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Uniform limit theorems for wavelet density estimators

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joint Statistics/Probability Seminar

The linear wavelet density estimator of a bounded density f consists of a truncated wavelet expansion with the coefficients for the expansion of f replaced by their empirical counterparts. The optimal number of terms of the expansion, obtained by balancing bias and variance, depends on the degree of smoothness of f, typically unknown. Donoho-Johnstone-Kerkyacharian-Picard (1996) introduced the `hard thresholding’ wavelet density estimator  -where part of the empirical coefficients are set equal to zero if they are smaller than a certain threshold- in order to obtain an estimator which is rate adaptive in L_p norm loss to the smoothness of f, up to a logarithmic factor. The sup-norm behavior of wavelet density estimators (thresholded or not) had not been considered before, and we use empirical process theory to close this gap, thus deriving optimal results first for the linear and then for the thresholded estimator. This is joint work with Richard Nickl.

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