Ordinal Pattern Dependence in the Context of Long-Range Dependence

Abstract

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

Ordinal pattern dependence as a multivariate dependence measure

“…These time series arise for example in the context of hydrology, if discharge data of a river is considered for two different locations. For a real-world data analysis see [10]. Using this approach, (time-shifted) ordinal pattern dependence of 1 is detected, since all patterns coincide if we reshift the second time series.…”

“…Originally, axiomatic systems are disregarded by the notion of ordinal pattern dependence, which is naturally interpreted as the degree of co-monotonic behavior of two time series. Against the background of this approach, limit theorems have been proved in the time series setting (see [16] for the SRD case and [11] for the LRD case).…”

“…We simulate (X i ), i ≥ 1, and (Y i ), i ≥ 1, as sequences of independent, multivariate normal random vectors with values in R 3 and a joint normal distribution with expectation 0 and covariance matrix(10).…”

Time Series Modelling

“…The final article presented in the group of real-valued time series also constitutes a bridge to the next group of articles—namely, to those on discrete-valued time series. Nüßgen and Schnurr [ 9 ] consider a multivariate long-range dependent Gaussian time series, but they analyze its dependence structure based on discrete ordinal patterns derived thereof. The estimators of ordinal pattern dependence are analyzed asymptotically and within a simulation study.…”