Comments on \ Wavelets in Statistics : A Review

I would like to congratulate Professor Antoniadis for successfully outlining the current state-of-art of wavelet applications in statistics. Since wavelet techniques were introduced to statistics in the early 90's, the applications of wavelet techniques have mushroomed. There is a vast forest of wavelet theory and techniques in statistics and one can nd himself easily lost in the jungle. The review by Antoniadis, ranging from linear wavelets to nonlinear wavelets, addressing both theoretical issues and practical relevance, gives in-depth coverage of the wavelet applications in statistics and guides one entering easily into the realm of wavelets. 1 Variable smoothing and spatial adaptation Wavelets are a family of orthonormal bases having ability of representing functions that are local in both time and frequency domains (subjects to the contraints as in Heisenberg's uncertainty principle). These properties allow one to compress eeciently a wide array of classes of signals, reducing dimensionality from n \highly-correlated" dimensions to much smaller \nearly-independent dimen-sions", without introducing large approximation errors. These classes of signals include piecewise smooth functions and functions with variable degrees of smoothness and functions with variable local frequencies. The adaptation properties of nonlinear wavelet methods to the Besov classes of functions are thoroughly studied by Donoho, Johnstone, Kerkyacharian and Picard 37, 38]. The adaptation properties of nonlinear wavelet methods to the functions with variable frequencies can be found in Fan, Hall, Martin and Patil 3]. The time and frequency localization of wavelet functions permits nonlinear wavelets methods to conduct automatically variable smoothing: diierent location uses a diierent value of smoothing parameter. This feature enables nonlinear wavelet estimators to recover functions with diierent degrees of smoothness and diierent local frequencies. Namely, nonlinear wavelet estimators possess spatial adaptation property. As pointed out in Donoho, Johnstone, Kerkyacharian and Picard 37], linear estimators, including linear wavelet, kernel and spline methods, can not eeciently estimate functions with variable degrees of smoothness. A natural question is if the traditional methods can be modiied to ee-ciently estimate the functions with variable degrees of smoothness. The answer is positive. To recover functions with diierent degrees of smoothness, variable bandwidth schemes have to be incorporated into kernel or local polynomial estimators, resulting in highly nonlinear estimators. For example, with the implementation of variable bandwidth as in Fan and Gijbels 2] , it is shown via 1