  We propose a wavelet-based technique for the nonparametric estimation of
functions contaminated with noise whose mean and variance are linked via a
possibly unknown variance function. Our method, termed the data-driven
wavelet-Fisz technique, consists of estimating the variance function via a
Nadaraya-Watson estimator, and then performing a wavelet thresholding procedure
which uses the estimated variance function and local means of the data to set
the thresholds at a suitable level. We demonstrate the mean-square
near-optimality of our wavelet estimator over the usual range of Besov classes.
To achieve this, we establish an exponential inequality for the Nadaraya-Watson
variance function estimator. We discuss various implementation issues
concerning our wavelet estimator, and demonstrate its good practical
performance. We also show how it leads to a new wavelet-domain data-driven
variance-stabilising transform. Our estimator can be applied to a variety of
problems, including the estimation of volatilities, spectral densities and
Poisson intensities, as well as to a range of problems in which the
distribution of the noise is unknown.
