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Added by
Lutz D\\\"umbgen
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
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Let $(X,Y)$ be a random variable consisting of an observed feature vector $Xin mathcal{X}$ and an unobserved class label $Yin {1,2,...,L}$ 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 $theta =1,2,...,L$ a p-value $pi_{theta}(X,mathcal{D})$ for the null hypothesis that $Y=theta$, 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.
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Goods formula and Fishers method are frequently used for combining independent P-values. Interestingly, the equivalent of Goods formula already emerged in 1910 and mathematical expressions relevant to even more general situations have been repeatedly derived, albeit in different context. We provide here a novel derivation and show how the analytic formula obtained reduces to the two aforementioned ones as special cases. The main novelty of this paper, however, is the explicit treatment of nearly degenerate weights, which are known to cause numerical instabilities. We derive a controlled expansion, in powers of differences in inverse weights, that provides both accurate statistics and stable numerics.
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