On the consistency of kernel density estimates under modality constraints

Nonparametric density estimation under modality constraints has attracted considerable interest. With classical kernel density estimates, the number of modes depends on the chosen bandwidth. We consider the Gaussian kernel and prove for almost arbitrary and possibly non-smooth densities that there is a sequence of ranges of bandwidths leading to consistent estimates while still guaranteeing either the correct number of modes k or a correct upper bound on k. More precisely, a bandwidth sequence that is selected by any method from our proposed sequence of ranges will lead to estimates that are consistent at continuity points.