The Annals of Statistics

Best Possible Constant for Bandwidth Selection

Jianqing Fan and James S. Marron

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For the data based choice of the bandwidth of a kernel density estimator, several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth. In particular the relative rate of convergence is the square root of the sample size, which is known to be the best possible. The point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient, that is, an analog of the usual Fisher information, in this convergence. This establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform. It is seen that some existing methods attain the bound, others do not.

Article information

Ann. Statist., Volume 20, Number 4 (1992), 2057-2070.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G07: Density estimation
Secondary: 62B10: Information-theoretic topics [See also 94A17] 62C20: Minimax procedures

Bandwidth selection efficient bounds kernel density estimator semiparametric methods


Fan, Jianqing; Marron, James S. Best Possible Constant for Bandwidth Selection. Ann. Statist. 20 (1992), no. 4, 2057--2070. doi:10.1214/aos/1176348902.

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