ASYMPTOTICS FOR THE TRANSFORMATION KERNEL DENSITY ESTIMATOR

An asymptotic expansion is provided for the transformation kernel dens ity estimator introduced by Ruppert and Cline. Let h(k) be the bandwid th used in the kth iteration, k = 1, 2,..., t. If all bandwidths are o f the same order, the leading bias term of the lth derivative of the t th iterate of the density estimator has the form (b) over bar(t)((l))( x)Pi(k=1)(t) h(k)(2), where the bias factor (b) over bar(t)(x) depends on the second moment of the kernel K, as well as on all derivatives o f the density f up to order 2t. In particular, the leading bias term i s of the same order as when using an ordinary kernel density estimator with a kernel of order 2t. The leading stochastic term involves a ker nel of order 2t that depends on K, h(l) and h(k)/f(x), k = 2,..., t.