Locally adaptive estimation of evolutionary wavelet spectra
Sébastien Van Bellegem and Rainer von Sachs
Source: Ann. Statist. Volume 36, Number 4 (2008), 1879-1924.
Abstract
We introduce a wavelet-based model of local stationarity. This model enlarges the class of locally stationary wavelet processes and contains processes whose spectral density function may change very suddenly in time. A notion of time-varying wavelet spectrum is uniquely defined as a wavelet-type transform of the autocovariance function with respect to so-called autocorrelation wavelets. This leads to a natural representation of the autocovariance which is localized on scales. We propose a pointwise adaptive estimator of the time-varying spectrum. The behavior of the estimator studied in homogeneous and inhomogeneous regions of the wavelet spectrum.
Primary Subjects: 62M10
Secondary Subjects: 60G15, 62G10, 62G05
Keywords: Local stationarity; nonstationary time series; wavelet spectrum; autocorrelation wavelet; change-point; pointwise adaptive estimation; quadratic form; regularization
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1216237303
Digital Object Identifier: doi:10.1214/07-AOS524
References
Brillinger, D. (1975). Time Series. Data Analysis and Theory. Holt, Rinehart and Winston, New York.
Mathematical Reviews (MathSciNet):
MR443257
Zentralblatt MATH:
0321.62004
Brown, L. D. and Low, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 2524–2535.
Mathematical Reviews (MathSciNet):
MR1425965
Digital Object Identifier: doi:10.1214/aos/1032181166
Project Euclid: euclid.aos/1032181166
Comte, F. (2001). Adaptive estimation of the spectrum of a stationary Gaussian sequence. Bernoulli 7 267–298.
Mathematical Reviews (MathSciNet):
MR1828506
Digital Object Identifier: doi:10.2307/3318739
Project Euclid: euclid.bj/1080222090
Dahlhaus, R. (1988). Empirical spectral process and their applications to time series analysis. Stochastic Process. Appl. 30 69–83.
Mathematical Reviews (MathSciNet):
MR968166
Digital Object Identifier: doi:10.1016/0304-4149(88)90076-2
Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
Mathematical Reviews (MathSciNet):
MR1429916
Digital Object Identifier: doi:10.1214/aos/1034276620
Project Euclid: euclid.aos/1034276620
Dahlhaus, R. (2000). A likelihood approximation for locally stationary processes. Ann. Statist. 28 1762–1794.
Mathematical Reviews (MathSciNet):
MR1835040
Digital Object Identifier: doi:10.1214/aos/1015957480
Project Euclid: euclid.aos/1015957480
Dahlhaus, R. and Polonik, W. (2002). Empirical spectral processes and nonparametric maximum likelihood estimation for time series. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 275–298. Birkhäuser, Boston.
Dahlhaus, R. and Polonik, W. (2006). Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes. Ann. Statist. 34 2790-2824.
Mathematical Reviews (MathSciNet):
MR2329468
Digital Object Identifier: doi:10.1214/009053606000000867
Project Euclid: euclid.aos/1179935065
Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet):
MR1162107
Zentralblatt MATH:
0776.42018
Fryźlewicz, P. and Nason, G. P. (2006). Haar–Fisz estimation of evolutionary wavelet spectra. J. Roy. Statist. Soc. Ser. B 68 611–634.
Fryźlewicz, P., Van Bellegem, S. and von Sachs, R. (2003). Forecasting non-stationary time series by wavelet process modelling. Ann. Inst. Statist. Math. 55 737–764.
Mathematical Reviews (MathSciNet):
MR2028615
Digital Object Identifier: doi:10.1007/BF02523391
Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximations, and Statistical Applications. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1618204
Laurent, B. and Massart, P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 1302–1338.
Mathematical Reviews (MathSciNet):
MR1805785
Digital Object Identifier: doi:10.1214/aos/1015957395
Project Euclid: euclid.aos/1015957395
Lepski, O. (1990). On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454–470.
Mathematical Reviews (MathSciNet):
MR1091202
Lepski, O. and Spokoiny, V. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512–2546.
Mathematical Reviews (MathSciNet):
MR1604408
Digital Object Identifier: doi:10.1214/aos/1030741083
Project Euclid: euclid.aos/1030741083
Lii, K.-S. and Rosenblatt, M. (2002). Spectral analysis for harmonizable processes. Ann. Statist. 30 258–297.
