A Normal Test for Independence via Generalized Mutual Information

TL;DR

This paper introduces a new normality-based test for independence using generalized mutual information, effective for large or sparse contingency tables, with proven asymptotic properties and faster convergence than traditional methods.

Contribution

It proposes a novel independence test based on generalized mutual information that is asymptotically normal and consistent, suitable for large or sparse contingency data.

Findings

Test statistic is asymptotically normal under independence.

The test is consistent and detects any dependence.

Converges faster than Pearson's chi-squared test in large or sparse tables.

Abstract

Testing hypothesis of independence between two random elements on a joint alphabet is a fundamental exercise in statistics. Pearson's chi-squared test is an effective test for such a situation when the contingency table is relatively small. General statistical tools are lacking when the contingency data tables are large or sparse. A test based on generalized mutual information is derived and proposed in this article. The new test has two desired theoretical properties. First, the test statistic is asymptotically normal under the hypothesis of independence; consequently it does not require the knowledge of the row and column sizes of the contingency table. Second, the test is consistent and therefore it would detect any form of dependence structure in the general alternative space given a sufficiently large sample. In addition, simulation studies show that the proposed test converges faster than Pearson's chi-squared test when the contingency table is large or sparse.