Nonparametric estimation for i.i.d. Gaussian continuous time moving average models.
Résumé
We consider a Gaussian continuous time moving average model X(t) = t 0 a(t − s)dW (s) where W is a standard Brownian motion and a(.) a deterministic function locally square integrable on R +. Given N i.i.d. continuous time observations of (Xi(t)) t∈[0,T ] on [0, T ], for i = 1,. .. , N distributed like (X(t)) t∈[0,T ] , we propose nonparametric projection estimators of a 2 under different sets of assumptions, which authorize or not fractional models. We study the asymptotics in T, N (depending on the setup) ensuring their consistency, provide their nonparametric rates of convergence on functional regularity spaces. Then, we propose a data-driven method corresponding to each setup, for selecting the dimension of the projection space. The findings are illustrated through a simulation study.
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