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Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

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  • Hoffmann, Michael
  • Vetter, Mathias
  • Dette, Holger

Abstract

In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localization of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analysed by deriving new results on the weak convergence of a sequential empirical tail integral process and a corresponding multiplier bootstrap procedure.

Suggested Citation

  • Hoffmann, Michael & Vetter, Mathias & Dette, Holger, 2018. "Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3679-3723.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:11:p:3679-3723
    DOI: 10.1016/j.spa.2017.12.005
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    References listed on IDEAS

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    1. Kosorok, Michael R., 2003. "Bootstraps of sums of independent but not identically distributed stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 299-318, February.
    2. Alexander Aue & Lajos Horváth, 2013. "Structural breaks in time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(1), pages 1-16, January.
    3. Figueroa-López, José E. & Houdré, Christian, 2009. "Small-time expansions for the transition distributions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3862-3889, November.
    4. Aue, Alexander & Steinebach, Josef, 2002. "A note on estimating the change-point of a gradually changing stochastic process," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 177-191, January.
    5. A. F. Bissell, 1984. "The Performance of Control Charts and Cusums Under Linear Trend," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(2), pages 145-151, June.
    6. Hoffmann, Michael & Vetter, Mathias, 2017. "Weak convergence of the empirical truncated distribution function of the Lévy measure of an Itō semimartingale," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1517-1543.
    7. Hušková M. & Steinebach J., 2002. "Asymptotic Tests For Gradual Changes," Statistics & Risk Modeling, De Gruyter, vol. 20(1-4), pages 137-152, April.
    8. Venkata Jandhyala & Stergios Fotopoulos & Ian MacNeill & Pengyu Liu, 2013. "Inference for single and multiple change-points in time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(4), pages 423-446, July.
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    Cited by:

    1. Marie Hušková & Zuzana Prášková & Josef G. Steinebach, 2022. "Estimating a gradual parameter change in an AR(1)-process," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(7), pages 771-808, October.

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