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Modified sequential change point procedures based on estimating functions. (English) Zbl 1392.62241

Summary: A large class of sequential change point tests are based on estimating functions where estimation is computationally efficient as (possibly numeric) optimization is restricted to an initial estimation. This includes examples as diverse as mean changes, linear or non-linear autoregressive and binary models. While the standard cumulative-sum-detector (CUSUM) has recently been considered in this general setup, we consider several modifications that have faster detection rates in particular if changes do occur late in the monitoring period. More presicely, we use three different types of detector statistics based on partial sums of a monitoring function, namely the modified moving-sum-statistic (mMOSUM), Page’s cumulative-sum-statistic (Page-CUSUM) and the standard moving-sum-statistic (MOSUM). The statistics only differ in the number of observations included in the partial sum. The mMOSUM uses a bandwidth parameter which multiplicatively scales the lower bound of the moving sum. The MOSUM uses a constant bandwidth parameter, while Page-CUSUM chooses the maximum over all possible lower bounds for the partial sums. So far, the first two schemes have only been studied in a linear model, the MOSUM only for a mean change.
We develop the asymptotics under the null hypothesis and alternatives under mild regularity conditions for each test statistic, which include the existing theory but also many new examples. In a simulation study we compare all four types of test procedures in terms of their size, power and run length. Additionally we illustrate their behavior by applications to exchange rate data as well as the Boston homicide data.

MSC:

62L10 Sequential statistical analysis
62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)

Software:

strucchange
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Full Text: DOI Euclid

References:

[1] E. Andreou and E. Ghysels. Monitoring disruptions in financial markets., Journal of Econometrics, 135:77-124, 2006. · Zbl 1418.62372
[2] A. Aue and L. Horváth. Delay time in sequential detection of change., Statistics and Probability Letters, 67:221-231, 2004. · Zbl 1059.62085
[3] A. Aue, L. Horváth, M. Husková, and P. Kokoszka. Change-point monitoring in linear models., Econometrics Journal, 9(3):373-403, 2006. · Zbl 1106.62067
[4] A. Aue, L. Horváth, M. Kühn, and J. Steinebach. On the reaction time of moving sum detectors., Journal of Statistical Planning and Inference, 27(2):126-140, 2008.
[5] A. Aue, S. Hörmann, L. Horváth, and M. Hušková. Sequential testing for the stability of portfolio betas., Econometric Theory, 2011.
[6] I. Berkes, E. Gombay, L. Horváth, and P. Kokoszka. Sequential change-point detection in GARCH(p,q) models., Econometric theory, 20 :1140-1167, 2004. · Zbl 1069.62058
[7] Z. Chen and Z. Tian. Modified procedures for change point monitoring in linear models., Mathematics and computers in simulation, 81(1):62-75, 2010. · Zbl 1201.62073
[8] C.-S.J. Chu, M. Stinchcombe, and H. White. Monitoring structural change., Econometrica, 64 :1045-1065, 1996. · Zbl 0856.90027
[9] G. Ciupera. Two tests for sequential detection of a change-point in a nonlinear model., Journal of Statistical Planning and Inference, 143(10) :1719-1743, 2013. · Zbl 1432.62272
[10] M. Csörgö and P. Révész., Strong Approximations in Probability and Statistics. Academic Press, 1981. · Zbl 0539.60029
[11] B. Eichinger and C. Kirch. A mosum procedure for the estimation of multiple random change points., Bernoulli, 24(1):526-564, 2018. · Zbl 1388.62251
[12] S. Fremdt. Page’s sequential procedure for change-point detection in time series regression., Statistics: A Journal of Theoretical and Applied Statistics, 48(4):1-28, 2014.
[13] R. Fried and M. Imhoff. On the online detection of monotonic trends in time series., Biometrical Journal, 46:90-102, 2004.
[14] L. Horváth, M. Hušková, P. Kokoszka, and J. Steinebach. Monitoring changes in linear models., Journal of statistical and inference, 126(1):225-251, 2004.
[15] L. Horváth, M. Kühn, and J. Steinebach. On the performance of the fluctuation test for structural change., Sequential Analysis, 27(2):126-140, 2008.
[16] M. Hušková and C. Kirch. Bootstrapping sequential change-point tests for linear regression., Metrika, 75(5):673-708, 2012. · Zbl 1362.62161
[17] M. Hušková and A. Koubková. Monitoring jump changes in linear models., Journal on Statistical Research, 39:51-70, 2005.
[18] C. Kirch. Bootstrapping sequential change-point tests., Sequential Analysis, 27(3):330-349, 2008. · Zbl 1145.62060
[19] C. Kirch and J. Tadjuidje Kamgaing. Testing for parameter stability in nonlinear autoregressive models., Journal of Time Series Analysis, 33:365-385, 2012. · Zbl 1301.62088
[20] C. Kirch and J. Tadjuidje Kamgaing. On the use of estimating functions in monitoring time series for change points., Journal of Statistical Planning and Inference, 161:25-49, 2015. · Zbl 1311.62122
[21] C. Kirch and J. Tadjuidje Kamgaing. Detection of change points in discrete-valued time series. In Davis, R.A., Holan, S.A., Lund, R.B., and Ravishanker, N., editors, Handbook of Discrete Valued Time series, pages 219-244. CRC Press, 2016.
[22] C. Kirch, B. Muhsal, and H. Ombao. Detection of changes in multivariate time series with application to EEG data., Journal of the American Statistical Association, 110 :1197-1216, 2015. · Zbl 1378.62072
[23] S. Weber., Change-Point Procedures for Multivariate Dependent Data. PhD thesis, Karlsruhe Institute of Technology (KIT), 2017. URN: urn:nbn:de:swb:90-689812.
[24] A. Zeileis. Implementing a class of structural change tests: An econometric computing approach., Computational Statistics & Data Analysis, 50(11) :2987-3008, 2006. · Zbl 1445.62316
[25] A. Zeileis, F. Leisch, K. Hornik, and C. Kleiber. strucchange: An R package for testing for structural change in linear regression models., Journal of Statistical Software, 7(2):1-38, 2002. URL http://www.jstatsoft.org/v07/i02/.
[26] A. Zeileis, A. Shah, and I. Patnaik. Testing, monitoring, and dating structural changes in exchange rate regimes., Computational Statistics & Data Analysis, 54(6) :1696-1706, 2010. · Zbl 1284.91596
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