On near neighbour estimates of a multivariate density

A recent paper by Mack and Rosenblatt (J. Multivar. Anal. 9 (1979), 1-15) has shown that near neighbour estimators of a density may perform more poorly than other kernel-type estimators, particularly for x values in the tail of a distribution. In order to overcome the difficulties discovered by Mack and Rosenblatt, a generalized type of near neighbour estimator is proposed. Here the window size, or bandwidth, is chosen as a function of near neighbour distances, rather than actually equal to one of the distances. Two forms for this function are suggested and it is proved that for large samples the resulting estimator does not suffer the drawbacks of the usual near neighbour estimator.