The Annals of Statistics

On the Asymptotic Probability of Error in Nonparametric Discrimination

Luc Devroye

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Abstract

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.

Article information

Source
Ann. Statist., Volume 9, Number 6 (1981), 1320-1327.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345648

Digital Object Identifier
doi:10.1214/aos/1176345648

Mathematical Reviews number (MathSciNet)
MR630114

Zentralblatt MATH identifier
0481.62051

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation

Keywords
Nonparametric discrimination pattern recognition inequality of Cover and Hart nearest neighbor rule probability of error

Citation

Devroye, Luc. On the Asymptotic Probability of Error in Nonparametric Discrimination. Ann. Statist. 9 (1981), no. 6, 1320--1327. doi:10.1214/aos/1176345648. https://projecteuclid.org/euclid.aos/1176345648


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