On the asymptotic properties of a simple estimate of the Mode

Christophe Abraham, Gérard Biau, Benoît Cadre
2004 E S A I M: Probability & Statistics  
We consider an estimate of the mode θ of a multivariate probability density f with support in R d using a kernel estimate fn drawn from a sample X1, . . . , Xn. The estimate θn is defined as any x in {X1, . . . , Xn} such that fn(x) = maxi=1,...,n fn(Xi). It is shown that θn behaves asymptotically as any maximizerθn of fn. More precisely, we prove that for any sequence (rn) n≥1 of positive real numbers such that rn → ∞ and r d n log n/n → 0, one has rn θn −θn → 0 in probability. The asymptotic
more » ... ormality of θn follows without further work. Mathematics Subject Classification. 62G05.
doi:10.1051/ps:2003015 fatcat:xjz4d4fotrdyfawmlrdbnrx4yq