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

This paper deals with the problem of estimating the level sets $L©= {F(x) \geq c }$,with$c \in (0,1)$,ofanunknowndistributionfunction$F$ 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 $L_n(c)=\{F_n(x)\ge c\}$. 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.