Estimation in white Gaussian noise by means of finite number of linear statistics

G. K. Golubev ; R. Z. Has'minskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 278-290

Résumé

We consider the asymptotical properties of estimators $T$, which are measurable in respect to statistics $$ Y_i=\int_0^1 \psi_i(t)\,dX_\varepsilon(t) $$ if the observed process $X_\varepsilon(t)$ is determined by (1). The problem is to find the best «filters» $\psi_1(t),\dots,\psi_N(t)$ for subsequent estimation of $\theta$. It is proved that the best in minimax sense are the functions $\psi_i$ which determine the $N$-dimensional projector on the subspace, which is the tightest one to $\partial S/\partial\theta$ in some sense. More precisely it is necessary to consider the tightest projector among the admissible (in the sense of (11)) projectors. The examples, for which the optimal filters $\psi_i$ can be found, are considered.