On the optimal choice of conditions of observations in a problem of estimation of the mean. (Russian) Zbl 0567.62071

Dorogovtsev, A. Ya.; Kukush, A. G.

Teor. Veroyatn. Mat. Stat. 30, 38-45 (1984).

The authors study the problem of finding an unbiased linear estimator with minimum variance for the parameter \(\alpha\) from the observations \(\Psi_ i=\int^{i\Delta}_{i\Delta -\Delta}X(t)dt,\) \(1\leq i\leq 2n\), where X(t) is the process satisfying the stochastic differential equation \[ dX(t)=\alpha dt+u(t)[\phi (t)dt+\sigma (t)dW(t)],\quad t\geq 0, \] where \(\phi\) is the nuisance parameter, u a known periodic function with known period \(2\Delta\) \((\Delta >0\) fixed), \(\sigma\) (t) known and W(.) is the standard Wiener process. Suppose u(.) is of the form: \[ u_ x(t)=-1,\quad 0<t<x,\quad and\quad u_ x(t)=1\quad for\quad x<t<2\Delta, \] where x is the switch over point. Let \({\hat \alpha}{}_ x\) be the least squares estimator corresponding to \(u_ x\). \(x_ 0\in (0,2\Delta)\) is said to be an optimal switch over point if \[ Var[{\hat \alpha}_{x_ 0}]=\min \{Var[{\hat \alpha}_ x],\quad 0<x<2\Delta \}. \] Let \(x_ 0^{(n)}\) and \({\hat \alpha}{}_{x_ 0}^{(n)}\) be the corresponding functions when \(T=2n\Delta\). The point \(x_ 0^{(\infty)}\in [0,2\Delta]\) is said to be an asymptotically optimal switch over point if \(\lim_{n\to \infty}x_ 0^{(n)}=x_ 0^{(\infty)}\) and in this case \({\hat \alpha}{}_{x_ 0}^{(n)}(n)\) is said to be optimal if \(\lim_{n}Var {\hat \alpha}_{x_ 0}^{(n)}=0.\) The authors study conditions under which there exists an asymptotically optimal switch over point for optimal estimation of \(\alpha\).

Reviewer: B.L.S.Prakasa Rao

MSC:

62M05	Markov processes: estimation; hidden Markov models
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

unbiased linear estimator; minimum variance; standard Wiener process; least squares estimator; asymptotically optimal switch over point; optimal estimation