Data Sharpening as a Prelude to Density Estimation

Research Report SRR99-001

Data Sharpening as a Prelude to Density Estimation

Edwin Choi and Peter Hall

Abstract: We introduce a simple data-perturbation method as a means of reducing bias of a wide variety of density estimators, in univariate, multivariate, spatial and spherical data settings. The method involves `sharpening' each datum by moving it a small distance along an estimate of the projection into the sample space of the line of steepest ascent up the surface defined by the density; and then computing the estimator in the usual way, but from the sharpened data rather than the original data. The transformation depends in a very simple, explicit way on the smoothing parameter employed for the density estimator, which may be based on classical kernel methods, orthogonal series, histosplines, singular integrals or other linear or approximately-linear methods. Bias is reduced by an order of magnitude, at the expense of a constant-factor increase in variance. In its simplest form, appropriate for a classical multivariate kernel estimator, the sharpening transformation amounts exactly to a standard Nadaraya-Watson kernel estimator in which the data simultaneously play the roles of treatment and response variables. The sharpening transformation may be iterated, and other generalisations and variations are possible.

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