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

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.\par This book is an advanced study and accessibility to a broader readership could be improved.
[A. Akutowicz (Berlin)]

MSC 2000:: *93-02 Research monographs (systems and control)
93E10 Estimation and detection in stochastic control
93E24 Least squares and related methods in stochastic control
93E35 Stochastic learning and adaptive control
60J10 Markov chains with discrete parameter
62B10 Statistical information theory
62G99 Nonparametric inference
62M05 Markov processes: estimation
68T10 Pattern recognition
90C15 Stochastic programming

Keywords: statistical learning; Gibbs estimators; Markov chains; adaptive classification; least-squares regression; probability distribution; risk function; Kullback divergence; oracle inequalities; estimators; coding theory; pattern recognition; simulated annealing

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