If the regression design is random, an automatic solution is not available. For such non equispaced designs we propose an estimator that is a projection onto a multiresolution subspace in an associated multiresolution analysis. For defining scaling empirical coefficients in the proposed wavelet series estimator our method utilizes a probabilistic model on the design of independent variables. The paper deals with theoretical aspects of the estimator, in particular MSE convergence rates.

Key words and phrases: Irregular design, NES regression, nonparametric statistical procedures, projection estimators, wavelets.