On asymptotically optimal wavelet estimation of trend functions under long-range dependence. (English) Zbl 1235.62124
Summary: We consider data-adaptive wavelet estimation of a trend function in a time series model with strongly dependent Gaussian residuals. Asymptotic expressions for the optimal mean integrated squared error and corresponding optimal smoothing and resolution parameters are derived. Due to adaptation to the properties of the underlying trend function, the approach shows very good performance for smooth trend functions while remaining competitive with minimax wavelet estimation for functions with discontinuities. Simulations illustrate the asymptotic results and finite-sample behavior.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G08 | Nonparametric regression and quantile regression |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 62G20 | Asymptotic properties of nonparametric inference |
| 65C60 | Computational problems in statistics (MSC2010) |
References:
| [1] | Abramovich, F., Sapatinas, T. and Silverman, B.W. (1998). Wavelet thresholding via a Bayesian approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 725-749. · Zbl 0910.62031 · doi:10.1111/1467-9868.00151 |
| [2] | Abry, P. and Veitch, D. (1998). Wavelet analysis of long-range-dependent traffic. IEEE Trans. Inform. Theory 44 2-15. · Zbl 0905.94006 · doi:10.1109/18.650984 |
| [3] | Bardet, J.M., Lang, G., Moulines, E. and Soulier, P. (2000). Wavelet estimator of long-range dependent processes. Stat. Inference Stoch. Process. 3 85-99. · Zbl 1054.62579 · doi:10.1023/A:1009953000763 |
| [4] | Beran, J. (1994). Statistics for Long-Memory Processes . London: Chapman and Hall. · Zbl 0869.60045 |
| [5] | Beran, J. (1995). Maximum likelihood estimation of the differencing parameter for invertible short- and long-memory ARIMA models. J. R. Stat. Soc. Ser. B Stat. Methodol. 57 659-672. · Zbl 0827.62088 |
| [6] | Beran, J., Bhansali, R.J. and Ocker, D. (1998). On unified model selection for stationary and nonstationary short- and long-memory autoregressive processes. Biometrika 85 921-934. · Zbl 1054.62581 · doi:10.1093/biomet/85.4.921 |
| [7] | Beran, J. and Feng, Y. (2002). SEMIFAR models - a semiparametric framework for modelling trends, long-range dependence and nonstationarity. Comput. Statist. Data Anal. 40 393-419. · Zbl 0993.62079 · doi:10.1016/S0167-9473(02)00007-5 |
| [8] | Beran, J. and Feng, Y. (2002). Data driven bandwidth choice for SEMIFAR models. J. Comput. Graph. Statist. 11 690-713. · doi:10.1198/106186002420 |
| [9] | Beran, J. and Feng, Y. (2002). Local polynomial fitting with long memory, short memory and antipersistent errors. Ann. Inst. Statist. Math. 54 291-311. · Zbl 1012.62033 · doi:10.1023/A:1022469818068 |
| [10] | Beran, J., Feng, Y., Ghosh, S. and Sibbertsen, P. (2002) On robust local polynomial estimation with long-memory errors. Int. J. Forecasting 18 227-241. |
| [11] | Brillinger, D. (1994). Uses of cumulants in wavelet analysis. Proc. SPIE 2296 2-18. |
| [12] | Brillinger, D. (1996). Some uses of cumulants in wavelet analysis. Nonparametr. Statist. 6 93-114. · Zbl 0879.62027 · doi:10.1080/10485259608832666 |
| [13] | Cohen, A., Daubechies, I. and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmonic Anal. 1 54-81. · Zbl 0795.42018 · doi:10.1006/acha.1993.1005 |
| [14] | Csörgö, S. and Mielniczuk, J. (1995). Nonparametric regression under long-range dependent normal errors. Ann. Statist. 23 1000-1014. · Zbl 0852.62035 · doi:10.1214/aos/1176324633 |
| [15] | Csörgö, S. and Mielniczuk, J. (1999). Random-design regression under long-range dependent errors. Bernoulli 5 209-224. · Zbl 0946.62084 · doi:10.2307/3318432 |
| [16] | Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766. · Zbl 0703.62091 · doi:10.1214/aos/1176347393 |
| [17] | Daubechies, I. (1992). Ten lectures on wavelets. In CBMS-NSF Regional Conference Series in Applied Mathematics 61 . Philadelphia, PA: SIAM. · Zbl 0776.42018 |
| [18] | Donoho, D.L. and Johnstone, I.M. (1994). Ideal spatial adaption by wavelet shrinkage. Biometrika 81 425-455. · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425 |
| [19] | Donoho, D.L. and Johnstone, I.M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879-921. · Zbl 0935.62041 · doi:10.1214/aos/1024691081 |
| [20] | Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 57 301-369. · Zbl 0827.62035 |
| [21] | Donoho, D.L. and Johnstone, I.M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224. · Zbl 0869.62024 · doi:10.2307/2291512 |
