Abstract
We discuss a new way of evaluating the performance of a statistical estimation procedure. This consists of investigating the maximal set where a given procedure has a given rate of convergence. Although the setting is not vastly different from the minimax context, it is in a sense less pessimistic and provides a functional set which is authentically connected to the procedure and the model. We also investigate more traditional concerns about procedures: oracle inequalities. Difficulties arise in the practical definition of this notion when the loss function is not the L2 norm. We explain these difficulties and suggest a new definition in the cases of Lp norms and pointwise estimation. We investigate the connections between maxisets and local oracle inequalities, and prove that verifying a local oracle inequality implies that the maxiset automatically contains a prescribed set linked with the oracle inequality. We have investigated the consequences of this statement on well-known efficient adaptive methods: wavelet thresholding and local bandwidth selection. We prove local oracle inequalities for these methods and draw conclusions about the maxisets associated with them.
Citation
Gérard Kerkyacharian. Dominique Picard. "Minimax or maxisets?." Bernoulli 8 (2) 219 - 253, April 2002.
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