Statistical estimation in a scheme of random variables on a Markov chain in case of incomplete observations. (Russian) Zbl 0639.62080

Voĭna, A. A.

Teor. Veroyatn. Mat. Stat., Kiev 37, 16-26 (1987).

The author considers an ergodic stationary Markov chain x(t), \(t\in N\), walking over a sequence of i.i.d. random vectors \((\xi_ i^{(t)}\); \(i=1,...,k)\). Let the composition \(\eta (t)=\xi^{(t)}_{x(t)}\) be observable only, and the transition probabilities \(p_{ij}\) of x and distribution functions \(F_ i\) of \(\xi_ i^{(1)}\) depend on an unknown vector parameter \(\theta\) with true value \(\theta\) 0.
Using a modification of a posteriori probability principle, under smoothness and nondegeneracy assumptions on \(p_{ij}^{(\theta)}\) and \(F_ i^{(\theta)}\), one establishes consistency and asymptotic normality of the estimators for \(\theta\) 0, and for the first moments, and the quantiles of \(F_ i^{(\theta \quad 0)}\).

Reviewer: E.I.Trofimov

MSC:

62M05	Markov processes: estimation; hidden Markov models
62F12	Asymptotic properties of parametric estimators
62E20	Asymptotic distribution theory in statistics

Keywords:

ergodic stationary Markov chain; transition probabilities; a posteriori probability principle; consistency; asymptotic normality; moments; quantiles