×

A test for second-order stationarity of a time series based on the maximum of Anderson-Darling statistics. (English) Zbl 1477.62144

Summary: This paper is concerned with testing the second-order stationarity of a time series. By using a blockwise scheme, the test is transformed to compare local spectra of different segments of the blocked time series. Based on periodogram-ratios of each pair of segments at the same frequency points, an Anderson-Darling-like statistic is constructed to compare their spectra. By maximizing several Anderson-Darling-like statistics, a test statistic is proposed for testing second-order stationarity. Under the null, the probability distribution of the proposed statistic can be approximated by simulation. Extensive simulation examples show that the proposed test approach achieves good performance.

MSC:

62H15 Hypothesis testing in multivariate analysis
62M15 Inference from stochastic processes and spectral analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)

Software:

BayesSpec
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, T. W.; Darling, D. A., Asymptotic theory of certain ‘goodness of fit’ criteria based on stochastic processes, Ann. Math. Stat., 23, 193-212 (1952) · Zbl 0048.11301
[2] Anderson, T. W.; Darling, D. A., A test of goodness of fit, J. Amer. Statist. Assoc., 49, 765-769 (1954) · Zbl 0059.13302
[3] Andreou, E.; Ghysels, E., Structural breaks in financial time series, (Anderson, T. G.; Davis, R. A.; Kreiss, J. P.; Mikosch, T., Handbook of Fiancial Time Series (2008), Springer: Springer Berlin), 839-866 · Zbl 1178.91217
[4] Bandyopadhyay, S.; Subba Rao, S., A test for stationarity for irregularly spaced spatial data, J. R. Stat. Soc. Ser. B Stat. Methodol., 79, 95-123 (2017) · Zbl 1414.62398
[5] van Bellegem, S.; von Sachs, R., Locally adaptive estimation of evoluationary wavelet spectra, Ann. Statist., 36, 1879-1924 (2008) · Zbl 1142.62067
[6] Billingsley, P., Probability and Measure (1995), John Wiley & Sons: John Wiley & Sons New York · Zbl 0822.60002
[7] Brillinger, D. R., Time Series: Data Analysis and Theory (2001), SIAM: SIAM Philadephia · Zbl 0983.62056
[8] Brockwell, P.; Davis, R. A., Time Series: Theory and Methods (1991), Springer: Springer New York · Zbl 0709.62080
[9] Chambers, J. M.; Mallows, C. L.; Stuck, B. W., A method for simulating stable random variables, J. Amer. Statist. Assoc., 71, 340-344 (1976) · Zbl 0341.65003
[10] Dahlhaus, R., Fitting time series models to nonstationary processes, Ann. Statist., 25, 1-37 (1997) · Zbl 0871.62080
[11] Dahlhaus, R., Local inference for locally stationary time series based on the empirical spectral measure, J. Econometrics, 151, 101-112 (2009) · Zbl 1431.62362
[12] DasGupta, A., Asymptotic Theory of Statistics and Probability (2008), Springer: Springer New York · Zbl 1154.62001
[13] Dette, H.; Preuss, P.; Vetter, M., A measure of stationarity in locally stationary processes with applications to testing, J. Amer. Statist. Assoc., 106, 1113-1124 (2011) · Zbl 1229.62119
[14] Dwivedi, Y.; Subba Rao, S., A test for second-order stationarity of a time series based on the discrete fourier transform, J. Time Series Anal., 32, 68-91 (2011) · Zbl 1290.62059
[15] Fuller, W. A., Introduction to Statistical Time Series (1996), John Wiley: John Wiley New York · Zbl 0851.62057
[16] Götze, F., On the rate of convergence in the multivariate CLT, Ann. Probab., 19, 724-739 (1991) · Zbl 0729.62051
[17] Guo, W.; Dai, M., Multivariate time-dependent spectral analysis using Cholesky decomposition, Statist. Sinica, 16, 825-845 (2006) · Zbl 1107.62098
[18] Horváth, L.; Kokoszka, P.; Rice, G., Testing stationarity of functional time series, J. Econom., 179, 66-82 (2014) · Zbl 1293.62186
[19] Hušková, M.; Prášková, Z.; Steinebach, J., On the detection of changes in autoregressive time series, I. asymptotics, J. Statist. Plann. Inference, 137, 1243-1259 (2007) · Zbl 1107.62090
[20] Jentsch, C.; Subba Rao, S., A test for second-order stationarity of a multivariate time series, J. Econom., 185, 124-161 (2015) · Zbl 1332.62328
[21] Kirch, C.; Muhsal, B.; Ombao, H., Detection of changes in multivariate time series with application to EEG data, J. Amer. Statist. Assoc., 110, 1197-1216 (2015) · Zbl 1378.62072
[22] Li, Z.; Krafty, R. T., Adaptive Bayesian time-frequency analysis of multivariate time series, J. Amer. Statist. Assoc., 114, 453-465 (2019) · Zbl 1478.62262
[23] Paparoditis, E., Testing temporal constancy of the spectral structure of a time series, Bernoulli, 15, 1190-1221 (2009) · Zbl 1200.62049
[24] Paparoditis, E.; Preuß, P., On local power properties of frequency domain-based tests for stationarity, Scand. J. Stat., 43, 664-682 (2016) · Zbl 1468.62298
[25] Preuss, P.; Puchstein, R.; Dette, H., Detection of multiple structural breaks in multivariate time series, J. Amer. Statist. Assoc., 110, 654-668 (2015) · Zbl 1373.62454
[26] Preuß, P.; Vetter, M.; Dette, H., A test for stationarity based on empirical processes, Bernoulli, 19, 2715-2749 (2013) · Zbl 1281.62183
[27] Rosen, O.; Wood, S.; Stoffer, D., AdaptSPEC: Adaptive spectral estimation for nonstationary time series, J. Amer. Statist. Assoc., 107, 1575-1589 (2012) · Zbl 1258.62093
[28] von Sachs, R.; Neumann, M. H., A wavelet-based test for stationarity, J. Time Ser. Anal., 21, 597-613 (1999) · Zbl 0972.62085
[29] Zhang, S.; Tu, X. M., Tests for comparing time-invariant and time-varying spectra based on the Pearson statistic, J. Time Ser. Anal., 39, 709-730 (2018) · Zbl 1401.62183
[30] Zhang, S.; Tu, X. M., Tests for comparing time-invariant and time-varying spectra based on the Anderson-Darling statistic (2019), http://arxiv.org/abs/1705.04821
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.