Electronic Journal of Statistics

On detecting changes in the jumps of arbitrary size of a time-continuous stochastic process

Michael Hoffmann and Holger Dette

Full-text: Open access

Abstract

This paper introduces test and estimation procedures for abrupt and gradual changes in the entire jump behaviour of a discretely observed Itō semimartingale. In contrast to existing work we analyse jumps of arbitrary size which are not restricted to a minimum height. Our methods are based on weak convergence of a truncated sequential empirical distribution function of the jump characteristic of the underlying Itō semimartingale. Critical values for the new tests are obtained by a multiplier bootstrap approach and we investigate the performance of the tests also under local alternatives. An extensive simulation study shows the finite-sample properties of the new procedures.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 3654-3709.

Dates
Received: February 2018
First available in Project Euclid: 1 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1569895285

Digital Object Identifier
doi:10.1214/19-EJS1610

Mathematical Reviews number (MathSciNet)
MR4013748

Zentralblatt MATH identifier
07113728

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; Lévy processes 62G10: Hypothesis testing 62M99: None of the above, but in this section

Keywords
Lévy measure jump compensator transition kernel empirical processes weak convergence multiplier bootstrap change points gradual changes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hoffmann, Michael; Dette, Holger. On detecting changes in the jumps of arbitrary size of a time-continuous stochastic process. Electron. J. Statist. 13 (2019), no. 2, 3654--3709. doi:10.1214/19-EJS1610. https://projecteuclid.org/euclid.ejs/1569895285


