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November, 1981 On the Asymptotic Probability of Error in Nonparametric Discrimination
Luc Devroye
Ann. Statist. 9(6): 1320-1327 (November, 1981). DOI: 10.1214/aos/1176345648


Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times \{0, 1\}$, and let $\hat{Y}$ be the $k$-nearest neighbor estimate of $Y$ from $X$ and the $(X_i, Y_i)$'s. We show that for all distributions of $(X, Y)$, the limit of $L_n = P(\hat{Y} \neq Y)$ exists and satisfies $\lim_{n\rightarrow\infty} L_n \leq (1 + a_k)R^\ast, a_k = \frac{\alpha \sqrt k}{k - 3.25}\big(1 + \frac{\beta}{\sqrt{k-3}}\big), k \text{odd}, k \geq 5,$ where $R^\ast$ is the Bayes probability of error and $\alpha = 0.3399 \cdots$ and $\beta = 0.9749 \cdots$ are universal constants. This bound is shown to be best possible in a certain sense.


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Luc Devroye. "On the Asymptotic Probability of Error in Nonparametric Discrimination." Ann. Statist. 9 (6) 1320 - 1327, November, 1981.


Published: November, 1981
First available in Project Euclid: 12 April 2007

zbMATH: 0481.62051
MathSciNet: MR630114
Digital Object Identifier: 10.1214/aos/1176345648

Primary: 62G05

Keywords: inequality of Cover and Hart , nearest neighbor rule , nonparametric discrimination , pattern recognition , probability of error

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 6 • November, 1981
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