This overview article motivates the use of wavelets in statistics and introduces the basic mathematics behind the construction of wavelets. Topics covered include the continuous and discrete wavelet transforms, multiresolution analysis, and the non-decimated wavelet transform. We describe the basic mechanics of nonparametric function estimation via wavelets, emphasizing the concepts of sparsity and thresholding. A simple proof of the mean-square consistency of the wavelet estimator is also included. The article ends with two special topics: function estimation with unbalanced Haar wavelets and variance stabilization via the Haar–Fisz transformation. WIREs Comp Stat 2010 2 654–667 DOI: 10.1002/wics.124
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