Abstract
It is well established that one can improve performance of kernel density estimates by varying the bandwidth with the location and/or the sample data at hand. Our interest in this paper is in the data-based selection of a variable bandwidth within an appropriate parameterized class of functions. We present an automatic selection procedure inspired by the combinatorial tools developed in Devroye and Lugosi (2001). It is shown that the expected L 1 error of the corresponding selected estimate is up to a given constant multiple of the best possible error plus an additive term which tends to zero under mild assumptions.