Nonparametric estimation for i.i.d. Gaussian continuous time moving average models
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Nonparametric estimation for i.i.d. Gaussian continuous time moving average models Fabienne Comte1
· Valentine Genon-Catalot1
Received: 16 April 2020 / Accepted: 19 September 2020 © Springer Nature B.V. 2020
Abstract t We consider a Gaussian continuous time moving average model X (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 (X i (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. Keywords Continuous time moving average · Gaussian processes · Model selection · Nonparametric estimation · Projection estimators Mathematics Subject Classification 62G05 · 62M09
1 Introduction Samples of infinite dimensional data, especially of data recorded continuously over a time interval are now a commonly encountered type of data due to the possibilities of modern technology. They arise in many fields of applications, e.g. in econometrics where authors rather speak of panel data and supply the field of functional data analysis (FDA) whose scope is no more to be demonstrated (see, for general ideas and lots of examples, Hsiao (2003), Ramsay and Silverman (2007), Wang et al. (2015)). Parametric models are most often proposed to deal with FDA. However, nonparametric approaches allow for more flexibility and robustness.
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Fabienne Comte [email protected] Valentine Genon-Catalot [email protected]
1
Université de Paris, CNRS, MAP5 UMR 8145, 75006 Paris, France
123
Statistical Inference for Stochastic Processes
In the present contribution, we consider i.i.d. observations (X i (t), t ∈ [0, T ], i = 1, . . . , N ) of the continuous time moving average (CMA) process t X (t) = a(t − s)dW (s) (1) 0
where (W (t), t ≥ 0) is a Wiener process and a : R+ → R is a deterministic square integrable function. Our aim is to study the new and challenging question of the nonparametric estimation of the function g = a 2 from these observations under very general conditions on the function a(t). Our assumptions include in particular the classical CARMA processes (continuous ARMA) but also more complicated processes such as the continuous time fractionally integrated process of order d (see (3)), defined in (Comte and Renault 1996, Definition 2) which is linked with Brownian motion with Hurst index H = d + (1/2). CMA processes have been the subject of a huge number of contributions concerned with modelling properties. Estimation procedures rely on the observation of a unique sa
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