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

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)