Asymptotic normality in density support estimation. (English) Zbl 1185.62071

Biau, Gérard; Cadre, Benoît; Mason, David M.; Pelletier, Bruno

Electron. J. Probab. 14, 2617-2635 (2009).

Summary: Let \(X_1,\dots,X_n\) be \(n\) independent observations drawn from a multivariate probability density \(f\) with compact support \(S_f\). This paper is devoted to the study of the estimator \(\widehat{S}_n\) of \(S_f\) defined as the union of balls centered at the \(X_i\) and with common radius \(r_n\). Using tools from Riemannian geometry, and under mild assumptions on \(f\) and the sequence \((r_n)\), we prove a central limit theorem for \(\lambda (S_n \Delta S_f)\), where \(\lambda\) denotes the Lebesgue measure on \(\mathbb R^d\) and \(\Delta\) the symmetric difference operation.

MSC:

62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference
60F05	Central limit and other weak theorems

Keywords:

support estimation; nonparametric statistics; central limit theorem; tubular neighborhood