Semiparametric estimation of the long-range parameter. (English) Zbl 1047.62083
Summary: We study two estimators of the long-range parameter of a covariance stationary linear process. We show that one of the estimators achieves the optimal semiparametric rate of convergence, whereas the other has a rate of convergence as close as desired to the optimal rate. Moreover, we show that the estimators are asymptotically normal with a variance, which does not depend on any unknown parameter, smaller than others suggested in the literature. Finally, a small Monte Carlo study is included to illustrate the finite sample relative performance of our estimators compared to other suggested semiparametric estimators. More specifically, the Monte-Carlo experiment shows the superiority of the proposed estimators in terms of the mean squared error.
MSC:
| 62M15 | Inference from stochastic processes and spectral analysis |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G05 | Nonparametric estimation |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
References:
| [1] | Brillinger, D. R. (1981).Time Series: Data Analysis and Theory, McGraw-Hill, New York. · Zbl 0486.62095 |
| [2] | Chen, Z.-G. and Hannan, E. J. (1980). The distribution of periodogram ordinates,J. Time Ser. Anal.,1, 73–82. · Zbl 0499.62085 · doi:10.1111/j.1467-9892.1980.tb00301.x |
| [3] | Cheung, Y.-W. and Diebold, F. X. (1994). On maximum-likelihood estimation of the differencing parameter of fractionally-integrated noise with unknown mean,J. Econometrics,62, 317–350. · doi:10.1016/0304-4076(94)90026-4 |
| [4] | Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes,Ann. Statist.,17, 1749–1766. · Zbl 0703.62091 · doi:10.1214/aos/1176347393 |
| [5] | Davies, R. B. and Harte, D. S. (1987). Tests for Hurst effect,Biometrica,74, 95–101. · Zbl 0612.62123 · doi:10.1093/biomet/74.1.95 |
| [6] | Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series,Ann. Statist.,14, 517–532. · Zbl 0606.62096 · doi:10.1214/aos/1176349936 |
| [7] | Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long-memory time series models,J. Time Ser. Anal.,4, 221–238. · Zbl 0534.62062 · doi:10.1111/j.1467-9892.1983.tb00371.x |
| [8] | Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle’s estimate,Probab. Theory Related Fields,86, 87–104. · Zbl 0717.62015 · doi:10.1007/BF01207515 |
| [9] | Giraitis, L., Robinson, P. M. and Samarov, A. (1997). Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence,J. Time Ser. Anal.,18, 49–60. · Zbl 0870.62073 · doi:10.1111/1467-9892.00038 |
| [10] | Hidalgo, J. (1996). Spectral analysis for bivariate time series with long memory,Econometric Theory,12, 773–792. · doi:10.1017/S0266466600007155 |
| [11] | Hidalgo, J. (2002). Semiparametric estimation for stationary processes whose spectra have an unknown pole (preprint). |
| [12] | Hidalgo, J. and Robinson, P. M. (2002). Adapting to unknown disturbance autocorrelation in regression with long memory,Econometrica,70, 1545–1581. · Zbl 1141.62340 · doi:10.1111/1468-0262.00341 |
| [13] | Hosoya, Y. (1997). A limit theory with long-range dependence and statistical inference on related models,Ann. Statist.,25, 105–137. · Zbl 0873.62096 · doi:10.1214/aos/1034276623 |
| [14] | Hurvich, C. M. and Beltrao, K. I. (1993). Asymptotics for the low-frequency ordinates of the periodogram of a long memory time series,J. Time Ser. Anal.,14, 455–472. · Zbl 0782.62086 · doi:10.1111/j.1467-9892.1993.tb00157.x |
| [15] | Künsch, H. R. (1986). Discrimination between monotonic trends and long-range dependence,J. Appl. Probab.,23, 1025–1030. · Zbl 0623.62085 · doi:10.2307/3214476 |
| [16] | Künsch, H. R. (1987) Statistical aspects of self-similar processes,Proceedings of the 1st World Congress of the Bernoulli Society, 67–74, VNU Science Press, Utrecht. · Zbl 0673.62073 |
| [17] | Lobato, I. and Robinson, P. M. (1996). Average periodogram estimation of long memory,J. Econometrics,73, 303–324. · Zbl 0854.62088 · doi:10.1016/0304-4076(95)01742-9 |
| [18] | Parzen, E. (1986). Quantile spectral analysis and long-memory time series,J. Appl. Probab., Special Vol.23A, 41–54. · Zbl 0586.62150 · doi:10.2307/3214341 |
| [19] | Robinson, P. M. (1994). Semiparametric analysis of long-memory time series,Ann. Statist.,22, 515–539. · Zbl 0795.62082 · doi:10.1214/aos/1176325382 |
| [20] | Robinson, P. M. (1995a). Log-periodogram regression of time series with long range dependence,Ann. Statist.,23, 1048–1072. · Zbl 0838.62085 · doi:10.1214/aos/1176324636 |
| [21] | Robinson, P. M. (1995b). Gaussian semiparametric estimation of long range dependence,Ann. Statist.,23, 1630–1661. · Zbl 0843.62092 · doi:10.1214/aos/1176324317 |
| [22] | Stout, W. F. (1974).Almost Sure Convergence, Academic Press, New York. · Zbl 0321.60022 |
| [23] | Taqqu, M. S. (1975). Weak convergence to fractional Brownian to the Rosenblatt process,Z. Wahrsch. Verw. Gebiete,31, 203–238. · Zbl 0303.60033 · doi:10.1007/BF00532868 |
| [24] | 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 |
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