Conference Agenda

The ordinal structure of long-range dependent time series is analyzed. To this end, so-called ordinal patterns are used, which describe the relative position of consecutive data points. Two estimators are provided for the probabilities of ordinal patterns and we prove limit theorems in different settings, namely for funtions of Hermite Rank 1 and 2. In the second setting, a Rosenblatt distribution in the limit is encountered. In the context of fractional Gaussian noise, the limit distribution is derived for an estimation of the Hurst parameter H if it is higher than 3/4. Thus, the theorems complement results for lower values of H, which can be found in the literature.

The use of ordinal patterns (OPs) for analyzing the dependence structure of univariate and continuously distributed processes has gained popularity in recent years. Here, we go one step further and consider the transcripts being computed from successive OPs in the time series. Transcripts constitute a kind of "difference" between successive OPs and thus naturally relate to two algebraic distances between OPs, the Cayley and Kendall distance. We transform the original time series into a sequence of transcripts or distances, respectively, and derive important stochastic properties thereof. We show that these properties differ substantially between different types of original process. This motivates to develop various statistics based on transcripts and algebraic distances in order to investigate the dependence structure of the original process. In particular, we derive the asymptotic distribution of these statistics under the null hypothesis of serial independence, which is then used to develop nonparametric tests for serial dependence. A simulation study shows that these novel dependence tests have appealing power properties, often outperforming the former OP-based dependence tests. We conclude with a real-world data example, where we illustrate the application and interpretaion of the proposed approaches in practice.