On finite additivity, non-conglomerability and statistical paradoxes.
Some statistical paradoxes arising from the use of non-conglomerable finitely additive distributions are discussed.
On Invariant Parametric Covariance Families.
On maximum entropy priors and a most likely likelihood in auditing.
There are two basic questions auditors and accountants must consider when developing test and estimation applications using Bayes' Theorem: What prior probability function should be used and what likelihood function should be used. In this paper we propose to use a maximum entropy prior probability function MEP with the most likely likelihood function MLL in the Quasi-Bayesian QB model introduced by McCray (1984). It is defined on an adequate parameter. Thus procedure only needs an expected value...
On prior distributions which give rise to a dominated Bayesian experiment.
On probability of first order formulas in a given model
On the amount of information resulting from empirical and theoretical knowledge.
We present a mathematical model allowing formally define the concepts of empirical and theoretical knowledge. The model consists of a finite set P of predicates and a probability space (Ω, S, P) over a finite set Ω called ontology which consists of objects ω for which the predicates π ∈ P are either valid (π(ω) = 1) or not valid (π(ω) = 0). Since this is a first step in this area, our approach is as simple as possible, but still nontrivial, as it is demonstrated by examples. More realistic approach...
On the continuity of invariant statistics
The aim of this paper is to establish theorems on the absolute continuity of translation as well as scale invariant statistics in general, from which the related results by Hodges-Lehmann and Puri-Sen follow. The continuity relations between the joint cdf of a random vector and its marginal cdf's are also considered.
On the equivalence of invariance and almost-invariance from a Bayesian point of view.
On the foundations of statistics and decision theory.
An elementary axiomatic foundation for decision theory is presented at a general enough level to cover standard applications of Bayesian methods. The intuitive meaning of both axioms and results is stressed. It is argued that statistical inference is a particular decision problem to which the axiomatic argument fully applies.
On the frequentist and Bayesian approaches to hypothesis testing.
Hypothesis testing is a model selection problem for which the solution proposed by the two main statistical streams of thought, frequentists and Bayesians, substantially differ. One may think that this fact might be due to the prior chosen in the Bayesian analysis and that a convenient prior selection may reconcile both approaches. However, the Bayesian robustness viewpoint has shown that, in general, this is not so and hence a profound disagreement between both approaches exists. In this paper...
On the Heyde theorem for discrete Abelian groups
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 . 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.
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.
On the relation between gnostical and probability theories
On Third Order Rotatability.
Optimal and Robust Invariant Designs for Cubic Multiple Regression.
Optimal Designs for Models with Block-Block Resp. Treatment-Treatment Correlations.
Optimal estimators in learning theory
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 and outputs , 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 on the base of given data that approximates well the regression function of...
Pivotal inference and the Bayesian controversy.
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.
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...
Principe d'invariance sur le processus de vraisemblance