Unbalanced Haar-Fisz Methodology for Function Estimation and Variance Stabilisation

Abstract

The Discrete Unbalanced Haar (DUH) transform is a decomposition of 1D signals with respect to an orthonormal Haar-like basis where jumps in the basis vectors do not necessarily occur in the middle of their support. We introduce a procedure for estimation in Gaussian noise which consists of three steps: a DUH transform, thresholding of the decomposition coefficients, and the inverse DUH transform. We show that our estimator is mean-square consistent with near- optimal rates for a wide range of functions, uniformly over DUH bases which are not "too unbalanced". An important ingredient of our approach is basis selection. We choose each basis vector so that it best matches the data at a specific scale and location, where the latter parameters are determined by the "parent" basis vector. Our estimator performs well and is computable in O(n log n) operations. Modifications to the above procedure are needed for other types of noise. We consider Poisson intensity estimation, as well as a multiplicative regression set-up occurring in e.g. spectrum or volatility estimation. To account for the heterogeneity of the data, both the "basis selection" and "thresholding" steps of our algorithm use the so-called Fisz variance stabilising transform, whose main idea is to divide a given Unbalanced Haar coefficient by an appropriate function of the corresponding "smooth" coefficient. With a small modification, the resulting Unbalanced Haar-Fisz estimation algorithm can also be used to stabilise the variance of heterogeneous data and bring their distribution closer to normality.