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References

  • Aït-Sahalia, Y. and Jacod, J. (2009a). Estimating the degree of activity of jumps in high frequency data., The Annals of Statistics, 37(5):2202–2244.
  • Aït-Sahalia, Y. and Jacod, J. (2009b). Testing for jumps in a discretely observed process., The Annals of Statistics, 37(1):184–222.
  • Aït-Sahalia, Y. and Jacod, J. (2010). Is Brownian motion necessary to model high-frequency data?, The Annals of Statistics, 38:3093–3128.
  • Aït-Sahalia, Y. and Jacod, J. (2014)., High-Frequency Financial Econometrics. Princeton University Press.
  • Andreou, E. and Ghysels, E. (2009). Structural breaks in financial time series. In Mikosch, T., Kreiß, J.-P., Davis, R. A., and Andersen, T. G., editors, Handbook of Financial Time Series, pages 839–870. Springer Berlin Heidelberg.
  • Aue, A. and Horváth, L. (2013). Structural breaks in time series., Journal of Time Series Analysis, 34(1):1–16.
  • Aue, A. and Steinebach, J. (2002). A note on estimating the change-point of a gradually changing stochastic process., Statistics & Probability Letters, 56:177–191.
  • Bissell, A. F. (1984). The performance of control charts and cusums under linear trend., Applied Statistics, 33:145–151.
  • Bücher, A., Hoffmann, M., Vetter, M., and Dette, H. (2017). Nonparametric tests for detecting breaks in the jump behaviour of a time-continuous process., Bernoulli, 23(2):1335–1364. DOI: 10.3150/15-BEJ780.
  • Bücher, A. and Kojadinovic, I. (2016). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing., Bernoulli, 22(2):927–968.
  • Chen, L. and Shao, Q.-M. (2001). A non-uniform Berry-Esseen bound via Stein’s method., Probability Theory and Related Fields, 120:236–254.
  • Cont, R. and Tankov, P. (2004)., Financial Modelling with Jump Processes. Chapman and Hall/CRC. ISBN: 1-58488-413-4.
  • Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing., Mathematische Annalen, 300:463–520.
  • Gaenssler, P., Molnár, P., and Rost, D. (2007). On continuity and strict increase of the CDF for the sup-functional of a Gaussian process with applications to statistics., Results in Mathematics, 51:51–60.
  • Gan, F. F. (1991). Ewma control chart under linear drift., Journal of Statistical Computation and Simulation, 38:181–200.
  • Hartogs, F. (1906). Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten., Mathematische Annalen, 62:1–88.
  • Hoffmann, M. and Vetter, M. (2017). Weak convergence of the empirical truncated distribution function of the Lévy measure of an Itō semimartingale., Stochastic Processes and their Applications, 127(5):1517–1543.
  • Hoffmann, M., Vetter, M., and Dette, H. (2017). Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes., to appear: Stochastic Processes and their Applications. arXiv:1704.04040.
  • Hoffmann, M., and Dette, H. (2019). Supplement to “On detecting changes in the jumps of arbitrary size of a time-continuous stochastic process.” DOI:, 10.1214/19-EJS1610SUPP.
  • Hušková, M. (1999). Gradual changes versus abrupt changes., Journal of Statistical Planning and Inference, 76:109–125.
  • Hušková, M. and Steinebach, J. (2002). Asymptotic tests for gradual changes., Statistics & Decisions, 20:137–151.
  • Inoue, A. (2001). Testing for distributional change in time series., Econometric Theory, 17(1):156–187.
  • Jacod, J. and Protter, P. (2012)., Discretization of Processes. Springer. ISBN: 978-3-642-24126-0.
  • Jandhyala, V., Fotopoulos, S., MacNeill, I., and Liu, P. (2013). Inference for single and multiple change-points in time series., Journal of Time Series Analysis, 34(4):423–446. doi: 10.1111/jtsa.12035.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics., The Annals of Statistics, 18(1):191–219.
  • Kosorok, M. (2008)., Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. Springer. ISBN: 978-0-387-74977-8.
  • Madan, D. B., Carr, P. P., and Chang, E. C. (1998). The variance gamma process and option pricing., European Finance Review, 2:79–105.
  • Mallik, A., Banerjee, M., and Sen, B. (2013). Asymptotics for $p$-value based threshold estimation in regression settings., Electronic Journal of Statistics, 7:2477–2515.
  • Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps., Scandinavian Journal of Statistics, 36:270–296.
  • Nickl, R. and Reiß, M. (2012). A Donsker theorem for Lévy measures., Journal of Functional Analysis, 263:3306–3332.
  • Nickl, R., Reiß, M., Söhl, J., and Trabs, M. (2016). High-frequency Donsker theorems for Lévy measures., Probability Theory and Related Fields, 164:61–108.
  • Page, E. (1954). Continuous inspection schemes., Biometrika, 41(1-2):100–115.
  • Page, E. (1955). A test for a change in a parameter occurring at an unknown point., Biometrika, 42:523–527.
  • Perron, P. (2006). Dealing with structural breaks. In Patterson, K. and Mills, T., editors, Palgrave Handbook of Econometrics, volume 1, pages 278–352. Palgrave Macmillan.
  • Reeves, J., Chen, J., Wang, X., Lund, R., and Lu, Q. (2007). A review and comparison of changepoint detection techniques for climate data., Journal of Applied Meteorology and Climatology, 46:900–915.
  • Scheidemann, V. (2005)., Introduction to Complex Analysis in Several Variables. Birkhäuser. ISBN: 3-7643-7490-X.
  • Siegmund, D. O. and Zhang, H. (1994). Confidence regions in broken line regression. In Carlstein, E., Müller, H.-G., and Siegmund, D., editors, Change-point problems, volume 23, pages 292–316. Institute of Mathematical Statistics.
  • Stoumbos, Z., Reynolds, J. M., Ryan, T., and Woodall, W. (2000). The state of statistical process control as we proceed into the 21st century., Journal of the American Statistical Association, 95:992–998.
  • Van der Vaart, A. and Wellner, J. (1996)., Weak Convergence and Empirical Processes. Springer. ISBN: 0-387-94640-3.
  • van Kampen, N. G. (2007)., Stochastic Processes in Physics and Chemistry. Elsevier, 3 edition. ISBN-10: 0-444-52965-9.
  • Vogt, M. and Dette, H. (2015). Detecting gradual changes in locally stationary processes., The Annals of Statistics, 43(2):713–740.
  • Vostrikova, L. (1981). Detecting disorder in multidimensional random processes., Soviet Mathematics Doklady, 24:55–59.

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