[Paper Review] Nonsymmetric Dependence Measures: the Discrete Case

Hui Li|arXiv (Cornell University)|Dec 25, 2015

Financial Risk and Volatility Modeling11 references3 citations

TL;DR

This paper introduces nonsymmetric dependence measures for discrete random variables using transition matrices, extending copula-based methods from the continuous case. It establishes that the product operation on copulas becomes matrix multiplication of transition matrices, proving the data processing inequality (DPI) and enabling detection of group and conditional dependence, though values depend on marginal distributions.

ABSTRACT

Following our previous work on copula-based nonsymmetric dependence measures, we introduce similar measures for discrete random variables. The measures cover the range between two extremes: independence and complete dependence, which take minimum value exactly on independence and take maximum value exactly on complete dependence. We find that the * product on copulas in the continuous case reduces to matrix product of transition matrices in the discrete case and we use it to prove the DPI condition. The measures can also be extended to detect dependence between groups of discrete random variables or conditional dependence. Unlike the continuous case, one drawback is that the value of the measures depends on marginal distributions.

Motivation & Objective

• To extend copula-based nonsymmetric dependence measures to discrete random variables.
• To establish a mathematical framework for dependence in discrete settings using transition matrices.
• To prove the data processing inequality (DPI) in the discrete case using matrix operations.
• To enable detection of dependence between groups or conditionally dependent discrete variables.
• To address the limitation that dependence measure values depend on marginal distributions in the discrete case.

Proposed method

• Uses transition matrices to represent conditional distributions of discrete random variables.
• Replaces the copula product operation in the continuous case with matrix multiplication of transition matrices in the discrete case.
• Defines dependence measures that range from independence (minimum value) to complete dependence (maximum value).
• Applies the matrix product framework to prove the data processing inequality (DPI) for discrete dependence measures.
• Extends the framework to detect dependence between groups of discrete variables and conditional dependence structures.
• Demonstrates that the value of the dependence measure is sensitive to marginal distributions, a key distinction from the continuous case.

Experimental results

Research questions

• RQ1 How can nonsymmetric dependence measures be adapted from continuous to discrete random variables?
• RQ2 What algebraic operation replaces the copula product in the discrete case, and how does it preserve key properties like DPI?
• RQ3 Can the proposed discrete dependence measures detect dependence between groups of discrete variables?
• RQ4 How does the dependence measure behave under conditioning, and can it be used to detect conditional dependence?
• RQ5 Why does the value of the dependence measure depend on marginal distributions in the discrete case, and what are the implications?

Key findings

• The copula product in the continuous case is replaced by matrix multiplication of transition matrices in the discrete case.
• The proposed dependence measures achieve their minimum value exactly at independence and maximum value exactly at complete dependence.
• The data processing inequality (DPI) is proven to hold in the discrete setting using the matrix product framework.
• The framework allows for the detection of dependence between groups of discrete random variables.
• The dependence measure's value is sensitive to marginal distributions, a notable difference from the continuous case.
• The method enables the detection of conditional dependence by extending the transition matrix approach to conditional structures.