Mohapl, Jaroslav Maximum likelihood estimation in linear infinite dimensional models. (English) Zbl 0815.62057 Commun. Stat., Stochastic Models 10, No. 4, 781-794 (1994). Summary: Consider a martingale \((c(t), {\mathcal F}_ t )_{t\geq 0}\) with values in the strong dual \({\mathcal D}'\) of a nuclear space \({\mathcal D}\). Let \(c(t)\) satisfy the functional equation \[ c(t)- c(0)= \int_ 0^ t A'(\theta) c(s) ds+ B'W(t) \] in which \((W(t), {\mathcal F}_ t )_{t\geq 0}\) is a \({\mathcal D}'\) valued Gaussian white noise process and \(B':{\mathcal D}'\to {\mathcal D}'\) and \(A' (\theta): {\mathcal D}'\to {\mathcal D}'\), \(\theta\in \Theta \subset (-\infty, \infty)^ K\), are continuous linear operators. It is shown that under suitable assumptions the initial condition \(c(0)\) can be chosen in such a way that \((c(t), {\mathcal F}_ t )_{t\geq 0}\) becomes an ergodic stationary Markov process and the unknown parameter \(\theta\) can be estimated by the maximum likelihood method. The obtained estimator of \(\theta\) is strongly consistent and satisfies a version of the central limit theorem. Cited in 2 Documents MSC: 62M05 Markov processes: estimation; hidden Markov models 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F05 Central limit and other weak theorems 60G44 Martingales with continuous parameter 46N30 Applications of functional analysis in probability theory and statistics Keywords:maximum likelihood estimation; ergodic theorem; infinitesimal generator; infinite dimensional model; infinite dimensional stationary Markov process; nuclear space; Gaussian white noise process; ergodic stationary Markov process; central limit theorem PDFBibTeX XMLCite \textit{J. Mohapl}, Commun. Stat., Stochastic Models 10, No. 4, 781--794 (1994; Zbl 0815.62057) Full Text: DOI