Nonparametric estimation for i.i.d. Gaussian continuous time moving average models

Abstract

We consider a Gaussian continuous time moving average model $$X(t)=\int _0^t a(t-s)dW(s)$$ where W is a standard Brownian motion and a(.) a deterministic function locally square integrable on $${{\mathbb {R}}}^+$$ . Given N i.i.d. continuous time observations of $$(X_i(t))_{t\in [0,T]}$$ on [0, T], for $$i=1, \dots , N$$ distributed like $$(X(t))_{t\in [0,T]}$$ , 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.