Abstract
In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein–Uhlenbeck process which is defined as the solution of
and which is observed in continuous time. We derive an explicit representation of the generalized likelihood ratio test statistic assuming that the mean reversion function \(L(t)\) is a finite linear combination of known basis functions. In the case of a periodic mean reversion function, we determine the asymptotic distribution of the test statistic under the null hypothesis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beibel M (1997) Sequential change point detection in continuous time when the post-change drift is unknown. Bernoulli 3:457–478
Bradley R (2007) Introduction to strong mixing conditions. Kendrick Press, Heber City
Csörgő M, Horváth L (1997) Limit theorems in change-point analysis. Wiley, Chichester
Davis RA, Huang D, Yao YC (1995) Testing for a change in the parameter values and order of an autoregressive model. Ann Stat 23:282–304
Dehling H, Franke B, Kott T (2010) Drift estimation for a periodic mean reversion process. Stat Inference Stoch Process 13:175–192
Ethier S, Kurtz T (1986) Markov processes: characterization and convergence. Wiley, New York
Höpfner R, Kutoyants Y (2010) Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Stat Inference Stoch Process 13:193–230
Horváth L (1993) The maximum likelihood method for testing changes in the parameters of normal observations. Ann Stat 21:671–680
Kuelbs J, Philipp W (1980) Almost sure invariance principles for partial sums of mixing \( B \)-valued random variables. Ann Probabil 8:1003–1036
Kutoyants Y (2004) Statistical inference for Ergodic diffusion processes. Springer Verlag, London
Lee S, Nishiyama Y, Yoshida N (2006) Test for parameter change in diffusion processes by cusum statistics based on one-step estimators. Ann Inst Stat Math 58:211–222
Lipster RS, Shiryayev AN (1977) Statistics of random processes I. Springer-Verlag, Berlin
Mihalache S (2012) Strong approximations and sequential change analysis for diffusion processes. Stat Probabil Lett 82:464–472
Negri I, Nishiyama Y (2012) Asymptotically distribution free test for parameter change in a diffusion process model. Ann Inst Stat Math 64:911–918
Ornstein LS, Uhlenbeck GE (1930) On the theory of Brownian motion. Phys Rev 36:823–841
Siegmund D (1985) Sequential analysis. Tests and confidence intervals. Springer-Verlag, New York
Siegmund D, Venkatraman ES (1995) Using the generalized likelihood ratio statistic for sequential detection of a change point. Ann Stat 23:255–271
Acknowledgments
This work was partly supported by the Collaborative Research Project SFB 823 (Statistical modelling of nonlinear dynamic processes) of the German Research Foundation DFG. Thomas Kott was supported by the E.ON Ruhrgas AG. The authors wish to thank Martin Wendler for his help with the proof of Propostion 4.2, and two anonymous referees for their careful reading of the manuscript and for their comments that helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dehling, H., Franke, B., Kott, T. et al. Change point testing for the drift parameters of a periodic mean reversion process. Stat Inference Stoch Process 17, 1–18 (2014). https://doi.org/10.1007/s11203-014-9092-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11203-014-9092-7