Changepoint Analysis by Modified Empirical Likelihood Method in Two-phase Linear Regression Models

Hualing Zhao, Hanfeng Chen, Wei Ning

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio , USA.
Department of Statistics, School of Science, Wuhan University of Technology, Wuhan, Hubei , P.R. of China.

DOI: 10.4236/ojapps.2013.31B1001                

Abstract

A changepoint in statistical applications refers to an observational time point at which the structure pattern changes during a somewhat long-term experimentation process. In many cases, the change point time and cause are documented and it is reasonably straightforward to statistically adjust (homogenize) the series for the effects of the changepoint. Sadly many changepoint times are undocumented and the changepoint times themselves are the main purpose of study. In this article, the changepoint analysis in two-phrase linear regression models is developed and discussed. Following Liu and Qian (2010)'s idea in the segmented linear regression models, the modified empirical likelihood ratio statistic is proposed to test if there exists a changepoint during the long-term experiment and observation. The modified empirical likelihood ratio statistic is computation-friendly and its ρ-value can be easily approximated based on the large sample properties. The procedure is applied to the Old Faithful geyser eruption data in October 1980.

Keywords

Changepoint; Extreme-Value Distribution; Modified Empirical Likelihood Ratio; Segmented Linear Regression

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. Csorgo and L. Horvnth, “Limit Theorem in Change-Point Analysi,” Wiley Series in Probability and Statistics, John Wiley & Sons: New York, 1997.
[2]	I. Fiteni, “ τ-estimators of Regression Models with Structural Change of Unknown Location,” Journal of Econometrics , Vol. 119, No. 1, 2004, pp. 19-44. doi:10.1016/S0304-4076(03)00153-2
[3]	Z. Liu and L. Qian, “Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood,” Communications in Statistics--Simulation and Computation, Vol. 89, 2010, pp. 85-100.
[4]	L. H. Koul and L. F. Qian, “Asymptotics of Maximum Likelihood Estimator in a Two-phaselinear Regres-sion Model,”Journal of Statistical Planning and Inference, Vol. 108, No. 1-2, 2002, pp. 99-119. doi:10.1016/S0378-3758(02)00273-2
[5]	A. B. Owen, “Empirical Likelihood for Linear Models,” Annals of Statistics, Vol.19, No.19, 1991, pp. 1725-1747. doi:10.1214/aos/1176348368
[6]	A. B. Owen, “Empirical Likelihood,” New York: Chapman & Hall, 2001. doi:10.1201/9781420036152
[7]	R. Pastor and E. Guallar, “Use of Two-segmented Logistic Regression to Estimate Changepoints in Epidemiologic Studies,” American Journal of Epidemiology, Vol. 148, No. 7, 1998, pp. 631-642. doi:10.1093/aje/148.7.631
[8]	P. Perron and T. J. Vogelsang, “Testing for a Unit Root in a Time Series with a Changing Mean: Corrections and Extensions,” J. Business Econom. Statist.,Vol. 10,1992, pp. 467-470.
[9]	W. W. Piegorsch and A. J. Bailer,“Statistics for Environmental Biology and Toxicology,” London: Chapman and Hall, 1997.
[10]	A. M. F. Smith and D. G. Cook,“Straight Lines with a Change Point: A Bayesian Analysis of Some Renal Transplant Data,” Applied Statistics, Vol. 29, No. 2, 1980，pp. 180-189. doi:10.2307/2986304
[11]	S. Weisberg,“Applied Linear Regeression,” 3th Edition, John Wiley& Sons, Inc., Hoboken, New Jersey, 2005.