Asymptotics for the Transformation Kernel Density Estimator

Abstract

An asymptotic expansion is provided for the transformation kernel density estimator introduced by Ruppert and Cline. Let $h_k$ be the band-width used in the $k$th iteration, $k = 1,2,\ldots, t$. If all bandwidths are of the same order, the leading bias term of the $l$th derivative of the $t$th iterate of the density estimator has the form $\bar{b}^{(l)}_t(x) \pi^t_{k=1} h^2_k$, where the bias factor $\bar{b}_t(x)$ 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 $K, h_1$ and $h_k/f(x), k = 2,\ldots, t$.