*(English)*Zbl 1076.93002

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:

93-02 | 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 (discrete-time Markov processes on discrete state spaces) |

62B10 | Statistical information theory |

62G99 | Nonparametric inference |

62M05 | Markov processes: estimation |

68T10 | Pattern recognition, speech recognition |

90C15 | Stochastic programming |