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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.

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