×
Preuß, Philip; Vetter, Mathias; Dette, Holger

A test for stationarity based on empirical processes. (English) Zbl 1281.62183

Bernoulli 19, No. 5B, 2715-2749 (2013).
Summary: We investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov-Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic.
Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate \(g_{T}\to 0\) such that \(g_{T}T^{1/2}\to \infty\), where \(T\) denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature.

Cited in 16 Documents

MSC:

62M07 Non-Markovian processes: hypothesis testing
62G10 Nonparametric hypothesis testing
62M15 Inference from stochastic processes and spectral analysis
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)

Keywords:

bootstrap; empirical spectral measure; goodness-of-fit tests; integrated periodograms; locally stationary processes; non-stationary processeses; spectral density
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory ( Tsahkadsor , 1971) 267-281. Budapest: Akadémiai Kiadó. · Zbl 0283.62006
[2] Beltrão, K.I. and Bloomfield, P. (1987). Determining the bandwidth of a kernel spectrum estimate. J. Time Series Anal. 8 21-38. · Zbl 0608.62118 · doi:10.1111/j.1467-9892.1987.tb00418.x
[3] Berg, A., Paparoditis, E. and Politis, D.N. (2010). A bootstrap test for time series linearity. J. Statist. Plann. Inference 140 3841-3857. · Zbl 1327.62460 · doi:10.1016/j.jspi.2010.04.047
[4] Brillinger, D.R. (1981). Time Series : Data Analysis and Theory . New York: McGraw-Hill. · Zbl 0486.62095
[5] Chiann, C. and Morettin, P.A. (1999). Estimation of time varying linear systems. Stat. Inference Stoch. Process. 2 253-285. · Zbl 0967.62073 · doi:10.1023/A:1009999208631
[6] Dahlhaus, R. (1988). Empirical spectral processes and their applications to time series analysis. Stochastic Process. Appl. 30 69-83. · Zbl 0655.60033 · doi:10.1016/0304-4149(88)90076-2
[7] Dahlhaus, R. (1996). On the Kullback-Leibler information divergence of locally stationary processes. Stochastic Process. Appl. 62 139-168. · Zbl 0849.60032 · doi:10.1016/0304-4149(95)00090-9
[8] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37. · Zbl 0871.62080 · doi:10.1214/aos/1034276620
[9] Dahlhaus, R. (2009). Local inference for locally stationary time series based on the empirical spectral measure. J. Econometrics 151 101-112. · Zbl 1431.62362 · doi:10.1016/j.jeconom.2009.03.002
[10] Dahlhaus, R., Neumann, M.H. and von Sachs, R. (1999). Nonlinear wavelet estimation of time-varying autoregressive processes. Bernoulli 5 873-906. · Zbl 0954.62103 · doi:10.2307/3318448
[11] Dahlhaus, R. and Polonik, W. (2006). Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes. Ann. Statist. 34 2790-2824. · Zbl 1114.62034 · doi:10.1214/009053606000000867
[12] Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1-39. · Zbl 1204.62156 · doi:10.3150/08-BEJ137
[13] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075-1114. · Zbl 1113.62099 · doi:10.1214/009053606000000227
[14] Dette, H., Preuss, P. and Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing. J. Amer. Statist. Assoc. 106 1113-1124. · Zbl 1229.62119 · doi:10.1198/jasa.2011.tm10811
[15] Dwivedi, Y. and Subba Rao, S. (2011). A test for second-order stationarity of a time series based on the discrete Fourier transform. J. Time Series Anal. 32 68-91. · Zbl 1290.62059 · doi:10.1111/j.1467-9892.2010.00685.x
[16] Hannan, E. and Kavalieris, L. (1986). Regression, autoregression models. J. Time Series Anal. 7 27-49. · Zbl 0588.62163 · doi:10.1111/j.1467-9892.1986.tb00484.x
[17] Kreiß, J.P. (1988). Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Fachbereich Mathematik, Univ. Hamburg.
[18] Kreiß, J.P. (1997). Asymptotical properties of residual bootstrap for autoregressions. Technical report, TU Braunschweig.
[19] Kreiss, J.P., Paparoditis, E. and Politis, D.N. (2011). On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 2103-2130. · Zbl 1227.62067 · doi:10.1214/11-AOS900
[20] Neumann, M.H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 38-76. · Zbl 0871.62081 · doi:10.1214/aos/1034276621
[21] Palma, W. and Olea, R. (2010). An efficient estimator for locally stationary Gaussian long-memory processes. Ann. Statist. 38 2958-2997. · Zbl 1200.62109 · doi:10.1214/10-AOS812
[22] Paparoditis, E. (2009). Testing temporal constancy of the spectral structure of a time series. Bernoulli 15 1190-1221. · Zbl 1200.62049 · doi:10.3150/08-BEJ179
[23] Paparoditis, E. (2010). Validating stationarity assumptions in time series analysis by rolling local periodograms. J. Amer. Statist. Assoc. 105 839-851. · Zbl 1392.62275 · doi:10.1198/jasa.2010.tm08243
[24] Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics . New York: Springer. · Zbl 0544.60045
[25] Priestley, M.B. (1965). Evolutionary spectra and non-stationary processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 204-237. · Zbl 0144.41001
[26] Sergides, M. and Paparoditis, E. (2009). Frequency domain tests of semi-parametric hypotheses for locally stationary processes. Scand. J. Stat. 36 800-821. · Zbl 1224.62070 · doi:10.1111/j.1467-9469.2009.00652.x
[27] Van Bellegem, S. and von Sachs, R. (2008). Locally adaptive estimation of evolutionary wavelet spectra. Ann. Statist. 36 1879-1924. · Zbl 1142.62067 · doi:10.1214/07-AOS524
[28] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics . New York: Springer. · Zbl 0862.60002
[29] von Sachs, R. and Neumann, M.H. (2000). A wavelet-based test for stationarity. J. Time Series Anal. 21 597-613. · Zbl 0972.62085 · doi:10.1111/1467-9892.00200
[30] Whittle, P. (1951). Hypothesis Testing in Time Series Analysis . Uppsala: Almqvist and Wiksell. · Zbl 0045.41301
[31] Whittle, P. (1952). Some results in time series analysis. Skand. Aktuarietidskr. 35 48-60. · Zbl 0049.37301
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.