On change-point estimation under Sobolev sparsity

Aurélie Fischer (1) , Dominique Picard (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

In this paper, we consider the estimation of a change-point for possibly high-dimensional data in a Gaussian model, using a k-means method. We prove that, up to a logarithmic term, this change-point estimator has a minimax rate of convergence. Then, considering the case of sparse data, with a Sobolev regularity, we propose a smoothing procedure based on Lepski's method and show that the resulting estimator attains the optimal rate of convergence. Our results are illustrated by some simulations. As the theoretical statement relying on Lepski's method depends on some unknown constant, practical strategies are suggested to perform an optimal smoothing.

Mots clés: k-means; Clustering; change-point detection; minimax rates; high dimension; smoothing; Lepski's method

Domaines: Théorie [stat.TH]

Aurélie Fischer, Dominique Picard. On change-point estimation under Sobolev sparsity. Electronic Journal of Statistics , 2020, 14 (1), ⟨10.1214/20-EJS1692⟩. ⟨hal-01712690⟩