On visual distances in density estimation: the Hausdorff choice

We consider a "visual" metric between multivariate densities that is defined in terms of the Hausdorff distance between their hypographs. This distance has been first proposed and analyzed by Beer (1982) in the non-probabilistic context of approximation theory. We suggest the use of this distance in density estimation as a weaker, more flexible alternative to the supremum metric: it also has a direct visual interpretation but does not require very restrictive continuity assumptions. A further Hausdorff-based distance is also proposed and analyzed. We obtain consistency results, and a convergence rate, for the usual kernel density estimators with respect to these metrics provided that the underlying density is not too discontinuous. These results can be seen as a partial extension to the "qualitative smoothing" setup (see Marron and Tsybakov, 1995) of the classical analogous theorems with respect to the supremum metric.