Estimating level sets of a distribution function using a plug-in method: a multidimensional extension

Elena Di Bernadino (1) , Thomas Laloë (2)

1 SAF - Laboratoire de Sciences Actuarielle et Financière
2 JAD - Laboratoire Jean Alexandre Dieudonné

This paper deals with the problem of estimating the level sets $L(c)= \{F(x) \geq c \}$, with $c \in (0,1)$, of an unknown distribution function $F$ on \mathbb{R}^d_+$. A plug-in approach is followed. That is, given a consistent estimator $F_n$ of $F$, we estimate $L(c)$ by $L_n(c)= \{F_n(x) \geq c \}$. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. These results can be considered as generalizations of results previously obtained, in a bivariate framework, in Di Bernardino et al. (2011). Finally we investigate the effects of scaling data on our consistency results.

Mots clés: Level sets; multidimensional distribution function; plug-in estimation; Hausdorff distance

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

Elena Di Bernadino, Thomas Laloë. Estimating level sets of a distribution function using a plug-in method: a multidimensional extension. 2012. ⟨hal-00668317⟩