Grzegorz A Rempala | Searcharxiv

arXiv2025

Random Measures, ANOVA Models and Quantifying Uncertainty in Randomized Controlled Trials

The paper introduces a novel approach to global sensitivity analysis, grounded in the variance-covariance structure of random variables derived from random measures. The proposed methodology facilitates the application of information-theoretic rules for uncertainty quantification, offering several advantages. Specifically, the approach provides valuable insights into the decomposition of variance within discrete subspaces, similar to the standard ANOVA analysis. To illustrate this point, the method is applied to datasets obtained from the analysis of randomized controlled trials on evaluating the efficacy of the COVID-19 vaccine and assessing clinical endpoints in a lung cancer study.

stat.ME↗

arXiv2025

Orthogonal Dice

In this paper, we introduce a family of discrete rectangular uniform distributions on the natural numbers-referred to as orthogonal dice-characterized by the property that their means equal their variances. These distributions arise naturally in statistics and applied mathematics. We show that the orthogonal dice correspond to solutions of a quadratic Diophantine equation on the naturals, exhibiting divisibility properties tied to their dimensions, generating coprime arithmetic progressions, yielding disjoint partitions of the naturals, and displaying self-similarity. Their associated random counting measures (mixed binomial processes) exhibit interesting structural properties, including orthogonal splitting and convergence to Poisson limits. As a result, the orthogonal dice define canonical stochastic processes that that may be used to construct Brownian and geometric Brownian motions. More broadly, they serve as Poisson-like building blocks-natural substrates for modeling systems with bounded counts. Furthermore, they induce a trichotomy within the broader class of such distributions, partitioning them into three infinite subfamilies-negative, orthogonal, and positive-according to their mean-variance relationships.

math.PR↗