On Adaptive Function Estimation
نویسنده
چکیده
General results on adaptive function estimation are obtained with respect to a collection of estimation strategies for both density estimation and nonparametric regression under square L 2 loss. It is shown that without knowing which strategy in a given countable collection works best for the underlying function, a single strategy can be constructed by mixing the proposed ones so that it is adaptive in terms of an average cumulative risk. An implication is that under some mild conditions, a minimax-rate adaptive estimator exists for a given countable collection of function classes, i.e., a single estimator can be constructed to be simultaneously minimax optimal in terms of rates of convergence for all the function classes being considered. Examples for high-dimensional function estimation are provided.
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