ASYMPTOTIC BIAS AND VARIANCE FOR A GENERAL-CLASS OF VARYING BANDWIDTHDENSITY ESTIMATORS

We consider a general class of varying bandwidth estimators of a proba bility density function. The class includes the Abramson estimator, tr ansformation kernel density estimator (TKDE), Jones transformation ker nel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-LintonNielsen estimator (JLN), Taylor series approximations of T KDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinar y kernel estimator (f) over cap(1)$, it is possible to iterate and com pute a sequence of estimates (f) over cap(2)$,...,(f) over cap(t)$, us ing each estimate as a pilot estimator in the next step. The first mai n result is a formula for the bias order. If the bandwidths used in di fferent steps have a common order h = h(n), the bias of (f) over cap(k )$ is of order h(2k Lambda m), k = 1,...,t. Here h(m) is the bias orde r of the ideal estimator (defined by using the unknown f as pilot). Th e second main result is a recursive formula for the leading bias and s tochastic terms in an asymptotic expansion of the density estimates. I f in < infinity, it is possible to make (f) over cap(t)$ asymptoticall y equivalent to the ideal estimator.