arXiv subjects

Robert Zimmerman

Publications and source records attributed to Robert Zimmerman.

5 recordsLinked to original sources

arXiv2026

Primitive Sequences for Probability Measures on Compact Intervals

We introduce a sequence representation of a random variable $X$ supported on a compact interval $[a,b]$, which we call a primitive sequence. We construct this sequence by repeatedly antidifferentiating the associated cumulative distribution function of $X$ and evaluating the antiderivatives at the endpoint $b$. We show that the primitive sequence of $X$ can be identified as a factorially rescaled moment sequence of the reflected random variable $b-X$. Through this identification, we show that the primitive sequence transparently captures qualitative features of the distribution of $X$. We then connect primitive sequences directly to classical moment theory and exploit this connection to characterize admissible primitive sequences and to show that under natural topologies, the map from probability measures to primitive sequences is a homeomorphism. We end by examining the set of probability measures whose first $m$ primitive sequence terms are fixed, and thereby obtaining sharp upper and lower bounds on two functionals of those measures.

math.PR↗

arXiv2025

A Tractable Family of Smooth Copulas with Rotational Dependence: Properties, Inference, and Application

We introduce a new family of copula densities constructed from univariate distributions on $[0,1]$. Although our construction is structurally simple, the resulting family is versatile: it includes both smooth and irregular examples, and reveals clear links between properties of the underlying univariate distribution and the strength, direction, and form of multivariate dependence. The framework brings with it a range of explicit mathematical properties, including interpretable characterizations of dependence and transparent descriptions of how rotational forms arise. We propose model selection and inference methods in parametric and nonparametric settings, supported by asymptotic theory that reduces multivariate estimation to well-studied univariate problems. Simulation studies confirm the reliable recovery of structural features, and an application involving neural connectivity data illustrates how the family can yield a better fit than existing models.

math.ST↗

arXiv2024

Separating States in Astronomical Sources Using Hidden Markov Models: With a Case Study of Flaring and Quiescence on EV Lac

We present a new method to distinguish between different states (e.g., high and low, quiescent and flaring) in astronomical sources with count data. The method models the underlying physical process as latent variables following a continuous-space Markov chain that determines the expected Poisson counts in observed light curves in multiple passbands. For the underlying state process, we consider several autoregressive processes, yielding continuous-space hidden Markov models of varying complexity. Under these models, we can infer the state that the object is in at any given time. The continuous state predictions from these models are then dichotomized with the help of a finite mixture model to produce state classifications. We apply these techniques to X-ray data from the active dMe flare star EV Lac, splitting the data into quiescent and flaring states. We find that a first-order vector autoregressive process efficiently separates flaring from quiescence: flaring occurs over 30-40% of the observation durations, a well-defined persistent quiescent state can be identified, and the flaring state is characterized by higher plasma temperatures and emission measures.

astro-ph.SR↗

arXiv2022

Copula Modelling of Serially Correlated Multivariate Data with Hidden Structures

We propose a copula-based extension of the hidden Markov model (HMM) which applies when the observations recorded at each time in the sample are multivariate. The joint model produced by the copula extension allows decoding of the hidden states based on information from multiple observations. However, unlike the case of independent marginals, the copula dependence structure embedded into the likelihood poses additional computational challenges. We tackle the latter using a theoretically-justified variation of the EM algorithm developed within the framework of inference functions for margins. We illustrate the method using numerical experiments and an analysis of house occupancy.

stat.ME↗

arXiv2022

A new family of smooth copulas with arbitrarily irregular densities

Copulas are known to satisfy a number of regularity properties, and one might therefore believe that their densities, when they exist, admit a certain degree of regularity themselves. We show that this is not true in general by constructing a broad family of copulas which admit densities that can hardly be considered regular. The copula densities are constructed from arbitrary univariate densities supported on the unit interval, and we show by example that the copula densities can inherit pathological behaviour from the underlying univariate densities. In particular, we construct a nontrivial univariate density which is unbounded in every open set of the unit interval, and show that it induces a copula density which is finite everywhere but unbounded in every neighborhood of the unit hypercube. Nevertheless, all of our copulas are shown to enjoy attractive smoothness properties.

math.PR↗