Optimal L1 bandwidth selection for variable kernel density estimates

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. Combinatorial Methods in Density Estimation. Springer, New York]. It is shown that the expected L1 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.