  Density-based clustering relies on the idea of linking groups to some
specific features of the probability distribution underlying the data. The
reference to a true, yet unknown, population structure allows to frame the
clustering problem in a standard inferential setting, where the concept of
ideal population clustering is defined as the partition induced by the true
density function. The nonparametric formulation of this approach, known as
modal clustering, draws a correspondence between the groups and the domains of
attraction of the density modes. Operationally, a nonparametric density
estimate is required and a proper selection of the amount of smoothing,
governing the shape of the density and hence possibly the modal structure, is
crucial to identify the final partition. In this work, we address the issue of
density estimation for modal clustering from an asymptotic perspective. A
natural and easy to interpret metric to measure the distance between
density-based partitions is discussed, its asymptotic approximation explored,
and employed to study the problem of bandwidth selection for nonparametric
modal clustering.
