151 research outputs found
A Consistency Theorem for Regular Conditional Distributions
Let (omega, beta) be a measurable space, An in B a sub-sigma-field and Āµn a random probability measure, n >= 1. In various frameworks, one looks for a probability P on B such that Āµn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The conditions are quite simple when (omega, beta) is a compact Hausdorff space equipped with the Borel or the Bairesigma-field (as well as under other similar assumptions). Such conditions are then applied to Bayesian statistics.Posterior distribution, Random probability measure, Regular conditional distribution.
Complexity regularization via localized random penalties
In this article, model selection via penalized empirical loss minimization in
nonparametric classification problems is studied. Data-dependent penalties are
constructed, which are based on estimates of the complexity of a small subclass
of each model class, containing only those functions with small empirical loss.
The penalties are novel since those considered in the literature are typically
based on the entire model class. Oracle inequalities using these penalties are
established, and the advantage of the new penalties over those based on the
complexity of the whole model class is demonstrated.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000046
Modeling inequality and spread in multiple regression
We consider concepts and models for measuring inequality in the distribution
of resources with a focus on how inequality varies as a function of covariates.
Lorenz introduced a device for measuring inequality in the distribution of
income that indicates how much the incomes below the u quantile fall
short of the egalitarian situation where everyone has the same income. Gini
introduced a summary measure of inequality that is the average over u of the
difference between the Lorenz curve and its values in the egalitarian case.
More generally, measures of inequality are useful for other response variables
in addition to income, e.g. wealth, sales, dividends, taxes, market share and
test scores. In this paper we show that a generalized van Zwet type dispersion
ordering for distributions of positive random variables induces an ordering on
the Lorenz curve, the Gini coefficient and other measures of inequality. We use
this result and distributional orderings based on transformations of
distributions to motivate parametric and semiparametric models whose regression
coefficients measure effects of covariates on inequality. In particular, we
extend a parametric Pareto regression model to a flexible semiparametric
regression model and give partial likelihood estimates of the regression
coefficients and a baseline distribution that can be used to construct
estimates of the various conditional measures of inequality.Comment: Published at http://dx.doi.org/10.1214/074921706000000428 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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