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Tests for scale changes based on pairwise differences. (English) Zbl 1441.62223
Summary: In many applications it is important to know whether the amount of fluctuation in a series of observations changes over time. In this article, we investigate different tests for detecting changes in the scale of mean-stationary time series. The classical approach, based on the CUSUM test applied to the squared centered observations, is very vulnerable to outliers and impractical for heavy-tailed data, which leads us to contemplate test statistics based on alternative, less outlier-sensitive scale estimators. It turns out that the tests based on Gini’s mean difference (the average of all pairwise distances) and generalized \(Q_n\) estimators (sample quantiles of all pairwise distances) are very suitable candidates. They improve upon the classical test not only under heavy tails or in the presence of outliers, but also under normality. We use recent results on the process convergence of \(U\)-statistics and \(U\)-quantiles for dependent sequences to derive the limiting distribution of the test statistics and propose estimators for the long-run variance. We show the consistency of the tests and demonstrate the applicability of the new change-point detection methods at two real-life data examples from hydrology and finance.
62M07 Non-Markovian processes: hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
R; robustbase
Full Text: DOI
[1] Andrews, D. W., “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation,”, Econometrica, 59, 817-858 (1991) · Zbl 0732.62052
[2] Aue, A.; Hörmann, S.; Horváth, L.; Reimherr, M., “Break Detection in the Covariance Structure of Multivariate Time Series Models,”, Annals of Statistics, 37, 4046-4087 (2009) · Zbl 1191.62143
[3] Bücher, A.; Kojadinovic, I., “A Dependent Multiplier Bootstrap for the Sequential Empirical Copula Process Under Strong Mixing,”, Bernoulli, 22, 927-968 (2016) · Zbl 1388.62123
[4] Bücher, A.; Kojadinovic, I., “Dependent Multiplier Bootstraps for Non-Degenerate U-Statistics Under Mixing Conditions With Applications,”, Journal of Statistical Planning and Inference, 170, 83-105 (2016) · Zbl 1383.62126
[5] Carlstein, E., “The Use of Subseries Values for Estimating the Variance of a General Statistic From a Stationary Sequence,”, Annals of Statistics, 14, 1171-1179 (1986) · Zbl 0602.62029
[6] de Jong, R. M.; Davidson, J., “Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices,”, Econometrica, 68, 407-424 (2000) · Zbl 1016.62030
[7] Dehling, H.; Fried, R.; Sharipov, O. S.; Vogel, D.; Wornowizki, M., “Estimation of the Variance of Partial Sums of Dependent Processes,”, Statistics & Probability Letters, 83, 141-147 (2013) · Zbl 06130776
[8] Dehling, H.; Vogel, D.; Wendler, M.; Wied, D., “Testing for Changes in Kendall’s Tau,”, Econometric Theory, 33, 1352-1386 (2017) · Zbl 1396.62202
[9] Dehling, H.; Wendler, M., “Central Limit Theorem and the Bootstrap for U-Statistics of Strongly Mixing Data,”, Journal of Multivariate Analysis, 101, 126-137 (2010) · Zbl 1177.62056
[10] Doukhan, P.; Lang, G.; Leucht, A.; Neumann, M. H., “Dependent Wild Bootstrap for the Empirical Process,”, Journal of Time Series Analysis, 36, 290-314 (2015) · Zbl 1325.62065
[11] Gerstenberger, C.; Vogel, D., “On the Efficiency of Gini’s Mean Difference,”, Statistical Methods & Applications, 24, 569-596 (2015) · Zbl 1328.62299
[12] Gombay, E.; Horváth, L.; Husková, M., “Estimators and Tests for Change in Variances,”, Statistics & Risk Modeling, 14, 145-160 (1996) · Zbl 0864.62030
[13] Hampel, F. R., “The Influence Curve and Its Role in Robust Estimation,”, Journal of the American Statistical Association, 69, 383-393 (1974) · Zbl 0305.62031
[14] Huber, P. J.; Ronchetti, E. M., Robust Statistics (2009), Hoboken, NJ: Wiley, Hoboken, NJ · Zbl 1276.62022
[15] Hyndman, R. J.; Fan, Y., “Sample Quantiles in Statistical Packages,”, The American Statistician, 50, 361-365 (1996)
