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Estimation with missing data. (English) Zbl 0940.93069
The paper is a clearly written review of estimation problems with missing data. A hidden Markov model is considered, \[ x_{t+1}= Ax_t+ \nu_t;\quad y_t= Cx_t+ w_t. \] Here \(x_t\in \mathbb{R}^n\) form an unobserved state sequence, \(y_t\in \mathbb{R}^m\) are measured data, \(A\) and \(C\) are constant matrices, \(\{\nu_t\}\) and \(\{w_t\}\) are mutually independent i.i.d. sequences. Two problems are investigated:
(i) the estimation of a state given measured data and model parameters, and
(ii) the estimation of model parameters given the measured data only. Parameter estimation is based on the Expectation Maximization (EM) algorithm. In the case of missing measurements, the fixed interval smoothing technique is employed to complete the Expectation-step in order to ‘fill-in’ the missing data. More mathematical details can be found in R. J. Elliott et al. [Hidden Markov models: estimation and control, Springer-Verlag (1995; Zbl 0819.60045)].

MSC:
93E10 Estimation and detection in stochastic control theory
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