ASYMPTOTICS FOR THE TRANSFORMATION KERNEL DENSITY ESTIMATOR

An asymptotic expansion is provided for the transformation kernel density estimator introduced by Ruppert and Cline. Let h(k) be the bandwidth used in the kth iteration, k = 1, 2,..., t. If all bandwidths are of the same order, the leading bias term of the lth derivative of the tth 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 of the density f up to order 2t. In particular, the leading bias term is of the same order as when using an ordinary kernel density estimator with a kernel of order 2t. The leading stochastic term involves a kernel of order 2t that depends on K, h(l) and h(k)/f(x), k = 2,..., t.