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June 1998 Asymptotic expansions of the $k$ nearest neighbor risk
Robert R. Snapp, Santosh S. Venkatesh
Ann. Statist. 26(3): 850-878 (June 1998). DOI: 10.1214/aos/1024691080


The finite-sample risk of the $k$ nearest neighbor classifier that uses a weighted $L^p$-metric as a measure of class similarity is examined. For a family of classification problems with smooth distributions in $mathbb{R}^n$, an asymptotic expansion for the risk is obtained in decreasing fractional powers of the reference sample size. An analysis of the leading expansion coefficients reveals that the optimal weighted $L^p$-metric, that is, the metric that minimizes the finite-sample risk, tends to a weighted Euclidean (i.e., $L^2$) metric as the sample size is increased. Numerical simulations corroborate this finding for a pattern recognition problem with normal class-conditional densities.


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Robert R. Snapp. Santosh S. Venkatesh. "Asymptotic expansions of the $k$ nearest neighbor risk." Ann. Statist. 26 (3) 850 - 878, June 1998.


Published: June 1998
First available in Project Euclid: 21 June 2002

zbMATH: 0929.62070
MathSciNet: MR1635410
Digital Object Identifier: 10.1214/aos/1024691080

Primary: 41A60 , 62G20 , 62H30

Keywords: $k$ nearest neighbor classifier , asymptotic expansions , finite-sample risk , Laplace’s method

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • June 1998
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