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On the Heyde theorem for discrete Abelian groups

G. M. Feldman (2006)

Studia Mathematica

Let X be a countable discrete Abelian group, Aut(X) the set of automorphisms of X, and I(X) the set of idempotent distributions on X. Assume that α₁, α₂, β₁, β₂ ∈ Aut(X) satisfy β ₁ α ₁ - 1 ± β ₂ α ₂ - 1 ∈ A u t ( X ) . Let ξ₁, ξ₂ be independent random variables with values in X and distributions μ₁, μ₂. We prove that the symmetry of the conditional distribution of L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ implies that μ₁, μ₂ ∈ I(X) if and only if the group X contains no elements of order two. This theorem can be considered as an analogue...

On the logical development of statistical models.

Daniel Peña (1988)

Trabajos de Estadística

This paper presents a classification of statistical models using a simple and logical framework. Some remarks are made about the historical appearance of each type of model and the practical problems that motivated them. It is argued that the current stages of the statistical methodology for model building have arisen in response to the needs for more sophisticated procedures for building dynamic-explicative types of models. Some potentially important topics for future research are included.

Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs x i and outputs y i , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator f z on the base of given data z : = ( ( x ₁ , y ₁ ) , . . . , ( x m , y m ) ) that approximates well the regression function f ρ of...

Pivotal inference and the Bayesian controversy.

George A. Barnard (1980)

Trabajos de Estadística e Investigación Operativa

The theory of pivotal inference applies when parameters are defined by reference to their effect on observations rather than their effect on distributions. It is shown that pivotal inference embraces both Bayesian and frequentist reasoning.

Posterior odds ratios for selected regression hypotheses.

Arnold Zellner, Aloysius Siow (1980)

Trabajos de Estadística e Investigación Operativa

Bayesian posterior odds ratios for frequently encountered hypotheses about parameters of the normal linear multiple regression model are derived and discussed. For the particular prior distributions utilized, it is found that the posterior odds ratios can be well approximated by functions that are monotonic in usual sampling theory F statistics. Some implications of these finding and the relation of our work to the pioneering work of Jeffreys and others are considered. Tabulations of odd ratios...

Quasi-Bayesian behaviour: a more realistic approach to decision making?

Francisco Javier Girón, Sixto Ríos (1980)

Trabajos de Estadística e Investigación Operativa

In this paper the theoretical and practical implications of dropping -from the basic Bayesian coherence principles- the assumption of comparability of every pair of acts is examined. The resulting theory is shown to be still perfectly coherent and has Bayesian theory as a particular case. In particular we question the need of weakening or ruling out some of the axioms that constitute the coherence principles; what are their practical implications; how this drive to the notion of partial information...

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