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Bivariate hard thresholding in wavelet function estimation
by Piotr Fryzlewicz
We propose a generic bivariate hard thresholding estimator of the discrete
wavelet coefficients of a function contaminated with i.i.d. Gaussian noise. We
demonstrate its good risk properties in a motivating example, and derive upper
bounds for its mean-square error. Motivated by the clustering of large wavelet
coefficients in real-life signals, we propose two wavelet denoising algorithms, both
of which use specific instances of our bivariate estimator. The BABTE algorithm
uses basis averaging, and the BITUP algorithm uses the coupling of "parents" and
"children" in the wavelet coefficient tree. We prove the L2 near-optimality of both
algorithms over the usual range of Besov spaces, and demonstrate their excellent
finite-sample performance. Finally, we propose a robust and effective technique for
choosing the parameters of BITUP in a data-driven way.
Key words: Chi-square, discrete wavelet transform, nonparametric
regression, translation-invariance, universal threshold, wavelet shrinkage.
Full text of the paper (pdf),
to appear in Statistica Sinica.
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