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April 2004 Multiscale likelihood analysis and complexity penalized estimation
Eric D. Kolaczyk, Robert D. Nowak
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Ann. Statist. 32(2): 500-527 (April 2004). DOI: 10.1214/009053604000000076


We describe here a framework for a certain class of multiscale likelihood factorizations wherein, in analogy to a wavelet decomposition of an L2 function, a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and the parameter vector localized in position and scale. The framework is developed as a set of sufficient conditions for the existence of such factorizations, formulated in analogy to those underlying a standard multiresolution analysis for wavelets, and hence can be viewed as a multiresolution analysis for likelihoods. We then consider the use of these factorizations in the task of nonparametric, complexity penalized likelihood estimation. We study the risk properties of certain thresholding and partitioning estimators, and demonstrate their adaptivity and near-optimality, in a minimax sense over a broad range of function spaces, based on squared Hellinger distance as a loss function. In particular, our results provide an illustration of how properties of classical wavelet-based estimators can be obtained in a single, unified framework that includes models for continuous, count and categorical data types.


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Eric D. Kolaczyk. Robert D. Nowak. "Multiscale likelihood analysis and complexity penalized estimation." Ann. Statist. 32 (2) 500 - 527, April 2004.


Published: April 2004
First available in Project Euclid: 28 April 2004

zbMATH: 1048.62036
MathSciNet: MR2060167
Digital Object Identifier: 10.1214/009053604000000076

Primary: 62C20 , 62G05
Secondary: 60E05

Keywords: factorization , Haar bases , Hellinger distance , Kullback–Leibler divergence , minimax , Model selection , multiresolution , recursive partitioning , thresholding estimators , Wavelets

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 2 • April 2004
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