VARIABLE LOCATION AND SCALE KERNEL DENSITY ESTIMATION

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

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

2Department of Statistics, University of British Columbia, 2021 West Mall, Vancouver, Canada V6T 1W5 3Department 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 → 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.