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Abstract
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The main use of wavelets in statistics is in nonparametric function estimation. Nonlinear wavelets estimators based on thresholding typically perform well even for highly irregular functions, but are restricted to stationary (and often Gaussian) noise.
In this talk, I will propose a technique for the wavelet estimation of signals contaminated with noise whose variance is a fixed (and possiblyunknown) function of the local level of the signal. This set-up arises, for example, in volatility estimation, periodogram smoothing, estimation of gene expression levels or Poisson intensity estimation.
The algorithm, termed the data-driven wavelet-Fisz method, proceeds in two stages and yields consistent estimators. The consistency proof relies on a new exponential inequality for Nadaraya-Watson estimators, which may be of independent interest. An associated data-driven wavelet-Fisz variance stabilising transform will also be demonstrated and discussed. Finally, if time permits, I will discuss some further practical aspects of the resulting wavelet-Fisz procedure in the special case of volatility estimation.
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