Strong consistency properties of the variance change point estimator based on strong-mixing samples

Mengmei Xi; Yi Wu; Xuejun Wang; Mengmei Xi; Yi Wu; Xuejun Wang

doi:10.3934/math.20241452

AIMS Mathematics

2024, Volume 9, Issue 11: 30059-30072. doi: 10.3934/math.20241452

Previous Article Next Article

Research article

Strong consistency properties of the variance change point estimator based on strong-mixing samples

1.
School of Big Data and Statistics, Anhui University, Hefei, 230026, China
2.
School of Big Data and Artificial Intelligence, Chizhou University, Chizhou, 247000, China

Received: 25 August 2024 Revised: 27 September 2024 Accepted: 08 October 2024 Published: 23 October 2024
MSC : 62F12

In this paper, our primary attention was centered on the issue of detecting the variance change point for strong-mixing samples. We delved into the cumulative sum (CUSUM) estimator of variance change model and established the strong convergence rate of the variance change point estimation. Furthermore, to corroborate the effectiveness of the CUSUM based methodology, we have conducted a series of simulations, the outcomes of which underscored its validity.
- variance change point,
- CUSUM,
- strong-mixing sequence,
- strong consistency
Citation: Mengmei Xi, Yi Wu, Xuejun Wang. Strong consistency properties of the variance change point estimator based on strong-mixing samples[J]. AIMS Mathematics, 2024, 9(11): 30059-30072. doi: 10.3934/math.20241452

Related Papers:

Abstract

In this paper, our primary attention was centered on the issue of detecting the variance change point for strong-mixing samples. We delved into the cumulative sum (CUSUM) estimator of variance change model and established the strong convergence rate of the variance change point estimation. Furthermore, to corroborate the effectiveness of the CUSUM based methodology, we have conducted a series of simulations, the outcomes of which underscored its validity.