[16] Inclan, C.; Tiao, G. C., “Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance,”, Journal of the American Statistical Association, 89, 913-923 (1994) · Zbl 0825.62678
[17] Kojadinovic, I.; Quessy, J.-F.; Rohmer, T., “Testing the Constancy of Spearman’s Rho in Multivariate Time Series,”, Annals of the Institute of Statistical Mathematics, 65, 292-954 (2015) · Zbl 1400.62184
[18] Künsch, H. R., “The Jackknife and the Bootstrap for General Stationary Observations,”, Annals of Statistics, 17, 1217-1241 (1989) · Zbl 0684.62035
[19] Lax, D. A., “Robust Estimators of Scale: Finite-Sample Performance in Long-Tailed Symmetric Distributions,”, Journal of the American Statistical Association, 80, 736-741 (1985)
[20] Lee, S.; Park, S., “The CUSUM of Squares Test for Scale Changes in Infinite Order Moving Average Processes,”, Scandinavian Journal of Statistics, 28, 625-644 (2001) · Zbl 1010.62079
[21] Leucht, A.; Neumann, M. H., “Dependent Wild Bootstrap for Degenerate U- and V-Statistics,”, Journal of Multivariate Analysis, 117, 257-280 (2013) · Zbl 1279.62102
[22] Maronna, R. A.; Martin, D. R.; Yohai, V. J., Robust Statistics: Theory and Methods, Wiley Series in Probability and Statistics (2006), Chichester: Wiley, Chichester
[23] Nair, U. S., “The Standard Error of Gini’s Mean Difference,”, Biometrika, 28, 428-436 (1936) · Zbl 0015.31102
[24] Paparoditis, E.; Politis, D. N., “Tapered Block Bootstrap,”, Biometrika, 88, 1105-1119 (2001) · Zbl 0987.62027
[25] Patton, A.; Politis, D. N.; White, H., “Correction to ‘Automatic Block-Length Selection for the Dependent Bootstrap’ by D. Politis and H. White,”, Econometric Reviews, 28, 372-375 (2009) · Zbl 1400.62193
[26] Politis, D. N., “Adaptive Bandwidth Choice,”, Journal of Nonparametric Statistics, 15, 517-533 (2003) · Zbl 1054.62038
[27] Politis, D. N., Higher-Order Accurate, Positive Semidefinite Estimation of Large-Sample Covariance and Spectral Density Matrices,”, Econometric Theory, 27, 703-744 (2011) · Zbl 1219.62144
[28] Politis, D. N.; Romano, J. P., “The Stationary Bootstrap,”, Journal of the American Statistical Association, 89, 1303-1313 (1994) · Zbl 0814.62023
[29] Politis, D. N.; White, H., “Automatic Block-Length Selection for the Dependent Bootstrap,”, Econometric Reviews, 23, 53-70 (2004) · Zbl 1082.62076
[30] R Core Team, R: A Language and Environment for Statistical Computing (2015), Vienna, Austria: R Foundation for Statistical Computing, Vienna, Austria
[31] Rousseeuw, P. J.; Croux, C.; Dodge, Y., L1-Statistical Analysis and Related Methods, 1, “Explicit Scale Estimators With High Breakdown Point,”, 77-92 (1992), Amsterdam: North-Holland, Amsterdam
[32] Rousseeuw, P. J.; Croux, C., “Alternatives to the Median Absolute Deviation,”, Journal of the American Statistical Association, 88, 1273-1283 (1993) · Zbl 0792.62025
[33] Shao, X., “The Dependent Wild Bootstrap,”, Journal of the American Statistical Association, 105, 218-235 (2010) · Zbl 1397.62121
[34] Shao, X.; Zhang, X., “Testing for Change Points in Time Series,”, Journal of the American Statistical Association, 105, 1228-1240 (2010) · Zbl 1390.62184
[35] Sharipov, O. S.; Wendler, M., “Normal Limits, Nonnormal Limits, and the Bootstrap for Quantiles of Dependent Data,”, Statistics & Probability Letters, 83, 1028-1035 (2013) · Zbl 06162770
[36] Sun, S.; Lahiri, S. N., “Bootstrapping the Sample Quantile of a Weakly Dependent Sequence,”, Sankhyā, 68, 130-166 (2006) · Zbl 1193.62076
[37] Tukey, J. W.; Olkin, I.; Ghurye, S. G.; Hoeffding, W.; Madow, W. G.; Mann, H. B., Contributions to Probability and Statistics. Essays in Honor of Harold Hotteling, “A Survey of Sampling From Contaminated Distributions, 448-485 (1960), Stanford, CA: Stanford University Press, Stanford, CA
[38] Vogel, D.; Wendler, M., “Studentized U-Quantile Processes Under Dependence With Applications to Change-Point Analysis,”, Bernoulli, 23, 3114-3144 (2017) · Zbl 1401.62046
[39] Wied, D.; Arnold, M.; Bissantz, N.; Ziggel, D., “A New Fluctuation Test for Constant Variances With Applications to Finance,”, Metrika, 75, 1111-1127 (2012) · Zbl 1254.62097
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