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

Abstract

This paper deals with the problem of estimating the level sets $L(c)=\{F(x)\ge c\}$, with $c\in (0,1)$, of an unknown distribution function $F$ on \mathbb{R}^d_+$.Aplug-inapproachisfollowed.Thatis,givenaconsistentestimator$F_n$of$F$,weestimate$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.