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November, 1974 A Norm Reducing Property for Isotonized Cauchy Mean Value Functions
Tim Robertson, F. T. Wright
Ann. Statist. 2(6): 1302-1307 (November, 1974). DOI: 10.1214/aos/1176342882


We consider functions $\alpha(\bullet)$ and $\hat{\alpha}(\bullet)$ on a finite set $S$ which correspond to a function $M(\bullet)$ on the nonempty subsets of $S$ which has the Cauchy mean value property (i.e., $M(A + B)$ is between $M(A)$ and $M(B)$ whenever $A$ and $B$ are nonempty disjoint subsets of $S$). $\hat{\alpha}(\bullet)$ is isotone with respect to a partial ordering on $S$ and is equal to $\alpha(\bullet)$ when $\alpha(\bullet)$ is isotone. It is shown that $\hat{\alpha}(\bullet)$ has the following norm reducing property: $\max_{s\in S} |\hat{\alpha}(s) - \theta(s)| \leqq \max_{s\in S} |\alpha(s) - \theta(s)|$ for all isotone $\theta(\bullet)$. Computation algorithms for $\hat{\alpha}(\bullet)$ are discussed and the norm reducing property is shown to give consistency results in several isotonic regression problems.


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Tim Robertson. F. T. Wright. "A Norm Reducing Property for Isotonized Cauchy Mean Value Functions." Ann. Statist. 2 (6) 1302 - 1307, November, 1974.


Published: November, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0295.62044
MathSciNet: MR408100
Digital Object Identifier: 10.1214/aos/1176342882

Primary: 62G05
Secondary: 60F15

Keywords: Cauchy mean value functions , isotonic estimation , norm reducing extrema

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 6 • November, 1974
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