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 propose a technique for the wavelet estimation of signals contaminated with noise whose variance is a fixed (and possibly unknown) 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.