Mathematical Reviews (MathSciNet):
MR1892664
Digital Object Identifier: doi:10.1214/aos/1015362193
Project Euclid: euclid.aos/1015362193
Los, C. A. (2000). Nonparametric efficiency testing of Asian stock markets using weekly data. Adv. in Econometrics 14 329–363.
Nason, G. P. (1998). Wavethresh3 software. Dept. Mathematics, Univ. Bristol, Bristol, UK.
Nason, G. P. and Sapatinas, T. (2002). Wavelet packet transfer function modelling of nonstationary time series. Statist. Comput. 12 45–56.
Mathematical Reviews (MathSciNet):
MR1877579
Digital Object Identifier: doi:10.1023/A:1013168221710
Nason, G. P. and Silverman, B. W. (1995). The stationary wavelet transform and some statistical applications. In Wavelets and Statistics (A. Antoniadis and G. Oppenheim, eds.) 103 281–299. Springer, New York.
Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of evolutionary wavelet spectra. J. Roy. Statist. Soc. Ser. B 62 271–292.
Mathematical Reviews (MathSciNet):
MR1749539
Digital Object Identifier: doi:10.1111/1467-9868.00231
JSTOR: links.jstor.org
Neumann, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. 17 601–633.
Mathematical Reviews (MathSciNet):
MR1424908
Digital Object Identifier: doi:10.1111/j.1467-9892.1996.tb00295.x
Oh, H.-S., Ammann, C. M., Naveau, P., Nychka, D. and Otto-Bliesner, B. L. (2003). Multi-resolution time series analysis applied to solar irradiance and climate reconstructions. J. Atmos. Solar-Terr. Phys. 65 191–201.
Ombao, H., Raz, J., von Sachs, R. and Guo, W. (2002). The SLEX Model of a non-stationary random process. Ann. Inst. Statist. Math. 54 171–200.
Mathematical Reviews (MathSciNet):
MR1893549
Digital Object Identifier: doi:10.1023/A:1016130108440
Priestley, M. (1965). Evolutionary spectra and non-stationary processes. J. Roy. Statist. Soc. Ser. B 27 204–237.
Mathematical Reviews (MathSciNet):
MR199886
JSTOR: links.jstor.org
Ross, S. M. (1998). A First course in Probability, 3rd ed. Macmillan, New York.
Mathematical Reviews (MathSciNet):
MR732623
Zentralblatt MATH:
0599.60001
Rudzkis, R. (1978). Large deviations for estimates of spectrum of stationary data. Lith. Math. J. 18 214–226.
Mathematical Reviews (MathSciNet):
MR519099
Spokoiny, V. (1998). Estimation of a function with discontinuities via local polynomial fit with an adaptive choice. Ann. Statist. 26 1356–1378.
Mathematical Reviews (MathSciNet):
MR1647669
Digital Object Identifier: doi:10.1214/aos/1024691246
Project Euclid: euclid.aos/1024691246
Spokoiny, V. (2001). Data driven testing the fit of linear models. Math. Methods Statist. 10 465–497.
Mathematical Reviews (MathSciNet):
MR1887343
Spokoiny, V. (2002). Variance estimation for high-dimensional regression models. J. Multivariate Anal. 82 111–133.
Mathematical Reviews (MathSciNet):
MR1918617
Digital Object Identifier: doi:10.1006/jmva.2001.2023
Swanson, N. R. and White, H. (1997). Forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models. Int. J. Forecasting 13 439–461.
Van Bellegem, S. and von Sachs, R. (2004). On adaptive estimation for locally stationary wavelet processes and its applications. Int. J. Wavelets Multiresolut. Inf. Process. 2 545–565.
Mathematical Reviews (MathSciNet):
MR2104894
Digital Object Identifier: doi:10.1142/S0219691304000603
van de Geer, S. (2002). On Hoeffding’s inequality for dependent random variables. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 161–170. Birkhäuser, Boston.
Vidakovic, B. (1999). Statistical Modeling by Wavelets. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR1681904
Zentralblatt MATH:
0924.62032
von Sachs, R., Nason, G. P. and Kroisandt, G. (1997). Adaptive estimation of the evolutionary wavelet spectrum. Technical Report No. 516, Dept. Statistics, Stanford Univ.
Woyte, A., Belmans, R. and Nijs, J. (2007). Fluctuations in instantaneous clearness index: Analysis and statistics. Sol. Energy. To appear.
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