On local likelihood density estimation

This paper considers a class of local likelihood methods introduced by Eguchi and Copas. Unified asymptotic results are presented in the usual smoothing context of the bandwidth, h, tending to zero as the sample size tends to infinity. We present our results pointwise in the univariate case, but then go on to extend them to global properties and to indicate how to cope with the multivariate case. Specific members of the class due to Copas, and Hjort and Jones are seen to be members of a subset of the whole class with the same, and best, small h behavior. Further comparisons between members of the class are alluded to based on the complementary large h asymptotic results of Eguchi and Copas.