Wavelet Methods for Curve Estimation

Wavelets in statistics: A review

“…Among the first to consider (linear) wavelet methods in statistics are Douhan and Leon [40], Antoniadis and Carmona [6], Kerkyacharian and Picard [56] and Waiter [87] for density estimation and Doukhand and Leon [40], Antoniadis, Gregoire and McKeague [7] for nonparametric regression. In the following subsection we will address first the performance of such wavelet estimators in the case of a single model for nonparametric regression in close analogy with the classical theory of curve estimation.…”

“…In the context of non-uniform stochastic design there is a variety of ways to construct a wavelet estimator of the unknown mean function g. In this case, the basic wavelet estimator considered in Antoniadis et al [7] is of the product of j g, which is then corrected by dividing by an estimator of the design density jwhich is constructed by a simple wavelet estimator or a kernel estimator. To simplify the exposition we will only review here the case of the fixed design model.…”

“…The selection rule used in Antoniadis et at. [7] is to choose J as the minimizer of the cross validation function where gult) is the leave-one-out estimator obtained by evaluating g as a function of J and t) with the ith data point removed. This gives reasonable results when applied to real and simulated data.…”

“…As with any asymptotic result, there remain doubts as to how well the asymptotic describe small sample behavior. These issues are addressed by Marron et al [59] using the tools of exact risk analysis, which was developed in Gasser and MUller [45], and first applied to wavelet estimators by Antoniadis et al [7]. Finite sample performance of thresholded wavelet estimators has also been studied by Bruce and Gao [19], where computationally efficient formulas for computing the exact pointwise bias, variance and L 2 risk of thresholded wavelet estimators in finite sample situations are derived, thus complementing the tools of simulation and asymptotic analysis.…”

“…Wavelet versions of estimators of a hazard rate function in the context of inference for a counting process multiplicative intensity model have been studied by Antoniadis et al [7]. See also Antoniadis, Gregoire and Nason [12] for a contribution to the methodology available for estimating the density and the hazard rate from randomly censored data.…”