TY - UNPB

T1 - Cube Root Weak Convergence of Empirical Estimators of a Density Level Set

AU - Berthet, Philippe

AU - Einmahl, John

N1 - CentER Discussion Paper Nr. 2020-015

PY - 2020/6/4

Y1 - 2020/6/4

N2 - Given n independent random vectors with common density f on Rd, we study the weak convergence of three empirical-measure based estimators of the convex λ-level set Lλ 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 Lλ. Since these set-valued estimators approach Lλ, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of Lλ 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 Lλ, and the set-valued argmax of a drifted Wiener process.

AB - Given n independent random vectors with common density f on Rd, we study the weak convergence of three empirical-measure based estimators of the convex λ-level set Lλ 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 Lλ. Since these set-valued estimators approach Lλ, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of Lλ 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 Lλ, and the set-valued argmax of a drifted Wiener process.

KW - Argmax drifted Wiener process

KW - cube root asymp- totics

KW - density level set

KW - excess mass

KW - local empirical process

KW - minimum vol- ume set

KW - set-valued estimator

M3 - Discussion paper

VL - 2020-015

T3 - CentER Discussion Paper

BT - Cube Root Weak Convergence of Empirical Estimators of a Density Level Set

PB - CentER, Center for Economic Research

CY - Tilburg

ER -