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Identification of Dynamical Systems with Small Noise

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Small noise is a good noise. In this work, we are interested in the problems of estimation theory concerned with observations of the diffusion-type process Xo = Xo, 0 ~ t ~ T, (0. 1) where W is a standard Wiener process and St(') is some nonanticipative smooth t function. By the observations X = {X , 0 ~ t ~ T} of this process, we will solve some t of the problems of identification, both parametric and nonparametric. If the trend S(-) is known up to the value of some finite-dimensional parameter St(X) = St((}, X), where (} E e c Rd , then we have a parametric case. The nonparametric problems arise if we know only the degree of smoothness of the function St(X), 0 ~ t ~ T with respect to time t. It is supposed that the diffusion coefficient c is always known. In the parametric case, we describe the asymptotical properties of maximum likelihood (MLE), Bayes (BE) and minimum distance (MDE) estimators as c --+ 0 and in the nonparametric situation, we investigate some kernel-type estimators of unknown functions (say, StO,O ~ t ~ T). The asymptotic in such problems of estimation for this scheme of observations was usually considered as T --+ 00 , because this limit is a direct analog to the traditional limit (n --+ 00) in the classical mathematical statistics of i. i. d. observations. The limit c --+ 0 in (0. 1) is interesting for the following reasons.


This volume studies parametric and nonparametric estimation through the observation of diffusion-type processes. The properties of maximum likelihood, Bayes, and minimum distance estimators are considered in the context of the asymptotics of low noise. It is shown that, under certain conditions relating to regularity, these estimators are consistent and asymptotically normal. Their properties in nonregular cases are also discussed. Here, nonregularity means the absence of derivatives with respect to parameters, random initial value, incorrectly specified observations, nonidentifiable models, etc. The book has seven chapters. The first presents some auxiliary results needed in the subsequent work. Chapter 2 is devoted to the asymptotic properties of estimators in standard and nonstandard situations. Chapter 3 considers expansions of the maximum likelihood estimator and the distribution function. Chapters 4 and 5 cover nonparametric estimation and the disorder problem. Chapter 6 discusses problems of parameter estimator for linear and nonlinear partially observed models. The final chapter studies the properties of a wide range of minimum distance estimators. The book concludes with a remarks section, references and index. The volume will be of interest to statisticians, researchers in probability theory and stochastic processes, systems theory and communication theory.


Introduction. 1. Auxiliary Results. 2. Asymptotic Properties of Estimator in Standard and Nonstandard Situations. 3. Expansions. 4. Nonparametric Estimation. 5. The Disorder Problem. 6. Partially Observed Systems. 7. Minimum Distance Estimation. Remarks. References. Index.


Titel: Identification of Dynamical Systems with Small Noise
EAN: 9780792330530
ISBN: 0792330536
Format: Fester Einband
Herausgeber: Springer Netherlands
Genre: Mathematik
Anzahl Seiten: 312
Gewicht: 635g
Größe: H241mm x B160mm x T21mm
Jahr: 1994
Auflage: 1994

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