Potential for automatic bandwidth choice in variations on kernel density estimation

Recently, much progress has been made on understanding the bandwidth selection problem in kernel density estimation. Here, analogous questions are considered for extensions to the basic problem, namely, for estimating derivatives, using 'better' kernel estimators, and for the multivariate case. In basic kernel density estimation, recent advances have resulted in considerable improvements being made over 'moderate' methods such as least squares cross-validation. Here, it is argued that, in the first two extension cases, the performance of moderate methods deteriorates even more, so that the necessity for 'improved' methods -- and indeed their potential in theory if not necessarily in practice -- is greatly increased. Rather extraordinary things happen, however, when higher dimensions are considered. This paper is essentially that of Jones (1991).