## The Annals of Statistics

### On the Asymptotic Probability of Error in Nonparametric Discrimination

Luc Devroye

#### 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

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