Theory of adaptive estimation

  • Oleg V. Lepski

    Aix-Marseille Université, Institut de Mathématiques de Marseille, 39, rue F. Joiliot Curie, 13453 Marseille, France
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The paper is an introduction to the modern theory of adaptive estimation. We introduce a universal estimation procedure based on a random choice from collections of estimators satisfying a few very general assumptions. In the framework of an abstract statistical model, we present an upper bound for the risk of the proposed estimator (-oracle inequality). The basic technical tools here are a commutativity property of some operators and upper functions for positive random functionals. Since the obtained result is not related to a particular observation scheme, many conclusions for various problems in different statistical models can be derived from the single -oracle inequality.