##
**Dirichlet process hidden Markov multiple change-point model.**
*(English)*
Zbl 1335.62052

Summary: This paper proposes a new Bayesian multiple change-point model which is based on the hidden Markov approach. The Dirichlet process hidden Markov model does not require the specification of the number of change-points a priori. Hence our model is robust to model specification in contrast to the fully parametric Bayesian model. We propose a general Markov chain Monte Carlo algorithm which only needs to sample the states around change-points. Simulations for a normal mean-shift model with known and unknown variance demonstrate advantages of our approach. Two applications, namely the coal-mining disaster data and the real United States Gross Domestic Product growth, are provided. We detect a single change-point for both the disaster data and US GDP growth. All the change-point locations and posterior inferences of the two applications are in line with existing methods.

### MSC:

62F15 | Bayesian inference |

62G10 | Nonparametric hypothesis testing |

62M02 | Markov processes: hypothesis testing |

62P20 | Applications of statistics to economics |

### Keywords:

change-point; Dirichlet process; hidden Markov model; Markov chain Monte Carlo; nonparametric Bayesian
PDFBibTeX
XMLCite

\textit{S. I. M. Ko} et al., Bayesian Anal. 10, No. 2, 275--296 (2015; Zbl 1335.62052)

### References:

[1] | Beal, M. J., Ghahramani, Z., and Rasmussen, C. E. (2002). “The Infinite Hidden Markov Model.” In Dietterich, T. G., Becker, S., and Ghahramani, Z. (eds.), Advances in Neural Information Processing Systems , 577-584. MIT Press. |

[2] | Blackwell, D. and MacQueen, J. B. (1973). “Ferguson Distributions Via Polya Urn Schemes.” The Annals of Statistics , 1(2): 353-355. · Zbl 0276.62010 |

[3] | Carlin, P. B., Gelfand, A. E., and Smith, A. F. M. (1992). “Hierarchical Bayesian Analysis of Changepoint Problems.” Journal of the Royal Statistical Society. Series C (Applied Statistics) , 41(2): 389-405. · Zbl 0825.62408 |

[4] | Chernoff, H. and Zacks, S. (1964). “Estimating the Current Mean of a Normal Distribution which is Subjected to Changes in Time.” The Annals of Mathematical Statistics , 35(3): pp. 999-1018. · Zbl 0218.62033 |

[5] | Chib, S. (1998). “Estimation and comparison of multiple change-point models.” Journal of Econometrics , 86(2): 221 - 241. · Zbl 1045.62510 |

[6] | Chong, T. T.-L. (2001). “Structural Change in AR(1) Models.” Econometric Theory , 17(1): 87-155. · Zbl 1009.62073 |

[7] | Connor, R. J. and Mosimann, J. E. (1969). “Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution.” Journal of the American Statistical Association , 64(325): pp. 194-206. · Zbl 0179.24101 |

[8] | Ferguson, T. S. (1973). “A Bayesian analysis of some nonparametric problems.” Annals of Statistics , 1: 209-230. · Zbl 0255.62037 |

[9] | Geweke, J. and Yu, J. (2011). “Inference and prediction in a multiple-structural-break model.” Journal of Econometrics , 163(2): 172-185. · Zbl 1441.62701 |

[10] | Giordani, P. and Kohn, R. (2008). “Efficient Bayesian Inference for Multiple Change-Point and Mixture Innovation Models.” Journal of Business and Economic Statistics , 26(1): 66-77. |

[11] | Jarrett, R. G. (1979). “A Note on the Intervals Between Coal-Mining Disasters.” Biometrika , 66(1): 191-193. |

[12] | Koop, G. and Potter, S. M. (2007). “Estimation and Forecasting in Models with Multiple Breaks.” The Review of Economic Studies , 74(3): pp. 763-789. · Zbl 1171.62342 |

[13] | Kozumi, H. and Hasegawa, H. (2000). “A Bayesian analysis of structural changes with an application to the displacement effect.” The Manchester School , 68(4): 476-490. |

[14] | Maheu, J. M. and Gordon, S. (2008). “Learning, forecasting and structural breaks.” Journal of Applied Econometrics , 23(5): 553-583. |

[15] | Neal, R. M. (1992). “The Infinite Hidden Markov Model.” In Smith, C. R., Erickson, G. J., and Neudorfer, P. O. (eds.), Proceedings of the Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis , 197-211. Kluwer Academic Publishers. |

[16] | - (2000). “Markov Sampling Methods for Dirichlet Process Mixture Models.” Journal of Computational and Graphical Statistics , 9(2): 249-265. |

[17] | Pesaran, M. H., Pettenuzzo, D., and Timmermann, A. (2006). “Forecasting Time Series Subject to Multiple Structural Breaks.” Review of Economic Studies , 73(4): 1057-1084. · Zbl 1201.91156 |

[18] | Sethuraman, J. (1994). “A Constructive Definition of Dirichlet Priors.” Statistica Sinica , 4(2): 639-650. · Zbl 0823.62007 |

[19] | Smith, A. F. M. (1975). “A Bayesian approach to inference about a change-point in a sequence of random variables.” Biometrika , 62: 407-416. · Zbl 0321.62041 |

[20] | Stephens, D. A. (1994). “Bayesian Retrospective Multiple-Changepoint Identification.” Journal of the Royal Statistical Society. Series C (Applied Statistics) , 43(1): 159-178. · Zbl 0825.62412 |

[21] | Wang, J. and Zivot, E. (2000). “A Bayesian Time Series Model of Multiple Structural Changes in Level, Trend, and Variance.” Journal of Business & Economic Statistics , 18(3): 374-386. |

[22] | Wong, T. (1998). “Generalized Dirichlet distribution in Bayesian analysis.” Applied Mathematics and Computation , 97(2-3): 165-181. · Zbl 0945.62036 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.