Change point testing for the drift parameters of a periodic mean reversion process

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

$$\begin{aligned} dX_t=(L(t)-\alpha X_t) dt + \sigma dB_t \end{aligned}$$

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.

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References

  1. Beibel M (1997) Sequential change point detection in continuous time when the post-change drift is unknown. Bernoulli 3:457–478

    Article  MATH  MathSciNet  Google Scholar 

  2. Bradley R (2007) Introduction to strong mixing conditions. Kendrick Press, Heber City

    Google Scholar 

  3. Csörgő M, Horváth L (1997) Limit theorems in change-point analysis. Wiley, Chichester

    Google Scholar 

  4. 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

    Article  MATH  MathSciNet  Google Scholar 

  5. Dehling H, Franke B, Kott T (2010) Drift estimation for a periodic mean reversion process. Stat Inference Stoch Process 13:175–192

    Article  MATH  MathSciNet  Google Scholar 

  6. Ethier S, Kurtz T (1986) Markov processes: characterization and convergence. Wiley, New York

    Book  MATH  Google Scholar 

  7. Höpfner R, Kutoyants Y (2010) Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Stat Inference Stoch Process 13:193–230

    Article  MATH  MathSciNet  Google Scholar 

  8. Horváth L (1993) The maximum likelihood method for testing changes in the parameters of normal observations. Ann Stat 21:671–680

    Article  MATH  Google Scholar 

  9. Kuelbs J, Philipp W (1980) Almost sure invariance principles for partial sums of mixing \( B \)-valued random variables. Ann Probabil 8:1003–1036

    Article  MATH  MathSciNet  Google Scholar 

  10. Kutoyants Y (2004) Statistical inference for Ergodic diffusion processes. Springer Verlag, London

    Book  MATH  Google Scholar 

  11. 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

    Article  MATH  MathSciNet  Google Scholar 

  12. Lipster RS, Shiryayev AN (1977) Statistics of random processes I. Springer-Verlag, Berlin

    Google Scholar 

  13. Mihalache S (2012) Strong approximations and sequential change analysis for diffusion processes. Stat Probabil Lett 82:464–472

    Article  MATH  MathSciNet  Google Scholar 

  14. Negri I, Nishiyama Y (2012) Asymptotically distribution free test for parameter change in a diffusion process model. Ann Inst Stat Math 64:911–918

    Article  MATH  MathSciNet  Google Scholar 

  15. Ornstein LS, Uhlenbeck GE (1930) On the theory of Brownian motion. Phys Rev 36:823–841

    Article  MATH  Google Scholar 

  16. Siegmund D (1985) Sequential analysis. Tests and confidence intervals. Springer-Verlag, New York

    MATH  Google Scholar 

  17. Siegmund D, Venkatraman ES (1995) Using the generalized likelihood ratio statistic for sequential detection of a change point. Ann Stat 23:255–271

    Article  MATH  MathSciNet  Google Scholar 

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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.

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Correspondence to Herold Dehling.

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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

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Keywords

  • Time-inhomogeneous diffusion process
  • Change point
  • Generalized likelihood ratio test