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Variable kernel density estimation. (English) Zbl 0763.62024

Summary: We investigate some of the possibilities for improvement of univariate and multivariate kernel density estimates by varying the window over the domain of estimation, pointwise and globally. Two general approaches are to vary the window width by the point of estimation and by point of the sample observation.
The first possibility is shown to be of little efficacy in one variable. In particular, nearest-neighbor estimators in all versions perform poorly in one and two dimensions, but begin to be useful in three or more variables.
The second possibility is more promising. We give some general properties and then focus on the popular I. S. Abramson estimator [ibid. 10, 1217-1223 (1982; Zbl 0507.62040)]. We show that in many practical situations, such as normal data, a nonlocality phenomenon limits the commonly applied version of the Abramson estimator to bias of \(O([h/\log h]^ 2)\) instead of the hoped for \(O(h^ 4)\).

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis

Citations:

Zbl 0507.62040
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