Electronic Journal of Statistics

Improved inference in generalized mean-reverting processes with multiple change-points

Sévérien Nkurunziza and Kang Fu

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In this paper, we consider inference problem about the drift parameter vector in generalized mean reverting processes with multiple and unknown change-points. In particular, we study the case where the parameter may satisfy uncertain restriction. As compared to the results in literature, we generalize some findings in five ways. First, we consider the model which incorporates the uncertain prior knowledge. Second, we derive the unrestricted estimator (UE) and the restricted estimator (RE) and we study their asymptotic properties. Third, we derive a test for testing the hypothesized restriction and we derive its asymptotic local power. We also prove that the proposed test is consistent. Fourth, we construct a class of shrinkage type estimators (SEs) which encloses the UE, the RE and classical SEs. Fifth, we derive the relative risk dominance of the proposed estimators. More precisely, we prove that the SEs dominate the UE. Finally, we present some simulation results which corroborate the established theoretical findings.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 1400-1442.

Received: May 2018
First available in Project Euclid: 16 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F30: Inference under constraints
Secondary: 62M02: Markov processes: hypothesis testing

ADR change-point drift-parameter mean-reverting process Ornstein-Uhleneck process testing SDE shrinkage estimators unrestricted estimator

Creative Commons Attribution 4.0 International License.


Nkurunziza, Sévérien; Fu, Kang. Improved inference in generalized mean-reverting processes with multiple change-points. Electron. J. Statist. 13 (2019), no. 1, 1400--1442. doi:10.1214/19-EJS1548. https://projecteuclid.org/euclid.ejs/1555380048

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  • [1] Akaike, H. (1973)., Information theory and an extension of the maximum likelihood principle, in B. Petrov, F. Csaki, 2nd International Symposium on Information Theory, Tsahkadsor, Armenia, USSR, September 2-8, 1971, Budapest: Akadémiai Kiadó, 267–281.
  • [2] Auger, I. & Lawrence, C. (1989). Algorithms for the optimal identification of segment neighborhoods., Bulletin of Mathematical Biology, 51(1), 39–54.
  • [3] Bai, J. & Perron, P. (1998). Estimating and testing linear models with multiple structural changes., Econometrica, 66(1), 47–78.
  • [4] Chen, F., Mamon, R., & Davison, M. (2017). Inference for a mean-reverting stochastic process with multiple change points., Electronic Journal of Statisitics, 11, 2199–2257.
  • [5] Chen, F. & Nkurunziza, S. (2015). Optimal method in multiple regression with structural changes., Bernoulli, 21 (4), 2217–2241.
  • [6] Dehling, H., Franke, B., & Kott, T. (2010). Drift estimation for a periodic mean reversion process., Statistical Inference for Stochastic Process (SISP), 13, 175–192.
  • [7] Dehling, H., Franke, B., Kott, T., & Kulperger, R. (2014). Change point testing for the drift parameters of a periodic mean reversion process., SISP, 17(1), 1–18.
  • [8] Jackson, B., Scargle, J., Barnes, D., Arabhi, S., Alt, A., Gioumousis, P., Gwin, E., Sangtrakulcharoen, P., Tan, L. & Tsai, T. (2002). An algorithm for optimal partitioning of data on an interval., IEEE Signal Processing Letters, 12, 105–108.
  • [9] Killick, R., Fearnhead, P.& Eckley, I. (2012). Optimal detection of change points with a linear computational cost., Journal of the American Statistical Association, 107(500), 1590–1598.
  • [10] Kutoyants, Y. A. (2004)., Statistical inference for ergodic diffusion processes. Springer-Verlag, London.
  • [11] Lansky, P., & Sacerdote, L. (2001). The Ornstein-Uhlenbeck neuronal model with signal-dependent noise., Physics Letters A, 285(3-4), 132–140.
  • [12] Liptser, R. S., & Shiryaev, A. N. (2001)., Statistics of Random Processes I: I. General Theory (Vol. 1). Springer-Verlag, Berlin Heidelberg.
  • [13] Mathai, A. M. & Provost, S. B. (1992)., Quadratic forms in random variables: theory and applications. CRC Press, New York.
  • [14] Nkurunziza, S. (2012). The risk of pretest and shrinkage estimators., Statistics, 46(3), 305–312.
  • [15] Nkurunziza, S. & Zhang P. (2018). Estimation and testing in generalized mean-reverting processes with change-point., SISP, 21, 191–215.
  • [16] Perron, P. & Qu, Z. (2006). Estimating restricted structural change models., Journal of Econometrics, 134, 373–399.
  • [17] Rohlfs, R., Harrigan, P., & Nielsen, R. (2010). Modeling gene expression evolution with an extended Ornstein-Uhlenbeck process accounting for within-species variation., Scandinavian Journal of Statistics, 37(2), 200–220.
  • [18] Saleh, AME. (2006)., Theory of preliminary test and Stein-type estimation with applications, 517. Wiley, New Jersey.
  • [19] Schwarz, G. (1978). Estimating the dimension of a model., The Annals of Statistics, 6(2), 461–464.
  • [20] Sen, PK. & Saleh AKME. (1987). On preliminary test and shrinkage M-estimation in linear models., The Annals of Statistics, 15(4), 1580–1592.