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Sequential estimation in exponential-type processes under random initial conditions. (English) Zbl 0691.62076

Consider a homogeneous strong Markov process \(\{\) (Z(t),S(t)): \(t\in T\}\), where \(T=[0,\infty)\) or \(\{\) 0,1,...,\(\}\), \(Z(t)=(Z_ 1(t),...,Z_ n(t))\) with the \(Z_ i(t)\) real valued and S(t)\(\geq 0\). Several additional assumptions are made on the \(Z_ i\) and S. It is assumed that at time t the distribution belongs to an exponential family of the form \[ \exp [\sum^{n}_{1}\theta_ iZ_ i(t)+\Phi (\theta)S(t)+\Psi (\theta)] \] times a measure depending on t but not on \(\theta =(\theta_ 1,....,\theta_ n)\), where \(\theta\) lies in an open convex subset of \({\mathbb{R}}^ n\). At \(t=0\), the \(Z_ i(0)\) and S(0) are allowed to be random. With efficient sequential estimation of a real- valued function h(\(\theta)\) is meant a sequential unbiased estimator that attains the Cramér-Rao lower bound.
Previous results of the second author [see Ann. Stat. 14, 1606-1611 (1986; Zbl 0617.62087)], valid for processes with fixed initial conditions, are extended to the present situation with (in general) random initial conditions. A classification is given of efficiently estimable functions h together with their stopping rules. Among the examples given are the Ornstein-Uhlenbeck velocity process and a two- dimensional Gaussian Markov process satisfying certain stochastic differential equations.
Reviewer: R.A.Wijsman

MSC:

62L12 Sequential estimation
62M05 Markov processes: estimation; hidden Markov models

Citations:

Zbl 0617.62087
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