Measuring Dispersion and Serial Dependence in Ordinal Time Series Based on the Cumulative Paired ϕ-Entropy

Christian Weiß

doi:10.3390/e24010042

Abstract

The family of cumulative paired -entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen’s , with the weights being related to the actual choice of . Again, the asymptotic distribution of the corresponding sample is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel -entropy-related measures are applied to an ordinal time series on the daily level of air quality.

Highlights

During the last years, ordinal data in general [1] and ordinal time series in particular [2]received a great amount of interest in research and applications
In the recent paper by Weiß [20] on the asymptotics of some well-known dispersion measures for nominal data, it turned out that the corresponding dispersion measures—if these are applied to time series data—are related to specific measures of serial dependence
In view of our previous findings, achieved when discussing the asymptotics plotted in Figures 3 and 4, we do not further consider the choice a = 1/2 < 1 for the entropy generating function (EGF) φa, but we use a = 5/2 > 2 instead

Summary

Introduction

Ordinal data in general [1] and ordinal time series in particular [2]. We take up several recent works on measures of dispersion and serial dependence in ordinal (time series) data. M ( f m is omitted in f as it necessarily equals one) They classify any one-point distribution on S as a scenario of minimal dispersion, i.e., if all probability mass concentrates on one category from S (maximal consensus): 11. Time series data, and to check the finite sample performance of CPE the resulting approximate distribution, see Sections 3 and 5. In the recent paper by Weiß [20] on the asymptotics of some well-known dispersion measures for nominal data (i.e., qualitative data without a natural ordering), it turned out that the corresponding dispersion measures—if these are applied to time series data—are related to specific measures of serial dependence.

The Family of Cumulative Paired φ-Entropies

Asymptotic Distribution of Sample CPEφ

E CPEφ f

Simulation Results

Data Application

Conclusions