A data-driven Haar-Fisz transform for multiscale vari- ance stabilization Technical Report 05-06 Statistics Group Department of Mathematics University of Bristol Piotr Fryzlewicz Department of Mathematics, Imperial College, London, UK V´ eronique Delouille Royal Observatory of Belgium, Brussels, Belgium Guy P. Nason Department of Mathematics, University of Bristol, Bristol, UK Summary. Transforming data so that its variance is stable and its distribution is taken closer to the Gaussian is the aim of many techniques (e.g. Anscombe, Box-Cox). Recently, new tech- niques based on the Haar-Fisz transform have been introduced that use a multiscale method to transform and stabilize data with a known mean-variance relationship, e.g. Poisson or Chi- square. In many practical cases, the variance of the data can be assumed to increase with the mean, but other characteristics of the distribution are unknown. We introduce a method, the data-driven Haar-Fisz transform (DDHFT), which uses Haar-Fisz but also estimates the mean-variance relationship. For known noise distributions, the DDHFT is shown to be compet- itive with the fixed Haar-Fisz methods. We show how the DDHFT effectively denoises a solar flux time series obtained from the XRS X-ray sensor on GOES8 satellite where other existing methods fail. Keywords: variance stabilization, Gaussianisation, Haar-Fisz. 1. Introduction In non-parametric regression, we are often faced with the problem of estimating a one-dimensional function f : [0, 1] → R from noisy observations X i taken on an equispaced grid: X i = f  i n  + ε i , i =1,...,n, where ε i ’s are random variables with E(ε i )=0. Various subclasses of the problem can be iden- tified, depending on the joint distribution of (ε i ) n i=1 and on the smoothness of f . In particular, in cases where f possesses irregular features (such as discontinuities or spikes), several authors advo- cate the use of nonlinear estimators based on wavelet shrinkage: see e.g. Donoho and Johnstone