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March, 1973 Deleted Estimates of the Bayes Risk
T. J. Wagner
Ann. Statist. 1(2): 359-362 (March, 1973). DOI: 10.1214/aos/1176342374


Consider the usual decision theoretic situation where one observes a random vector $X$ from which an estimate of its classification $\theta \in \{0, 1\}$ is to be made. If one knows the a priori probabilities for $\theta$ and the conditional densities of $X$ given $\theta$ then the smallest probability of error which can be achieved is called the Bayes risk and denoted by $R^\ast$. Assuming that the a priori probabilities and conditional densities are unknown we consider the problem of estimating $R^\ast$ from the independent observations $(X_1, \theta_1),\cdots, (X_n, \theta_n)$. Suppose $X$ has an unknown classification $\theta$ where $(X, \theta)$ is independent of the observations $(X_1, \theta_1),\cdots, (X_n, \theta_n)$. If $\{\delta_n\}$ is a sequence of decision procedures, where $\delta_n$ determines the estimate of $\theta$ from $X$ and $(X_1, \theta_1),\cdots, (X_n, \theta_n)$, then the notion of a deleted estimate of $R^\ast$ with $\delta_n$ is introduced and, under mild assumptions, is shown to be a consistent estimate of $R^\ast$.


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T. J. Wagner. "Deleted Estimates of the Bayes Risk." Ann. Statist. 1 (2) 359 - 362, March, 1973.


Published: March, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0259.62006
MathSciNet: MR362591
Digital Object Identifier: 10.1214/aos/1176342374

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 2 • March, 1973
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