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May, 1978 A Finite Sample Distribution-Free Performance Bound for Local Discrimination Rules
W. H. Rogers, T. J. Wagner
Ann. Statist. 6(3): 506-514 (May, 1978). DOI: 10.1214/aos/1176344196

Abstract

In the discrimination problem the random variable $\theta$, known to take values in $\{1,\cdots, M\}$, is estimated from the random vector $X$. All that is known about the joint distribution of $(X, \theta)$ is that which can be inferred from a sample $(X_1, \theta_1),\cdots, (X_n, \theta_n)$ of size $n$ drawn from that distribution. A discrimination rule is any procedure which determines a decision $\hat{\theta}$ for $\theta$ from $X$ and $(X_1, \theta_1),\cdots, (X_n, \theta_n)$. A rule is called $k$-local if the decision $\hat{\theta}$ depends only on $X$ and the pairs $(X_i, \theta_i)$ for which $X_i$ is one of the $k$-closest to $X$ from $X_1,\cdots, X_n$. It is shown that for any $k$-local discrimination rule, the mean-square difference between the probability of error for the rule and its deleted estimate is bounded by $A/n$ where $A$ is an explicitly given small constant which depends only on $M$ and $k$. Thus distribution-free confidence intervals can be placed about probability of error estimates for $k$-local discrimination rules.

Citation

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W. H. Rogers. T. J. Wagner. "A Finite Sample Distribution-Free Performance Bound for Local Discrimination Rules." Ann. Statist. 6 (3) 506 - 514, May, 1978. https://doi.org/10.1214/aos/1176344196

Information

Published: May, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0385.62041
MathSciNet: MR468042
Digital Object Identifier: 10.1214/aos/1176344196

Subjects:
Primary: 62H30
Secondary: 62G05 , 62G15

Keywords: deleted estimate , discrimination , distribution-free bound , error rate estimate , finite sample bound , local inference , nearest neighbor rules

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 3 • May, 1978
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