arXiv subjects

Paul W. Vos

Publications and source records attributed to Paul W. Vos.

6 recordsLinked to original sources

arXiv2026

A New Look at the Classical Estimation Problem

Bahadur's \emph{Lectures on the Theory of Estimation} develop the classical theory of point estimation inside the geometry of Hilbert space, and they record with unusual honesty where the theory strains: the locally best unbiased estimate depends on the parameter, a two-point parameter space yields an estimate Bahadur calls absurd, the odds ratio in binomial sampling has no unbiased estimate, and the virtues of maximum likelihood enter as heuristics and remain heuristics. We present a subset of the lectures, in Bahadur's notation and development, and at each strain make one small modification: for each value in the sample space, an estimate $τ$ becomes a function on the parameter space rather than a point in it, the continuum of null hypotheses that Fisher described in 1955. Bahadur's own definition of an estimate, square-integrable at every distribution in the family, already supplies the domain. The payoffs are tracked lecture by lecture: estimators that exist at boundary samples where point estimates do not; an elementary lemma showing that no pointwise criterion admits a uniformly optimal estimator, which explains why admissibility, minimaxity, Bayes averaging, and unbiasedness arose as responses; assessment by information, $Λ(τ)$, with the score attaining the Fisher information bound uniformly by a three-line argument; Cramér--Rao attainment and sufficiency recovered as equality cases of that bound under two maps from point estimators to generalized estimators; and the maximum likelihood heuristics converted into exact statements about the score. Nothing classical is overturned; the classical apparatus is explained using Fisher's characterization of estimation as a continuum of significance tests.

math.ST↗

arXiv2026

Rethinking Mean Square Error: Information, Generalized Estimation, and the James-Stein Paradox

The James-Stein estimator's dominance over maximum likelihood in mean square error has been called a paradox because maximum likelihood is known to be superior in many other respects. One response, due to Efron, is to question maximum likelihood. Another is to question MSE. We pursue the second and compare MSE with $Λ$-information (Vos and Wu, 2025) as criteria for assessing estimators. The comparison rests on two distinctions: between point estimators and generalized estimators -- functions of the sample and parameter jointly, with the score as archetype -- as inferential objects, and between pointwise and family-aware assessment criteria. An elementary lemma shows that no pointwise criterion, MSE or any other risk built from a loss function, admits a uniformly optimal estimator; $Λ$-information, which is family-aware and parameter-invariant, is uniformly maximized by the score. A point estimator is assessed through the generalized estimators it induces, and under the score map its $Λ$-efficiency is the fraction of Fisher information the statistic retains, placing the criterion in Fisher's information-loss tradition. On unbiased estimators, $Λ$-efficiency coincides with variance-based efficiency. Returning to James-Stein, the paradox dissolves: maximum likelihood is fully efficient because it is sufficient, while the James-Stein statistic is exactly two-to-one in the sample, and the information it destroys -- computed exactly -- is concentrated precisely where its MSE advantage is greatest. MSE retains its proper domain under genuine squared-error loss.

math.ST↗

arXiv2025

Semi-tail Units: A Universal Scale for Test Statistics and Efficiency

We introduce $ζ$- and $s$-values as quantile-based standardizations that are particularly suited for hypothesis testing. Unlike p-values, which express tail probabilities, $s$-values measure the number of semi-tail units into a distribution's tail, where each unit represents a halving of the tail area. This logarithmic scale provides intuitive interpretation: $s=3.3$ corresponds to the 10th percentile, $s=4.3$ to the 5th percentile, and $s=5.3$ to the 2.5th percentile. For two-tailed tests, $ζ$-values extend this concept symmetrically around the median. We demonstrate how these measures unify the interpretation of all test statistics on a common scale, eliminating the need for distribution-specific tables. The approach offers practical advantages: critical values follow simple arithmetic progressions, combining evidence from independent studies reduces to the addition of $s$-values, and semi-tail units provide the natural scale for expressing Bahadur slopes. This leads to a new asymptotic efficiency measure based on differences rather than ratios of slopes, where a difference of 0.15 semi-tail units means that the more efficient test moves samples 10\% farther into the tail. Through examples ranging from standardized test scores to poker hand rankings, we show how semi-tail units provide a natural and interpretable scale for quantifying extremeness in any ordered distribution.

stat.ME↗

arXiv2025

Random Variables aren't Random

This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We show how measure-theoretic probability provides a framework for modeling populations through distributions, leading to three key contributions. First, we establish that random variables, properly understood as measurable functions, can be fully characterized without appealing to infinite hypothetical samples. Second, we demonstrate how this perspective enables statistical inference through logical rather than probabilistic reasoning, extending the reductio ad absurdum argument from deductive to inductive inference. Third, we show how this framework naturally leads to information-based assessment of statistical procedures, replacing traditional inference metrics that emphasize bias and variance with information-based approaches that better describe the families of distributions used in parametric inference. This reformulation addresses long-standing debates in statistical inference while providing a more coherent theoretical foundation. Our approach offers an alternative to traditional frequentist inference that maintains mathematical rigor while avoiding the philosophical complications inherent in repeated sampling interpretations.

stat.OT↗

arXiv2022

Generalized Estimators, Slope, Efficiency, and Fisher Information Bounds

Point estimators may not exist, need not be unique, and their distributions are not parameter invariant. Generalized estimators provide distributions that are parameter invariant, unique, and exist when point estimates do not. Comparing point estimators using variance is less useful when estimators are biased. A squared slope $Λ$ is defined that can be used to compare both point and generalized estimators and is unaffected by bias. Fisher information $I$ and variance are fundamentally different quantities: the latter is defined at a distribution that need not belong to a family, while the former cannot be defined without a family of distributions, $M$. Fisher information and $Λ$ are similar quantities as both are defined on the tangent bundle $T\!M$ and $I$ provides an upper bound, $Λ\le I$, that holds for all sample sizes -- asymptotics are not required. Comparing estimators using $Λ$ rather than variance supports Fisher's claim that $I$ provides a bound even in small samples. $Λ$-efficiency is defined that extends the efficiency of unbiased estimators based on variance. While defined by the slope, $Λ$-efficiency is simply $ρ^{2}$, the square of the correlation between estimator and score function.

math.ST↗

arXiv2012

Computational information geometry: theory and practice

This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information geometry from a manifold based approach to one where the simplex is the fundamental geometrical object, thereby allowing applications to models which do not have a fixed dimension or support. Finally, it starts to build a computational framework which will act as a proxy for the 'space of all distributions' that can be used, in particular, to investigate model selection and model uncertainty. A varied set of substantive running examples is used to illustrate theoretical and practical aspects of the discussion. Further developments are briefly indicated.

math.ST↗