Concentration study of M-estimators using the influence function

Mathieu, Timothée

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1214/22-ejs2030

Title :

Concentration study of M-estimators using the influence function

Author(s) :

Mathieu, Timothée [Auteur]
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Scool [Scool]
Statistique mathématique et apprentissage [CELESTE]

Journal title :

Electronic Journal of Statistics

Pages :

3695-3750

Publisher :

Shaker Heights, OH : Institute of Mathematical Statistics

Publication date :

2022-01-01

ISSN :

1935-7524

English keyword(s) :

Robust Statistics
concentration inequalities
mean estimation

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

We present a new finite-sample analysis of M-estimators of locations in a Hilbert space using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its ...
Show more >We present a new finite-sample analysis of M-estimators of locations in a Hilbert space using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its influence function (or its score function) and then, we use concentration inequality on M-estimators to investigate the robust estimation of the mean in high dimension in a corrupted setting (adversarial corruption setting) for bounded and unbounded score functions. For a sample of size n and covariance matrix Σ, we attain the minimax speed T r(Σ)/n + Σ op log(1/δ)/n with probability larger than 1 − δ in a heavy-tailed setting. One of the major advantages of our approach compared to others recently proposed is that our estimator is tractable and fast to compute even in very high dimension with a complexity of O(nd log(T r(Σ))) where n is the sample size and Σ is the covariance matrix of the inliers and in the code that we make available for this article is tested to be very fast.Show less >

Language :

Anglais

Popular science :

Non

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