On the nonparametric estimation of a value of a linear functional in the Gaussian white noise

I. A. Ibragimov   ; R. Z. Has'minskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 19-32

Résumé

Suppose we observe a random process $X_ \varepsilon(t)$, $0\le t\le 1$ satisfying the equation \begin{equation} dX_\varepsilon(t)=s(t)\,dt +\varepsilon\,dw(t) \end{equation} where $w$ is the standard Wiener process and the unknown function $s$ is assumed to belong to some symmetric closed convex subset $\Sigma$ of the space $L_2(0,1)$. Let $L$ be a linear functional defined on $\Sigma$. We consider the problem of estimation of the value $L(s)$ of $L$ at a point $s$ when $X_\varepsilon(t)$, $0\le t\le 1$ is observed. Denote by $\mathscr M$ the set of all linear estimates of $L(s)$ i. e. estimates of the form $\displaystyle\int_0^1m(t)\,dX_\varepsilon(t)$. We proved that 1) $\displaystyle\inf_{\widehat L\in\mathscr M}\sup_{s\in\Sigma}\mathbf E_s(L(s)-\widehat L)^2 =\sup_{s\in\Sigma}\varepsilon^2\frac{L^2(s)} {\varepsilon^2+\|s\|^2}$. 2) If $\displaystyle\sup_{s\in\Sigma}\varepsilon^2\frac{L^2(s)}{\varepsilon^2+\|s\|^2} =\varepsilon^2\frac{L^2(s_\varepsilon)}{\varepsilon^2+\|s_\varepsilon\|^2}$ then $\displaystyle\int_0^1 m_\varepsilon(t)\,dX_\varepsilon(t)$, with $\displaystyle m_\varepsilon= s_ \varepsilon\frac{L(s_\varepsilon)}{\varepsilon^2+\|s_\varepsilon\|^2}$ is a minimax linear estimator. Several examples are considered.