| [22] | Fox, R. and Taqqu, M.S. (1986). Large sample properties of parameter estimates for strongly dependent stationary time series. Ann. Statist. 14 517-532. · Zbl 0606.62096 · doi:10.1214/aos/1176349936 |
| [23] | Gasser, T. and Müller, H.G. (1984). Estimating regression functions and their derivatives by the kernel method. Scand. J. Statist. 11 171-185. · Zbl 0548.62028 |
| [24] | Ghosh, S., Beran, J. and Innes, J. (1997). Nonparametric conditional quantile estimation in the presence of long memory. Student 2 109-117. |
| [25] | Ghosh, S. and Draghicescu, D. (2002). Predicting the distribution function for long-memory processes. Int. J. Forecasting 18 283-290. |
| [26] | Ghosh, S. and Draghicescu, D. (2002). An algorithm for optimal bandwidth selection for smooth nonparametric quantiles and distribution functions. In Statistics in Industry and Technology : Statistical Data Analysis Based on the L 1 -Norm and Related Methods 161-168. Basel, Switzerland: Birkhäuser. · Zbl 1145.62320 · doi:10.1007/978-3-0348-8201-9_13 |
| [27] | Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and application to asymptotical normality of Whittle’s estimate. Probab. Theory Related Fields 86 87-104. · Zbl 0717.62015 · doi:10.1007/BF01207515 |
| [28] | Hall, P. and Hart, J.D. (1990). Nonparametric regression with long-range dependence. Stochastic Process. Appl. 36 339-351. · Zbl 0713.62048 · doi:10.1016/0304-4149(90)90100-7 |
| [29] | Hall, P. and Patil, P. (1995). Formulae for mean integated squared error of non-linear wavelet-based density estimators. Ann. Statist. 23 905-928. · Zbl 0839.62042 · doi:10.1214/aos/1176324628 |
| [30] | Hall, P. and Patil, P. (1996a). On the choice of smoothing parameter, threshold and truncation in nonparametric regression by nonlinear wavelet methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 58 361-377. · Zbl 0853.62033 |
| [31] | Hall, P. and Patil, P. (1996b). Effect of threshold rules on performance of wavelet-based curve estimators. Statist. Sinica 6 331-345. · Zbl 0843.62039 |
| [32] | Johnstone, I.M. (1999). Wavelet threshold estimators for correlated data and inverse problems: Adaptivity results. Statist. Sinica 9 51-83. · Zbl 1065.62519 |
| [33] | Johnstone, I.M. and Silverman, B.W. (1997). Wavelet threshold estimators for data with correlated noise. J. R. Stat. Soc. Ser. B Methodol. 59 319-351. · Zbl 0886.62044 · doi:10.1111/1467-9868.00071 |
| [34] | Li, L. and Xiao, Y. (2007). Mean integrated squared error of nonlinear wavelet-based estimators with long memory data. Ann. Inst. Statist. Math. 59 299-324. · Zbl 1332.62134 · doi:10.1007/s10463-006-0048-6 |
| [35] | Nason, G.P. (1996). Wavelet shrinkage using cross-validation. J. R. Stat. Soc. Ser. B Stat. Methodol. 58 463-479. · Zbl 0853.62034 |
| [36] | Percival, D.P. and Walden, A.T. (2000). Wavelet Methods for Time Series Analysis . Cambridge: Cambridge Univ. Press. · Zbl 0963.62079 |
| [37] | Ray, B.K. and Tsay, R.S. (1997). Bandwidth selection for kernel regression with long-range dependent errors. Biometrika 84 791-802. · Zbl 1090.62536 · doi:10.1093/biomet/84.4.791 |
| [38] | Robinson, P.M. (1997). Large sample inference for nonparametric regression with dependent errors. Ann. Statist. 25 2054-2083. · Zbl 0882.62039 · doi:10.1214/aos/1069362387 |
| [39] | Truong, Y.K. and Patil, P.N. (2001). Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series. Ann. Inst. Statist. Math. 53 159-178. · Zbl 0995.62092 · doi:10.1023/A:1017928823619 |
| [40] | von Sachs, R. and Macgibbon, B. (2000). Non-parametric curve estimation by wavelet thresholding with locally stationary errors. Scand. J. Statist. 27 475-499. · Zbl 0976.62034 · doi:10.1111/1467-9469.00202 |
| [41] | Taqqu, M.S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab. Theory Related Fields 31 287-302. · Zbl 0303.60033 · doi:10.1007/BF00532868 |
| [42] | Vidakovic, B. (1999). Statistical Modeling by Wavelets . New York: Wiley. · Zbl 0924.62032 |
| [43] | Wang, Y. (1996). Function estimation via wavelet shrinkage for long-memory data. Ann. Statist. 24 466-484. · Zbl 0859.62042 · doi:10.1214/aos/1032894449 |
| [44] | Yajima, Y. (1985). On estimation of long-memory time series models. Austral. J. Statist. 27 303-320. · Zbl 0584.62142 · doi:10.1111/j.1467-842X.1985.tb00576.x |
| [45] | Yang, Y. (2001). Nonparametric regression with dependent errors. Bernoulli 7 633-655. · Zbl 1006.62041 · doi:10.2307/3318730 |
| [46] | Zorich, V.A. (2004). Mathematical Analysis 1 . Berlin: Springer. · Zbl 1071.00003 |
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.
