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Zbl 1187.62115
Cérou, Frédéric; Guyader, Arnaud
Nearest neighbor classification in infinite dimension. (English)
[J] ESAIM, Probab. Stat. 10, 340-355 (2006). ISSN 1292-8100; ISSN 1262-3318/e

Summary: Let $X$ be a random element in a metric space $({\cal F},d)$, and let $Y$ be a random variable with value 0 or 1. $Y$ is called the class, or the label, of $X$. Let $(X_i,Y_i)_{1\leq i\leq n}$ be an observed i.i.d. sample having the same law as ($X,Y)$. The problem of classification is to predict the label of a new random element $X$. The $k$-nearest neighbor classifier is the simple following rule: look at the $k$ nearest neighbors of $X$ in the trial sample and choose 0 or 1 for its label according to the majority vote. When $({\cal F},d)=(\mathbb{R} ^d,\vert\vert\cdot \vert\vert)$, {\it C. J. Stone} [Ann. Stat. 5, 595--645 (1977; Zbl 0366.62051)] proved the universal consistency of this classifier: its probability of error converges to the Bayes error, whatever the distribution of ($X,Y)$. We show in this paper that this result is no longer valid in general metric spaces. However, if $({\cal F},d)$ is separable and if some regularity condition is assumed, then the $k$-nearest neighbor classifier is weakly consistent.

MSC 2000:: *62H30 Statistical classification, etc.
62G20 Nonparametric asymptotic efficiency
62G05 Nonparametric estimation

Keywords: classification; consistency

Citations: Zbl 0366.62051

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