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Statistical learning theory and stochastic optimization. Ecole d’Eté de Probabilitiés de Saint-Flour XXXI – 2001. (English) Zbl 1076.93002
Lecture Notes in Mathematics 1851. Berlin: Springer (ISBN 3-540-22572-2/pbk). viii, 273 p. EUR 39.95/net; sFr 73.00; £ 30.50; $ 59.95 (2004).

The main topic of the book is to estimate a probability distribution from observed samples. A risk function of the Kullback divergence type is considered. Other types of risk functions are also considered. Oracle inequalities are associated to estimators restricted to a (possibly random) parametrization. The goal is to bound the unresticted risk by the restricted one plus a term quantifying the complexity of the parametric model. The first chapters give connections with coding theory, universal lossless data compression, and pattern recognition. Then several types of oracle inequalities are studied. Finally, some connections with simulated annealing are investigated.

90 references and an index are provided at the end.

This book is an advanced study and accessibility to a broader readership could be improved.

MSC:
93-02Research monographs (systems and control)
93E10Estimation and detection in stochastic control
93E24Least squares and related methods in stochastic control
93E35Stochastic learning and adaptive control
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
62B10Statistical information theory
62G99Nonparametric inference
62M05Markov processes: estimation
68T10Pattern recognition, speech recognition
90C15Stochastic programming