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:
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