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

TL;DR

This paper extends the estimation of distribution function level sets to multidimensional data using a plug-in method, providing consistency results and analyzing the impact of data scaling.

Contribution

It generalizes previous bivariate results to higher dimensions and investigates the effects of data scaling on the consistency of level set estimators.

Findings

Consistency results with respect to Hausdorff distance

Consistency results for volume of symmetric difference

Effects of data scaling on estimator consistency

Abstract

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.