Arbitrariness of the pilot estimator in adaptive kernel methods

Consider estimating a smooth p-variate density f at 0 using the classical kernel estimator fn(0) = n-1 [Sigma]ibn-pw(bn-1Xi) based on a sample {Xi} from f. Under familiar conditions, assigning bn = bn-1/(4 + p) gives the best MSE decay rate O(n-4/(4 + p), but the optimal b, b* say, depends on f through its second derivatives, raising a feasibility objection to its use. By prescribing a pilot estimate of b* based on the same sample, Woodroofe has shown that there need be asymptotically no loss as against knowing the constant exactly, but his proposal is critically dependent on achieving a certain consistency rate for b*. Admitting a minor change in the risk function, we show by a tightness argument applied to the error process that any consistent estimator of b* may be used to achieve the same performance.