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.
Publié le : 1974-11-14
Classification:
Cauchy mean value functions,
isotonic estimation,
norm reducing extrema,
62G05,
60F15
@article{1176342882,
author = {Robertson, Tim and Wright, F. T.},
title = {A Norm Reducing Property for Isotonized Cauchy Mean Value Functions},
journal = {Ann. Statist.},
volume = {2},
number = {1},
year = {1974},
pages = { 1302-1307},
language = {en},
url = {http://dml.mathdoc.fr/item/1176342882}
}
Robertson, Tim; Wright, F. T. A Norm Reducing Property for Isotonized Cauchy Mean Value Functions. Ann. Statist., Tome 2 (1974) no. 1, pp. 1302-1307. http://gdmltest.u-ga.fr/item/1176342882/