On a class of kernel density estimate bandwidth selectors

Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. Scandinavian Journal of Statistics, 19(4) pp. 337–349.

URL: http://www.jstor.org/stable/4616251

Abstract

A class of data-based bandwidth selection procedures for kernel density estimation is investigated. These procedures yield bandwidth estimates ĥ which have a (poor) relative error rate of convergence to h₀, the bandwidth which minimises mean integrated squared error, of order n^-1/10 as n → ∞, where n is the sample size. Various members of this class are identified--some are new, but they include the well-known least squares cross-validation--and the unified treatment they receive is novel. Relative error rate of convergence of these ĥ’s to ĥ₀ the bandwidth which minimises integrated squared error, is also examined. For this criterion, an n^-1/10 convergence rate is best possible, and these ĥ's still achieve it. Our treatment of the latter problem has the important spinoff of helping clarify the role of methods which are better estimates of h₀ (for which convergence rates as good as n^-1/2 can be attained) in estimating ĥ₀. Constant multipliers of n-1/10 are derived throughout and these provide theoretical rankings of methods within the class. A small simulation study provides information about which comparisons are of real practical consequence. An important conclusion is that methods that are good at estimating ĥ₀ are likely to be as good as any at estimating ĥ₀too.

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