Continuous indetermination and average likelihood minimization

TL;DR

This paper extends the concept of indetermination coupling to continuous probabilities, introduces an average likelihood minimization, and develops a statistical test to identify indetermination with efficiency analysis.

Contribution

It introduces a novel continuous indetermination coupling concept and a likelihood-based test, advancing the understanding of dependence structures beyond copulas.

Findings

Indetermination coupling cannot be fully captured by traditional copulas.

Average likelihood is minimized under indetermination.

The proposed test's efficiency is evaluated using Bahadur's slope.

Abstract

The authors transpose a discrete notion of indetermination coupling in the case of continuous probabilities. They show that this coupling, expressed on densities, cannot be captured by a specific copula which acts on cumulative distribution functions without a high dependence on the margins. Furthermore, they define a notion of average likelihood which extends the discrete notion of couple matchings and demonstrate it is minimal under indetermination. Eventually, they leverage this property to build up a statistical test to distinguish indetermination and estimate its efficiency using the Bahadur's slope.