On detecting changes in the jumps of arbitrary size of a time-continuous stochastic process. (English) Zbl 1466.60070

Summary: 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.


60F17 Functional limit theorems; invariance principles
60G51 Processes with independent increments; Lévy processes
62G10 Nonparametric hypothesis testing
62M99 Inference from stochastic processes
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[1] 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. · Zbl 1173.62060
[2] Aït-Sahalia, Y. and Jacod, J. (2009b). Testing for jumps in a discretely observed process., The Annals of Statistics, 37(1):184-222. · Zbl 1155.62057
[3] Aït-Sahalia, Y. and Jacod, J. (2010). Is Brownian motion necessary to model high-frequency data?, The Annals of Statistics, 38:3093-3128. · Zbl 1327.62118
[4] Aït-Sahalia, Y. and Jacod, J. (2014)., High-Frequency Financial Econometrics. Princeton University Press. · Zbl 1298.91018
[5] 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. · Zbl 1178.91217
[6] Aue, A. and Horváth, L. (2013). Structural breaks in time series., Journal of Time Series Analysis, 34(1):1-16. · Zbl 1274.62553
[7] 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. · Zbl 1065.62149
[8] Bissell, A. F. (1984). The performance of control charts and cusums under linear trend., Applied Statistics, 33:145-151. · Zbl 0561.62093
[9] 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. · Zbl 1459.62067
[10] Bücher, A. and Kojadinovic, I. (2016). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing., Bernoulli, 22(2):927-968. · Zbl 1388.62123
[11] 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. · Zbl 0996.60029
[12] Cont, R. and Tankov, P. (2004)., Financial Modelling with Jump Processes. Chapman and Hall/CRC. ISBN: 1-58488-413-4. · Zbl 1052.91043
[13] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing., Mathematische Annalen, 300:463-520. · Zbl 0865.90014
[14] 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. · Zbl 1136.62038
[15] Gan, F. F. (1991). Ewma control chart under linear drift., Journal of Statistical Computation and Simulation, 38:181-200. · Zbl 0800.62654
[16] 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. · JFM 37.0444.01
[17] 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. · Zbl 1364.60042
[18] 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. · Zbl 1401.60053
[19] Hušková, M. (1999). Gradual changes versus abrupt changes., Journal of Statistical Planning and Inference, 76:109-125. · Zbl 1054.62520
[20] Hušková, M. and Steinebach, J. (2002). Asymptotic tests for gradual changes., Statistics & Decisions, 20:137-151. · Zbl 0997.62017
[21] Inoue, A. (2001). Testing for distributional change in time series., Econometric Theory, 17(1):156-187. · Zbl 0976.62088
[22] Jacod, J. and Protter, P. (2012)., Discretization of Processes. Springer. ISBN: 978-3-642-24126-0. · Zbl 1259.60004
[23] 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. · Zbl 1275.62061
[24] Kim, J. and Pollard, D. (1990). Cube root asymptotics., The Annals of Statistics, 18(1):191-219. · Zbl 0703.62063
[25] Kosorok, M. (2008)., Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. Springer. ISBN: 978-0-387-74977-8. · Zbl 1180.62137
[26] Madan, D. B., Carr, P. P., and Chang, E. C. (1998). The variance gamma process and option pricing., European Finance Review, 2:79-105. · Zbl 0937.91052
[27] 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. · Zbl 1294.62106
[28] Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps., Scandinavian Journal of Statistics, 36:270-296. · Zbl 1198.62079
[29] Nickl, R. and Reiß, M. (2012). A Donsker theorem for Lévy measures., Journal of Functional Analysis, 263:3306-3332. · Zbl 1310.60056
[30] 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.
[31] Page, E. (1954). Continuous inspection schemes., Biometrika, 41(1-2):100-115. · Zbl 0056.38002
[32] Page, E. (1955). A test for a change in a parameter occurring at an unknown point., Biometrika, 42:523-527. · Zbl 0067.11602
[33] 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.
[34] 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.
[35] Scheidemann, V. (2005)., Introduction to Complex Analysis in Several Variables. Birkhäuser. ISBN: 3-7643-7490-X. · Zbl 1085.32001
[36] 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. · Zbl 1163.62311
[37] 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.
[38] Van der Vaart, A. and Wellner, J. (1996)., Weak Convergence and Empirical Processes. Springer. ISBN: 0-387-94640-3. · Zbl 0862.60002
[39] van Kampen, N. G. (2007)., Stochastic Processes in Physics and Chemistry. Elsevier, 3 edition. ISBN-10: 0-444-52965-9. · Zbl 0974.60020
[40] Vogt, M. and Dette, H. (2015). Detecting gradual changes in locally stationary processes., The Annals of Statistics, 43(2):713-740. · Zbl 1312.62045
[41] Vostrikova, L. (1981). Detecting disorder in multidimensional random processes., Soviet Mathematics Doklady, 24:55-59. · Zbl 0487.62072
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