Identification of a hereditary system with distributed delay. (English) Zbl 0568.93064
The maximum likelihood method for identifying the parameter function in a linear hereditary system is considered. The delay is allowed to be distributed and so the classical theory is not directly applicable, since even if the maximum likelihood method is valid the estimates obtained may not be consistent.
The system model is \[ dX_ t=\int^{0}_{-b}\alpha (\theta)X_{t+\theta}d\theta dt+dW_ t \] and \(\alpha\) is estimated by using finite-dimensional approximations in the space \(L^ 2[-b,0]\). This gives rise to the finite-dimensional maximum likelihood estimate \({\hat \alpha}{}^ T(\theta)\) for the process \(\{X_ t\), \(t_ 0\leq t\leq T\}\). Sufficient conditions are given for the consistency of the estimate, i.e. \[ \lim_{T\to \infty}\int^{T}_{t_ 0}| {\hat \alpha}^{(T)}(\theta)-\alpha (\theta)|^ 2d\theta =0, \] in probability.
The system model is \[ dX_ t=\int^{0}_{-b}\alpha (\theta)X_{t+\theta}d\theta dt+dW_ t \] and \(\alpha\) is estimated by using finite-dimensional approximations in the space \(L^ 2[-b,0]\). This gives rise to the finite-dimensional maximum likelihood estimate \({\hat \alpha}{}^ T(\theta)\) for the process \(\{X_ t\), \(t_ 0\leq t\leq T\}\). Sufficient conditions are given for the consistency of the estimate, i.e. \[ \lim_{T\to \infty}\int^{T}_{t_ 0}| {\hat \alpha}^{(T)}(\theta)-\alpha (\theta)|^ 2d\theta =0, \] in probability.
Reviewer: S.Banks
MSC:
93E12 | Identification in stochastic control theory |
34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
93E10 | Estimation and detection in stochastic control theory |
62A01 | Foundations and philosophical topics in statistics |
93C05 | Linear systems in control theory |
Keywords:
maximum likelihood method; linear hereditary system; finite-dimensional approximations; consistencyReferences:
[1] | Polis, M., The distributed system parameter identification problem: a survey of recent results, (Proceedings of the 3rd IFAC Symposium on Control of Distributed Parameter Systems. Proceedings of the 3rd IFAC Symposium on Control of Distributed Parameter Systems, Toulouse, France (1982)), (Preprints) · Zbl 0566.93014 |
[2] | Rozanov, Ju. A., Infinite Dimensional Gaussian Distributions (1971), American Mathematical Society: American Mathematical Society Providence, RI, (translation of the Russian original) |
[3] | Grenander, V., (Abstract Inference (1981), John Wiley: John Wiley New York) · Zbl 0505.62069 |
[4] | Nguyen, H. T.; Pham, T. D., Identification of nonstationary diffusion model by the method of sieves, SIAM J. Control Optim., 20, 603-611 (1982) · Zbl 0488.62062 |
[5] | Itô, K.; Nisio, M., On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4, 1-75 (1964) · Zbl 0131.16402 |
[6] | Lipster, R. S.; Shiryayev, A. M., Statistics of Random Processes, I (1977), Springer: Springer New York · Zbl 0364.60004 |
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