  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.
