Binary distributions of concentric rings

TL;DR

This paper introduces a new class of binary distributions based on concentric rings, characterized by symmetry and positive dependence, with computational advantages and applications in dependence measurement and parameter estimation.

Contribution

It presents a novel family of symmetric binary distributions on star graphs, with Kronecker product representations and methods for dependence analysis and maximum likelihood estimation.

Findings

Distributions relate to evenly-spaced concentric rings.

Kronecker product makes computations efficient for many variables.

Derived MLEs for observed and hidden inner node cases.

Abstract

We introduce families of jointly symmetric, binary distributions that are generated over directed star graphs whose nodes represent variables and whose edges indicate positive dependences. The families are parametrized in terms of a single parameter. It is an outstanding feature of these distributions that joint probabilities relate to evenly-spaced concentric rings. Kronecker product characterizations make them computationally attractive for a large number of variables. We study the behaviour of different measures of dependence and derive maximum likelihood estimates when all nodes are observed and when the inner node is hidden.