Asymptotic expansions of the $k$ nearest neighbor risk
Snapp, Robert R. ; Venkatesh, Santosh S.
Ann. Statist., Tome 26 (1998) no. 3, p. 850-878 / Harvested from Project Euclid
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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.
Publié le : 1998-06-14
Classification:  $k$ nearest neighbor classifier,  finite-sample risk,  asymptotic expansions,  Laplace’s method,  62G20,  62H30,  41A60 
@article{1024691080,
     author = {Snapp, Robert R. and Venkatesh, Santosh S.},
     title = {Asymptotic expansions of the $k$ nearest neighbor risk},
     journal = {Ann. Statist.},
     volume = {26},
     number = {3},
     year = {1998},
     pages = { 850-878},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1024691080}
}

Snapp, Robert R.; Venkatesh, Santosh S. Asymptotic expansions of the $k$ nearest neighbor risk. Ann. Statist., Tome 26 (1998) no. 3, pp.  850-878. http://gdmltest.u-ga.fr/item/1024691080/