The Joint Asymptotic Distribution of Entropy and Complexity

TL;DR

This paper derives the asymptotic distribution of entropy and complexity measures in time-series analysis, providing theoretical results, simulation methods, and a test for serial dependence to better understand long-term behavior.

Contribution

It introduces new asymptotic distribution results for entropy and complexity pairs, including distinctions between uniform and non-uniform distributions, and develops a serial dependence test.

Findings

Derived asymptotic distributions for entropy and complexity pairs.

Provided simulation-based approximation methods for covariance matrices.

Evaluated finite-sample performance of the serial dependence test.

Abstract

We derive the asymptotic distribution of ordinal-pattern frequencies under weak dependence conditions and investigate the long-run covariance matrix not only analytically for moving-average, Gaussian, and the novel generalized coin-tossing processes, but also approximately by a simulation-based approach. Then, we deduce the asymptotic distribution of the entropy-complexity pair, which emerged as a popular tool for summarizing the time-series dynamics. Here, we make the necessary distinction between a uniform and a non-uniform ordinal pattern distribution and, thus, obtain two different limit theorems. On this basis, we consider a test for serial dependence and check its finite-sample performance. Moreover, we use our asymptotic results to approximate the estimation uncertainty of entropy-complexity pairs.