| Dewey Class |
519.5 |
| Title |
Identification of Dynamical Systems with Small Noise ([EBook] /) / by Yu. Kutoyants. |
| Author |
Kutoyants, Yu |
| Other name(s) |
SpringerLink (Online service) |
| Publication |
Dordrecht : : Springer Netherlands : : Imprint: Springer, , 1994. |
| Physical Details |
VIII, 301 p. : online resource. |
| Series |
Mathematics and its applications ; 300 |
| ISBN |
9789401110204 |
| Summary Note |
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.: |
| Contents note |
1 Auxiliary Results -- 1.1 Some notions of probability theory -- 1.2 Stochastic integral -- 1.3 On asymptotic estimation theory -- 2 Asymptotic Properties of Estimators in Standard and Nonstandard Situations -- 2.1 LAM bound on the risks of estimators -- 2.2 Asymptotic behavior of estimators in the regular case -- 2.3 Parameter estimation for linear systems -- 2.4 Nondifferentiable and “too differentiable” trends -- 2.5 Random initial value -- 2.6 Misspecified models -- 2.7 Nonconsistent estimation -- 2.8 Boundary of the parametric set -- 3 Expansions -- 3.1 Expansion of the MLE -- 3.2 Possible generalizations -- 3.3 Expansion of the distribution function -- 4 Nonparametric Estimation -- 4.1 Trend estimation -- 4.2 Linear multiplier estimation -- 4.3 State estimation -- 5 The Disorder Problem -- 5.1 Simultaneous estimation of the smooth parameter and the moment of switching -- 5.2 Multidimensional disorder -- 5.3 Misspecified disorder -- 6 Partially Observed Systems -- 6.1 Kalman filter identification -- 6.2 Nonlinear systems -- 6.3 Disorder problem for Kalman filter -- 7 Minimum Distance Estimation -- 7.1 Definitions and examples of the MDE -- 7.2 Consistence and limit distributions -- 7.3 Linear systems -- 7.4 Nonstandard situations and other problems -- 7.5 Asymptotic efficiency of the MDE -- Remarks -- References. |
| System details note |
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users) |
| Internet Site |
http://dx.doi.org/10.1007/978-94-011-1020-4 |
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