Locally adaptive estimation of evolutionary wavelet spectra

Annals of Statistics

Volumn 36, Issue 4, 2008, Pages 1879-1924

  Van Bellegem, Sébastien ^a   Von Sachs, Rainer ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Autocor relation wavelet; Change point; Local stationarity; Nonstationary time series; Pointwise adaptive estimation; Quadratic form; Regularization; Wavelet spectrum

Indexed keywords

EID: 51049117978     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-AOS524     Document Type: Article

References (36)

1. BRILLINGER, D. (1975). Time Series. Data Analysis and Theory. Holt, Rinehart and Winston, New York. MR0443257
2. BROWN, L. D. and LOW, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 2524-2535. MR1425965
3. COMTE, F. (2001). Adaptive estimation of the spectrum of a stationary Gaussian sequence. Bernoulli 7 267-298. MR1828506
4. DAHLHAUS, R. (1988). Empirical spectral process and their applications to time series analysis. Stochastic Process. Appl. 30 69-83. MR0968166
5. DAHLHAUS, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37. MR 1429916
6. DAHLHAUS, R. (2000). A likelihood approximation for locally stationary processes. Ann. Statist. 28 1762-1794. MR1835040
7. 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. MR1958786
8. DAHLHAUS, R. and POLONIK, W. (2006). Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes. Ann. Statist. 34 2790-2824. MR2329468
9. DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. MR11621.07
10. FRYŹLEWICZ, P. and NASON, G. P. (2006). Haar-Fisz estimation of evolutionary wavelet spectra. J. Roy. Statist. Soc. Ser. B 68 611-634. MR2301011
11. 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. MR2028615
12. HÄRDLE, W., KERKYACHARIAN, G., PICARD, D. and TSYBAKOV, A. (1998). Wavelets, Approximations, and Statistical Applications. Springer, New York. MR1618204
13. LAURENT, B. and MASS ART, P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 1302-1338. MR1805785
14. LEPSKI, O. (1990). On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454-170. MR1091202
15. LEPSKI, O. and SPOKOINY, V. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512-2546. MR1604408
16. LII, K.-S. and ROSENBLATT, M. (2002). Spectral analysis for harmonizable processes. Ann. Statist. 30 258-297. MR 1892664
17. LOS, C. A. (2000). Nonparametric efficiency testing of Asian stock markets using weekly data. Adv. in Econometrics 14 329-363.
18. NASON, G. P. (1998). Wavethresh3 software. Dept. Mathematics, Univ. Bristol, Bristol, UK.
19. NASON, G. P. and SAPATINAS, T. (2002). Wavelet packet transfer function modelling of nonstationary time series. Statist. Comput. 12 45-56. MR1877579
20. 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.
21. 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. MR1749539
22. NEUMANN, M. H. (1996). Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series. J. Time Ser. Anal. 17 601-633. MR 1424908
23. 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.
24. 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. MR 1893549
25. PRIESTLEY, M. (1965). Evolutionary spectra and non-stationary processes. J. Roy. Statist. Soc. Ser. B 27 204-237. MR0199886
26. ROSS, S. M. (1998). A First course in Probability, 3rd ed. Macmillan, New York. MR0732623
27. RUDZKIS, R. (1978). Large deviations for estimates of spectrum of stationary data. Lith. Math. J. 18 214-226. MR0519099
28. SPOKOINY, V. (1998). Estimation of a function with discontinuities via local polynomial fit with an adaptive choice. Ann. Statist. 26 1356-1378. MR1647669
29. SPOKOINY, V. (2001). Data driven testing the fit of linear models. Math. Methods Statist. 10 465-497. MR1887343
30. SPOKOINY, V. (2002). Variance estimation for high-dimensional regression models. J. Multivariate Anal. 82 111-133. MR1918617
31. 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.
32. 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. MR2104894
33. 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.
34. VIDAKOVIC, B. (1999). Statistical Modeling by Wavelets. Wiley, New York. MR1681904
35. VON SACHS, R., NASON, G. P. and KROISANDT, G. (1997). Adaptive estimation of the evolution-ary wavelet spectrum. Technical Report No. 516, Dept. Statistics, Stanford Univ.
36. WOYTE, A., BELMANS, R. and NUS, J. (2007). Fluctuations in instantaneous clearness index: Analysis and statistics. Sol. Energy. To appear.