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On the optimal choice of conditions of observations in the problem of estimation of the mean in the presence of a nuisance parameter depending linearly on time. (Russian) Zbl 0564.62064

Observations are given in the form \(\psi_ i=\int^{i\Delta}_{(i- 1)\Delta}X(t)dt\), \(1\leq i\leq 2n\), where the random process X(t) has the stochastic differential \[ dX(t)=\alpha dt+u(t)[(a+bt)dt+\sigma (t)dw(t)],\quad t\in [0,+\infty); \] \(\alpha\) is the unknown parameter to be estimated; u(t) is a control function; a and b are nuisance parameters; w(t) is a Wiener process; \(\sigma\) (t) is intensity of noise. The function u(t) is periodic with the period \(2\Delta\). Within the period u(t) has the form: \(u_ x(t)=1\) for \(0<t<x\), and \(u_ x(t)=-1\) for \(x<t<2\Delta\), where x is the switching point.
The linear least squares estimator of \(\alpha\) is obtained. The authors also obtain optimal switching points for providing the lowest possible dispersion of estimation.
Reviewer: E.Gilbo

MSC:

62M05 Markov processes: estimation; hidden Markov models
62M09 Non-Markovian processes: estimation
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62F10 Point estimation
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