Electronic Journal of Statistics

P-values for classification

Lutz Dümbgen, Bernd-Wolfgang Igl, and Axel Munk

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Abstract

Let (X,Y) be a random variable consisting of an observed feature vector $X\in \mathcal{X}$ and an unobserved class label Y{1,2,,L} with unknown joint distribution. In addition, let $\mathcal{D}$ be a training data set consisting of n completely observed independent copies of (X,Y). Usual classification procedures provide point predictors (classifiers) $\widehat{Y}(X,\mathcal{D})$ of Y or estimate the conditional distribution of Y given X. In order to quantify the certainty of classifying X we propose to construct for each θ=1,2,,L a p-value $\pi_{\theta}(X,\mathcal{D})$ for the null hypothesis that Y=θ, treating Y temporarily as a fixed parameter. In other words, the point predictor $\widehat{Y}(X,\mathcal{D})$ is replaced with a prediction region for Y with a certain confidence. We argue that (i) this approach is advantageous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects.

Article information

Source
Electron. J. Statist. Volume 2 (2008), 468-493.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
http://projecteuclid.org/euclid.ejs/1214491852

Digital Object Identifier
doi:10.1214/08-EJS245

Mathematical Reviews number (MathSciNet)
MR2417390

Zentralblatt MATH identifier
1138.60313

Subjects
Primary: 62C05: General considerations 62F25: Tolerance and confidence regions 62G09: Resampling methods 62G15: Tolerance and confidence regions 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Keywords
nearest neighbors nonparametric optimality permutation test prediction region ROC curve typicality index validity

Citation

Dümbgen, Lutz; Igl, Bernd-Wolfgang; Munk, Axel. P-values for classification. Electron. J. Statist. 2 (2008), 468--493. doi:10.1214/08-EJS245. http://projecteuclid.org/euclid.ejs/1214491852.


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