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—Probability models are estimated by use of penalized log-likelihood criteria related to AIC and MDL. The accuracies of the density estimators are shown to be related to the trade-off between three terms: the accuracy of approximation, the model dimension, and the descriptive complexity of the model classes. The asymptotic risk is determined under(More)
We address the consistency property of cross validation (CV) for classification. Sufficient conditions are obtained on the data splitting ratio to ensure that the better classifier between two candidates will be favored by CV with probability approaching 1. Interestingly, it turns out that for comparing two general learning methods, the ratio of the(More)
— This paper studies minimax aspects of nonpara-metric classification. We first study minimax estimation of the conditional probability of a class label, given the feature variable. This function, say f, is assumed to be in a general nonparametric class. We show the minimax rate of convergence under square L 2 loss is determined by the massiveness of the(More)
|We study nonparametric estimation of a conditional probability for classiication based on a collection of nite-dimensional models. For the sake of exibility, diierent types of models, linear or nonlinear, are allowed as long as each satisses a dimensionality assumption. We show that with a suitable model selection criterion, the penalized maximum(More)
Efforts have been directed at obtaining flexible learning procedures that optimally adapt to various possible characteristics of the data generating mechanism. A question that addresses the issue of how far one can go in this direction is: Given a regression procedure, however sophisticated it is, how many regression functions are estimated accurately? In(More)
|We study minimax-rate adaptive estimation for density classes indexed by continuous hyper-parameters. The classes are assumed to be partially ordered in terms of inclusion relationship. Under a mild condition on the minimax risks, we show that a minimax-rate adaptive estimator can be constructed for the classes. 1 Problem of interest This paper concerns(More)
Consider q-hulls, 0 < q ≤ 1, from a dictionary of M functions in L p space for 1 ≤ p < ∞. Their precise metric entropy orders are derived. Sparse linear approximation bounds are obtained to characterize the number of terms needed to achieve accurate approximation of the best function in a q-hull that is closest to a target function. Furthermore, in the(More)