Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator, it is possible to iterate and compute a sequence of estimates, using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh^2km,k=1, ...,t. Hereh^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.