Unbalanced Haar technique for nonparametric function estimation

The discrete Unbalanced Haar (UH) transform is a decomposition of one-dimensional data 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 multiscale procedure for estimation in Gaussian noise which consists of three steps: a UH transform, thresholding of the decomposition coefficients, and the inverse UH transform. We show that our estimator is mean-square consistent with near-optimal rates for a wide range of functions, uniformly over UH bases which are not “too unbalanced”. A vital 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 is computable in O(n log n) operations. A simulation study demonstrates the good performance of our estimator in comparison with state-of-the-art competitors. We apply our method to the estimation of the mean intensity of the time series of earthquake counts occurring in Northern California. Our study appears to confirm the previous observation that seismicity rates increase after major earthquakes in sites which are located not necessarily close to the examined area.