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Unbalanced Haar technique for nonparametric function estimation

Fryzlewicz, Piotr ORCID: 0000-0002-9676-902X (2007) Unbalanced Haar technique for nonparametric function estimation. Journal of the American Statistical Association, 102 (480). pp. 1318-1327. ISSN 0162-1459

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Identification Number: 10.1198/016214507000000860


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 that consists of three steps: a UH transform, thresholding of the decomposition coefficients, and the inverse UH transform. We show that our estimator is mean squared consistent with near-optimal rates for a wide range of functions, uniformly over UH bases that 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 compared 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. We discuss extensions to image data and to smoother wavelets.

Item Type: Article
Official URL:
Additional Information: © 2007 The American Statistical Association
Divisions: Statistics
Subjects: H Social Sciences > HA Statistics
Date Deposited: 18 Sep 2009 09:13
Last Modified: 16 May 2024 00:35

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