Discussion on Antoniadis and Fan "Regularization of Wavelet Approximations"
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Anestis Antoniadis and Janqing Fan deserve congratulations for a wonderful and illuminating paper. Links among wavelet-based penalized function estimation, model selection, and now actively explored wavelet-shrinkage estimation, are intriguing and attracted attention of many researchers. Antoniadis and Fan provide numerous references. The nonlinear estimators resulting as optimal in the process of regularization, for some specific penalty functions, turn out to be the familiar hard- or soft-thresholding rules, or some of their sensible modifications. Simply speaking, the penalty function determines the estimation rule, and in many cases, a practicable and ad-hoc shrinkage rule can be linked to a regularization process under a reasonable penalty function. The authors explore the nature of penalty functions resulting in thresholding-type rules. They also show, that for a large class of penalty functions, corresponding shrinkage estimators are adaptively minimax and have other good sampling properties. My discussion will be directed toward the link of the regularization problem and Bayesian modeling and inference in the wavelet domain, which is only hinted by Antoniadis and Fan.