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VARIABLE LOCATION AND SCALE KERNEL

DENSITY ESTIMATION

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M. C. JONES^{1}, I. J. MCKAY^{2} AND T.-C. HU^{3}

^{1} *Department of Statistics, The Open University, Milton Keynes MK7 6AA, U.K.*

^{2} *Department of Statistics, University of British Columbia,*

2021 West Mall, Vancouver, Canada V6T 1W5

^{3} *Department of Mathematics, National Tsing Hua University,*

Hsinchu, Taiwan 30043, R.O.C.
(Received April 2, 1993; revised October 29, 1993)

**Abstract.**
Variable (bandwidth) kernel density estimation
(Abramson (1982, *Ann. Statist.*, **10**, 1217-1223)) and
a kernel estimator with varying locations (Samiuddin and El-Sayyad
(1990, *Biometrika*, **77**, 865-874)) are complementary
ideas which essentially both afford bias of order *h*^{4} as the overall
smoothing parameter *h* \rightarrow 0, sufficient differentiability of
the density permitting. These ideas are put in a more general framework
in this paper. This enables us to describe a variety of ways in which
scale and location variation may be extended and/or combined to good
theoretical effect. This particularly includes extending the basic ideas
to provide new kernel estimators with bias of order *h*^{6}. Technical
difficulties associated with potentially overly large variations are
fully accounted for in our theory.

*Key words and phrases*:
Bias reduction, smoothing, variable
bandwidth.

**Source**
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