ASYMPTOTICS FOR WAVELET BASED ESTIMATES OF PIECEWISE SMOOTH REGRESSION FOR STATIONARY TIME SERIES

YOUNG K. TRUONG^{1} AND PRAKASH N. PATIL^{2}

1Department of Biostatistics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7400, U.S.A., e-mail: truong@bios.unc.edu

and Department of Statistics and Applied Probability, National University of Singapore,

3 Science Drive 2, Singapore 117543, Singapore

2School of Mathematics and Statistics, The University of Birmingham, Birmingham B15 2TT, U.K.

(Received April 14, 2000; revised August 16, 2000)

Abstract. Wavelet methods are used to estimate density and (auto-) regression functions that are possibly discontinuous. For stationary time series that satisfy appropriate mixing conditions, we derive mean integrated squared errors (MISEs) of wavelet-based estimators. In contrast to the case for kernel methods, the MISEs of wavelet-based estimators are not affected by the presence of discontinuities in the curves. Applications of this approach to problems of identification of nonlinear time series models are discussed.

Key words and phrases: onvergence rate, density estimation, nonparametric regression, piecewise-smoothness, wavelet.