Change-Point Estimation in Long Memory Nonparametric Models with Applications

Abstract

We consider the estimation of a change point or discontinuity in a regression function for random design model with long memory errors. We provide several change-point estimators and investigate the consistency of the estimators. Using the fractional ARIMA process as an example of long memory process, we report a small Monte Carlo experiment to compare the performance of the estimators in finite samples. We finish by applying the method to a climatological data example.

Keywords:

• Change-point
• Kernel estimation
• Long memory
• Nearest-neighbor estimation
• Nonparametric regression
• Random design

Mathematics Subject Classification:

• 62G08
• 62G05
• 62G20

Acknowledgments

The author sincerely wishes to thank the referee for his/her queries and many insightful remarks and suggestions which have led to improving the presentation of the results.

This article is supported by NSFC Grants of Lihong Wang (No. 10501020) and Jinde Wang (No. 10671089) at Nanjing University and by SRF for ROCS, SEM Grant.

Notes

The biases and MSEs of  = , i = 1, 2, 3, 4, with the standard errors in the parentheses, for Case 1, fixed design, X _t  = t/n, e _t ∼ fractional ARIMA (0, 0.35, 0), k* = 3n/5, n = 125,250, and 500, respectively using 1000 replications.

The biases and MSEs of  = , i = 1,2, 3,4, with the standard errors in the parentheses, for Case 2, weakly dependent random design, X _t ∼ ARMA(1, 1), e _t ∼ fractional ARIMA (0, 0.35,0), k* = 3n/5, n = 125, 250, and 500, respectively using 1000 replications.

The biases and MSEs of  = , i = 1, 2, 3, 4, with the standard errors in the parentheses, for Case 3, strongly dependent random design, X _t ∼ fractional ARIMA (0, d, 0), for d = 0.05(0.1)0.45, respectively, e _t ∼ fractional ARIMA (0, 0.35, 0), k* = 300, n = 500 using 1000 replications.

Four change-point estimators for the global temperature data.