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A note on bias reduction in variable‐kernel density estimates
Author(s) -
Mckay Ian
Publication year - 1993
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315701
Subject(s) - reduction (mathematics) , kernel (algebra) , mathematics , omitted variable bias , statistics , variable (mathematics) , kernel density estimation , econometrics , mathematical analysis , combinatorics , geometry , estimator
It is well known that the inverse‐square‐root rule of Abramson (1982) for the bandwidth h of a variable‐kernel density estimator achieves a reduction in bias from the fixed‐bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O ( h 4 ) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O ( h 4 ) bias expression holds uniformly on any set where the target density is bounded away from zero.