The Annals of Statistics

Multiscale likelihood analysis and complexity penalized estimation

Eric D. Kolaczyk and Robert D. Nowak

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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.

Article information

Ann. Statist. Volume 32, Number 2 (2004), 500-527.

First available in Project Euclid: 28 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62G05: Estimation
Secondary: 60E05: Distributions: general theory

Factorization Haar bases Hellinger distance Kullback–Leibler divergence minimax model selection multiresolution recursive partitioning thresholding estimators wavelets


Kolaczyk, Eric D.; Nowak, Robert D. Multiscale likelihood analysis and complexity penalized estimation. Ann. Statist. 32 (2004), no. 2, 500--527. doi:10.1214/009053604000000076.

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