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Parameter estimation in linear filtering. (English) Zbl 0768.62086
Let a partially observable random process \((x_ t,y_ t)\), \(t\geq 0\), be given, where only the second component \((y_ t)\) is observed. Suppose that \((x_ t,y_ t)\) satisfy the following system of stochastic differential equations driven by independent Wiener processes \((W_ 1(t))\) and \((W_ 2(t))\): \[ dx_ t=-\beta x_ t dt+dW_ 1(t),\;x_ 0=0,\;dy_ t=\alpha x_ t dt+dW_ 2(t),\;y_ 0=0;\;\alpha,\beta\in(a,b),\;\alpha>0. \] The local asymptotic normality of the model is proved and a large deviation inequality for the maximum likelihood estimator of the parameter \(\theta=(\alpha,\beta)\) is obtained. This implies strong consistency, efficiency, asymptotic normality and the convergence of moments for the maximum likelihood estimator.

62M20 Inference from stochastic processes and prediction
62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
62M99 Inference from stochastic processes
60F10 Large deviations
Full Text: DOI
[1] Bagchi, A; Borkar, V, Parameter identification in infinite dimensional linear systems, Stochastics, 12, 201-213, (1984) · Zbl 0541.93072
[2] Balakrishnan, A.V, (), Lecture notes in Economics and Mathematical Systems
[3] Balakrishnan, A.V, ()
[4] Basawa, I.V; Prakasa Rao, B.L.S, ()
[5] Basawa, I.V; Scott, D.J, (), Lecture notes in statistics · Zbl 0494.62074
[6] Ibragimov, I.A; Hasminski, R.Z, ()
[7] Kallianpur, G, ()
[8] Kutoyants, Yu.A, (), R & E, Research and Exposition in Mathematics, No. 6
[9] Liptser, R.S; Shiryayev, A.N, ()
[10] Rao, C.R, ()
[11] Selukar, R.S, On estimation of Hilbert space valued parameters, () · Zbl 0768.62086
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