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Ordinal patterns in long‐range dependent time series

Author

Listed:
  • Annika Betken
  • Jannis Buchsteiner
  • Herold Dehling
  • Ines Münker
  • Alexander Schnurr
  • Jeannette H.C. Woerner

Abstract

We analyze the ordinal structure of long‐range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting, we encounter a Rosenblatt distribution in the limit. We prove more general limit theorems for functions with Hermite rank 1 and 2. We derive the limit distribution for an estimation of the Hurst parameter H if it is higher than 3/4. Thus, our theorems complement results for lower values of H which can be found in the literature. Finally, we provide some simulations that illustrate our theoretical results.

Suggested Citation

  • Annika Betken & Jannis Buchsteiner & Herold Dehling & Ines Münker & Alexander Schnurr & Jeannette H.C. Woerner, 2021. "Ordinal patterns in long‐range dependent time series," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 969-1000, September.
    • Handle: RePEc:bla:scjsta:v:48:y:2021:i:3:p:969-1000
    DOI: 10.1111/sjos.12478
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    References listed on IDEAS

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    6. Sinn, Mathieu & Keller, Karsten, 2011. "Estimation of ordinal pattern probabilities in Gaussian processes with stationary increments," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1781-1790, April.
    7. Alexander Schnurr, 2014. "An ordinal pattern approach to detect and to model leverage effects and dependence structures between financial time series," Statistical Papers, Springer, vol. 55(4), pages 919-931, November.
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