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Nonparametric estimation of functionals of stationary sequences distributions. (Neparametricheskoe otsenivanie funktsionalov ot raspredelenij statsionarnykh posledovatel’nostej.) (Russian. English summary) Zbl 1068.62042
Moskva: Nauka (ISBN 5-02-032651-8/hbk). 508 p. (2004).
In the first part of this book nonparametric kernel estimates of functionals from unknown distributions of observed sequences are studied. The samples may be dependent but satisfy a mixing condition. Methods of construction and asymptotic properties of the estimators are presented, with a lot of important examples and precise and rigorous proofs.
The second part deals with applications in system identification, signal processing, and actuarial mathematics. a) For complex dynamical systems, ideas of L. Ljung and T. Söderström [Theory and practice of recursive identification. (1983; Zbl 0548.93075)] are developed concerning predictions with guaranteed statistical quality. This problem is solved for non-random regression, stochastic regression, and autoregression. b) Filtration, interpolation, and prediction of stochastic signals are considered. The noise distribution is known up to some parameters and belongs to an exponential family. For the estimation procedures, the risk is studied. c) Netto-premiums in life-insurance are estimated. The monograph will be useful for specialists who deal with statistical problems under a priori uncertainty conditions.
Contents: Ch. 1, General theorems on convergence of functionals of statistics. Ch. 2, Kernel estimates of functionals under independent samples. Ch. 3, Estimation of functionals under dependant samples. Ch. 4, Nonparametric estimation of functionals from distributions of observed sequences with additive dependent errors. Ch. 5, Modeling and identification of dynamic systems by nonparametric methods. Ch. 6, Nonparametric filtration of stochastic sequences. Ch. 7, Nonparametric interpolation and prediction of random signals. Quality of estimating procedures. Ch. 8, Nonparametric methods of detection of jump processes. Ch. 9, Application of statistical procedures in actuarial mathematics.

62G07 Density estimation
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G20 Asymptotic properties of nonparametric inference
62P05 Applications of statistics to actuarial sciences and financial mathematics
93E12 Identification in stochastic control theory
93E20 Optimal stochastic control