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August, 1995 Asymptotics for the Transformation Kernel Density Estimator
Ola Hossjer, David Ruppert
Ann. Statist. 23(4): 1198-1222 (August, 1995). DOI: 10.1214/aos/1176324705


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 $h_k/f(x), k = 2,\ldots, t$.


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Ola Hossjer. David Ruppert. "Asymptotics for the Transformation Kernel Density Estimator." Ann. Statist. 23 (4) 1198 - 1222, August, 1995.


Published: August, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0839.62043
MathSciNet: MR1353502
Digital Object Identifier: 10.1214/aos/1176324705

Primary: 62G07
Secondary: 62G20

Keywords: bias reduction , higher order kernels , smoothed empirical distribution , transformation to uniform distribution , variable bandwidths

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 4 • August, 1995
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