Nonparametric adaptive time-dependent multivariate function estimation

Jérémie Bigot (1) , Theofanis Sapatinas (2)

1 DMIA - Département de Mathématiques, Informatique, Automatique
2 Department of Mathematics and Statistics

We consider the nonparametric estimation problem of time-dependent multivariate functions observed in a presence of additive cylindrical Gaussian white noise of a small intensity. We derive minimax lower bounds for the $L^2$-risk in the proposed spatio-temporal model as the intensity goes to zero, when the underlying unknown response function is assumed to belong to a ball of appropriately constructed inhomogeneous time-dependent multivariate functions, motivated by practical applications. Furthermore, we propose both non-adaptive linear and adaptive non-linear wavelet estimators that are asymptotically optimal (in the minimax sense) in a wide range of the so-constructed balls of inhomogeneous time-dependent multivariate functions. The usefulness of the suggested adaptive nonlinear wavelet estimator is illustrated with the help of simulated and real-data examples.

Mots clés: Adaptivity; Besov spaces; Block Thresholding; Minimax Estimators; Time-dependent image processing; Wavelet Analysis

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Jérémie Bigot, Theofanis Sapatinas. Nonparametric adaptive time-dependent multivariate function estimation. 2012. ⟨hal-00746516v2⟩