Kernel estimation of density level sets

Benoît Cadre (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $f$ be a multivariate density and $f_n$ be a kernel estimate of $f$ drawn from the $n$-sample $X_1,\cdots,X_n$ of i.i.d. random variables with density $f$. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the $t$-level set $\{f\geq t\}$ and its plug-in estimator $\{f_n\geq t\}$. As a corollary, we obtain the exact rate of convergence of a plug-in type estimate of the density level set corresponding to a fixed probability for the law induced by $f$.

Mots clés: Kernel estimate; Density level sets; Hausdorff measure; Hausdorff measure.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Benoît Cadre. Kernel estimation of density level sets. Journal of Multivariate Analysis, 2006, pp.999-1023. ⟨hal-00003898⟩