Lower bound for quadratic losses of estimation of infinite-dimensional parameter
The paper deals with the estimation of a parameter \(\theta\) belonging to a relatively compact and convex subset \(\Theta\) of a separable Hilbert space with norm \(\|\cdot\|\). The experiment is generated by the observations \(X^N\) having the distribution \(P_\theta^{(N)}\). Contrary to earlier investigations [\textit{R. Rudzkis} and the author, ibid. 33, No. 1, 56-75 (1993; Zbl 0806.62026)] no i.i.d. assumptions are made. For any \(U\subset \Theta\) the minimax risk is defined by \[ \delta_N(U)= \inf_{\widehat{\theta}} \sup_{\theta\in U} E_\theta \|\widehat{\theta}- \theta\|^2 \] where the infimum is taken over all estimators \(\widehat{\theta} (X^N)\) of the parameter \(\theta\). Following the common approach based on the notion of local asymptotic normality it is shown that there exists a functional \(\Delta_N (\theta,\Theta)\) such that \[ \delta_N (U(\theta, \varepsilon_N))\geq \Delta_N(\theta, \Theta) (1+o(1)), \quad N\to \infty. \] Some applications to nonparametric estimation problems are presented, too.
scientific article; zbMATH DE number 1135549