Cube root weak convergence of empirical estimators of a density level set

Philippe Berthet (1) , John H.J. Einmahl (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Tilburg University - Department of Econometrics and Operations Research

Given n independent random vectors with common density f on R d , we study the weak convergence of three empirical-measure based estimators of the convex λ-level set L λ of f , namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets A that contains L λ. Since these set-valued estimators approach L λ , even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of L λ and each of the three estimators, at rate n −1/3. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits "richer" limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of L λ , and the set-valued argmax of a drifted Wiener process. MSC 2010 subject classifications: Primary 62G05, 62G20; secondary 60F05, 60F17.

Mots clés: Argmax drifted Wiener process; Cube root asymptotics; Density level set; Excess mass; Local empirical process; Minimum volume set; Set-valued estimator

Domaines: Mathématiques [math]; Probabilités [math.PR]; Statistiques [math.ST]

Philippe Berthet, John H.J. Einmahl. Cube root weak convergence of empirical estimators of a density level set. Annals of Statistics, 2022, 50 (3), pp.1423-1446. ⟨hal-02732982v2⟩