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Optimal and full rates of functional estimators for continuous-time processes. (Vitesses optimales et superoptimales des estimateurs fonctionnels pour les processus à temps continu.) (French) Zbl 0791.60027
Summary: Let \((X_ t,t \in \mathbb{R})\) be an \(\mathbb{R}^ t\)-valued stochastic process observed on the time interval \([0,T]\). We assume that the \(X_ t\)’s have the same absolutely continuous distribution with density \(f\). Then under mild regularity and asymptotic independence conditions the quadratic error of an estimator of \(f\) is an \(O(T^{-4/(s+4)})\). We show that this optimal rate is actually reached. Under a local assumption we obtain the full rate \(T^{-1}\) in a more general setting than J. V. Castellana and M. R. Leadbetter [Stochastic Processes Appl. 21, 179-193 (1986; Zbl 0588.62156)]. Intermediate rates are also given. Finally the same rates (optimal, full intermediate) are obtained for the kernel regression estimator.

60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)