References

[1]	M. Basseville, Detecting changes in signals and systems-a survey, Automatica, 24 (1998), 309–326. https://doi.org/10.1016/0005-1098(88)90073-8 doi: 10.1016/0005-1098(88)90073-8
[2]	J. Chen, A. Gupta, Testing and locating variance change points with application to stock prices, Public. American Stat. Assoc., 92 (1997), 739–747. https://doi.org/10.1080/01621459.1997.10474026 doi: 10.1080/01621459.1997.10474026
[3]	S. J. Hawkins, A. J. Southward, M. J. Genner, Detection of environmental change in a marine ecosystem-evidence from the western English channel, Sci. Total Envir., 310 (2003), 245–256. https://doi.org/10.1016/S0048-9697(02)00645-9 doi: 10.1016/S0048-9697(02)00645-9
[4]	R. Sparks, T. Keighley, D. Muscatello, Early warning CUSUM plans for surveillance of negative binomial daily disease counts, J. Appl. Stat., 37 (2010), 1911–1930. https://doi.org/10.1080/02664760903186056 doi: 10.1080/02664760903186056
[5]	R. Ratnam, J. B. Goense, M. E. Nelson, Change-point detection in neuronal spike train activity, Neurocomputing, 52 (2003), 849–855. https://doi.org/10.1016/S0925-2312(02)00815-9 doi: 10.1016/S0925-2312(02)00815-9
[6]	K. Lena, A. Go, K. Jutta, Single and multiple change point detection in spike trains: Comparison of different CUSUM methods, Front. Syst. Neurosc., 10 (2016), 6909747. https://doi.org/10.3389/fnsys.2016.00051 doi: 10.3389/fnsys.2016.00051
[7]	J. Bai, Least squares estimation of a shift in linear processes, J. Time Series Anal., 15 (1994), 453–472. https://doi.org/10.1111/j.1467-9892.1994.tb00204.x doi: 10.1111/j.1467-9892.1994.tb00204.x
[8]	P. Fearnhead, Exact and efficient Bayesian inference for multiple change point problems, Stat. Comput., 16 (2006), 203–213. https://doi.org/10.1007/s11222-006-8450-8 doi: 10.1007/s11222-006-8450-8
[9]	C. Zou, G. Yin, L. Feng, Z. Wang, Nonparametric maximum likelihood approach to multiple change-point problems, Annal. Stat., 42 (2014), 970–1002. https://doi.org/10.1214/14-AOS1210 doi: 10.1214/14-AOS1210
[10]	D. S. Matteson, N. A. James, A nonparametric approach for multiple change point analysis of multivariate data, J. American Stat. Assoc., 109 (2014), 334–345. https://doi.org/10.1080/01621459.2013.849605 doi: 10.1080/01621459.2013.849605
[11]	K. Haynes, P. Fearnhead, I. A. Eckley, A computationally efficient nonparametric approach for change point detection, Stat. Comput., 27 (2017), 1293–1305. https://doi.org/10.1007/s11222-016-9687-5 doi: 10.1007/s11222-016-9687-5
[12]	E. Gombay, L. Horvath, M. Huskova, Estimators and tests for change in variances, Stat. Risk Model., 14 (1996), 145–159. https://doi.org/10.1524/strm.1996.14.2.145 doi: 10.1524/strm.1996.14.2.145
[13]	S. Lee, S. Park, The Cusum of squares test for scale changes in infinite order moving average processes, Scandinav. J. Stat., 28 (2001), 625–644. https://doi.org/10.1111/1467-9469.00259 doi: 10.1111/1467-9469.00259
[14]	S. Lee, J. Ha, O. Na, S. Na, The Cusum test for parameter change time series models, Scandinav. J. Stat., 30 (2003), 781–796. https://doi.org/10.1111/1467-9469.00364 doi: 10.1111/1467-9469.00364
[15]	M. Y. Xu, P. S. Zhong, W. Wang, Detecting variance change-points for blocked time series and dependent panel data, J. Busin. Economic Stat., 34 (2016), 213–226. https://doi.org/10.1080/07350015.2015.1026438 doi: 10.1080/07350015.2015.1026438
[16]	R. B. Qin, W. Liu, Z. Tian, A strong convergence rate of estimator of variance change in linear processes and its applications, Statistics, 51 (2017), 314–330. https://doi.org/10.1080/02331888.2016.1268614 doi: 10.1080/02331888.2016.1268614
[17]	M. Xu, Y. Wu, B. Jin, Detection of a change-point in variance by a weighted sum of powers of variances test, J. Appl. Stat., 46 (2019), 664–679. https://doi.org/10.1080/02664763.2018.1510475 doi: 10.1080/02664763.2018.1510475
[18]	Y. C. Yu, X. S. Liu, L. Liu, P. Zhao, Detection of multiple change points for linear processes under negatively super-additive dependence J. Inequal. Appl., 2019 (2019), 16. https://doi.org/10.1186/s13660-019-2169-5 doi: 10.1186/s13660-019-2169-5
[19]	R. C. Bradley, Basic properties of strong mixing conditions, Prog. Prob. Stat. depend. Prob. Stat., 2 (1986), 165–192. https://doi.org/10.1007/978-1-4615-8162-8_8 doi: 10.1007/978-1-4615-8162-8_8
[20]	P. Doukhan, Mixing properties and examples, 85 Eds, Berlin: Springer, 1994. 10.1007/978-1-4612-2642-0
[21]	J. Q. Fan, Q. W. Yao, Nonlinear time series: Nonparametric and parametric methods, New York: Springer, 2006.
[22]	M. Gao, S. S. Ding, S. P. Wu, W. Z. Yang, The asymptotic distribution of CUSUM estimator based on $\alpha$-mixing sequences, Commun. Stat.-Theory Meth., 51 (2022), 6101–6113. https://doi.org/10.1080/03610918.2020.1794006 doi: 10.1080/03610918.2020.1794006
[23]	M. Gao, X. P. Shi, X. J. Wang, W. Z. Yang, Combination test for mean shift and variance change, Symmetry, 2023 (2023). https://doi.org/10.3390/sym15111975 doi: 10.3390/sym15111975
[24]	A. Rosalsky, L. V. Thành, A note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbers. Stat. Probab. Lett., 178 (2021), 10. https://doi.org/10.1016/j.spl.2021.109181 doi: 10.1016/j.spl.2021.109181
[25]	S. C. Yang, Maximal moment inequality for partial sums of strong mixing sequences and application, Acta Math. Sinica. English Series, 23 (2007), 1013–1024.
[26]	W. Z. Yang, Y. W. Wang, S. H. Hu, Some probability inequalities of least-squares estimator in non linear regression model with strong mixing errors, Commun. Stat. Theory Meth., 46 (2017), 165–175. https://doi.org/10.1080/03610926.2014.988261 doi: 10.1080/03610926.2014.988261

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)