×

Wavelet thresholding: Beyond the Gaussian i.i.d. situation. (English) Zbl 0831.62071

Antoniadis, Anestis (ed.) et al., Wavelets and statistics. Proceedings of the 15th French-Belgian meeting of statisticians, held at Villard de Lans, France, November 16-18, 1994. New York, NY: Springer-Verlag. Lect. Notes Stat., Springer-Verlag. 103, 301-329 (1995).
Summary: With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.
In the second part, we apply these nonlinear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non- stationary time series setup. For the latter one, in a model of R. Dahlhaus [Fitting time series models to nonstationary processes. Preprint, Univ. Heidelberg (1993)] on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.
For the entire collection see [Zbl 0824.00042].

MSC:

62M15 Inference from stochastic processes and spectral analysis
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
PDFBibTeX XMLCite