Abstract
In this paper, we investigate the CUSUM statistic of change point under the negatively associated (NA) sequences. By establishing the consistency estimators for mean and covariance functions respectively, the limit distribution of the CUSUM statistic is proved to be a standard Brownian bridge, which extends the results obtained under the case of an independent normal sample and the moving average processes. Finally, the finite sample properties of the CUSUM statistic are given to show the efficiency of the method by simulation studies and an application on a real data analysis.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J Bai. Least squares estimation of a shift in linear processes, J Time Series Anal, 1994, 15(5): 453–472.
J Bai. Common breaks in means and variance for pannel data, J Econometrics, 2010, 157(1): 78–92.
P Billingsley. Convergence of probability measures, Wiley, New York 1999, 2nd.
A Bulinski, A Shaskin. Limit theorems for associated random fields and related systems, World Scientific, Singapore, 2007.
J Chen, A Gupta. Parametric statistical change point analysis with applications to genetics medicine and finance, 2nd edn Birkhäuser, Boston, 2012.
F Christian, L Horváth, J Zakoian. Variance targeting estimation of multivariate GARCH models, J Financ Economet, 2016, 14(2): 353–382.
M Csörgő, L Horváth. Limit theorems in change-point analysis, Wiley, Chichester, 1997.
L Horváth, M Hušková. Change-point detection in panel data, J Time Series Anal, 2012, 33(4): 631–648.
L Horváth, M Hušková, G Rice, J Wang. Asymptotic properties of the CUSUM estimator for the time of change in linear panel data models, Economet Theory, 2017, 33(2): 366–412.
L Horváth, G Rice. Extensions of some classical methods in change point analysis, TEST, 2014, 23(2): 219–255.
D Hsu. Detecting shifts of parameter in gamma sequences with applications to stock price and air traffic flow analysis, J Amer Statist Assoc, 1979, 74(365): 31–40.
C Inclaán, G Tiao. Use of cumulative sums of squares for retrospective detection of changes of variance, J Amer Statist Assoc, 1994, 89(427): 913–923.
K Joag-Dev, F Proschan. Negative association of random variables with applications, Ann Statist 1983, 11(1): 286–295.
P Kokoszka, R Leipus. Change-point in the mean of dependent observations, Statist Probab Lett, 1998, 40(4): 385–393.
S Lee, J Ha, O Na. The cusum test for parameter change in time series models, Scand J Statist, 2003, 30(4): 781–796.
S Lee, S Park. The cusum of squares test for scale changes in infinite order moving average processes, Scand J Statist, 2001, 28(4): 625–644.
O Na, Y Lee, S Lee. Monitoring parameter change in time series models, Stat Methods Appl, 2011, 20(2): 171–199.
H Oh, S Lee. On score vector-and residual-based CUSUM tests in ARMA-GARCH models, Stat Methods Appl, 2018, 27(3): 385–406.
P Oliveira. Asymptotics for associated random variables, Springer, Berlin, 2012.
B Prakasa Rao. Associated sequences, demimartingales and nonparametric inference, Birkhäuser, Springer, Basel, 2012.
G Roussas. Asymptotic normality of random fields of positively or negatively associated processes, J Multiv Anal, 1994, 50(1): 152–173.
Q Shao. A comparison theorem on moment inequalities between negatively associated and independent random variables, J Theoret Probab, 2000, 13(2): 343–356.
X Shi, Y Wu, B Miao. Strong convergence rate of estimators of change point and its application, Comput Statist Data Anal, 2009, 53(4): 990–998.
X Shi, Y Wu, C Rao. Consistent and powerful graph-based change-point test for high-dimensional data, Proc Natl Acad Sci USA, 2017, 114(15): 3873–3878.
X Shi, Y Wu, C Rao. Consistent and powerful non-Euclidean graph-based change-point test with applications to segmenting random interfered video data, Proc Natl Acad Sci USA, 2018, 115(23): 5914–5919.
Acknowledgement
The authors are deeply grateful to editor and anonymous referees for their careful reading and insightful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the NNSF of China (11701004, 11801003), NSSF of China (14ATJ005), NSF of Anhui Province (1808085QA03, 1808085QA17, 1808085QF212, 2008085MA14) and Provincial Natural Science Research Project of Anhui Colleges (KJ2019A0006, KJ2019A0021).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articles Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ling, J., Li, Xq., Yang, Wz. et al. The CUSUM statistic of change point under NA sequences. Appl. Math. J. Chin. Univ. 36, 512–520 (2021). https://doi.org/10.1007/s11766-021-4015-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-021-4015-z