Multivariate quantiles with both overall and directional probability interpretation

Abstract

The article introduces multivariate quantiles (or reference regions) that have both overall and directional probability interpretation and need not be necessarily convex. They are defined by means of univariate conditional quantiles along the rays starting at a suitable central point.

Their basic properties are investigated, their sample estimators and regression extensions are proposed, and their use is illustrated with both simulated and real data.

Keywords

• directional quantile
• kernel estimation
• multivariate quantile
• nonconvex quantile
• quantile regression
• reference region
• tolerance region