Minimax properties of Fréchet means of discretely sampled curves

Jérémie Bigot (1) , Xavier Gendre (2)

1 DMIA - Département de Mathématiques, Informatique, Automatique
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the problem of estimating a mean pattern from a set of similar curves in the setting where the variability in the data is due to random geometric deformations and additive noise. This problem requires to define non-Euclidean distances by using the action of a Lie group on an infinite dimensional space of curves. This approach leads to the construction of estimators based on the notion of Fréchet mean that is a generalization of the standard notion of averaging to non-Euclidean spaces. A recent research direction in nonparametric statistics is the study of the properties of the Fréchet mean in deformable models, and the development of consistent estimators of a mean pattern. Using such models, we show the links that exist between minimax theory in nonparametric statistics and the problem of estimating a mean pattern from a sequence of curves.

Mots clés: Fréchet mean; Non-Euclidean metric; Deformable models; Lie group action; Curve registration; Minimax rate of convergence; Sobolev balls; Functional data analysis

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

Jérémie Bigot, Xavier Gendre. Minimax properties of Fréchet means of discretely sampled curves. 2012. ⟨hal-00737560v1⟩