A comparison of Hurst exponent estimators in long-range dependent curve time series. (English) Zbl 1494.62019
Summary: The Hurst exponent is the simplest numerical summary of self-similar long-range dependent stochastic processes. We consider the estimation of Hurst exponent in long-range dependent curve time series. Our estimation method begins by constructing an estimate of the long-run covariance function, which we use, via dynamic functional principal component analysis, in estimating the orthonormal functions spanning the dominant sub-space of functional time series. Within the context of functional autoregressive fractionally integrated moving average (ARFIMA) models, we compare finite-sample bias, variance and mean square error among some time- and frequency-domain Hurst exponent estimators and make our recommendations.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62M15 | Inference from stochastic processes and spectral analysis |
| 62G32 | Statistics of extreme values; tail inference |
Keywords:
curve process; dynamic functional principal component analysis; functional ARFIMA; long-run covariance; long-range dependenceReferences:
| [1] | Andrews, D. 1991. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica 59 (3): 817-58, doi:10.1016/0304-4076(95)01788-7. · Zbl 0878.62103 · doi:10.1016/0304-4076(95)01788-7 |
| [2] | Andrews, D. W. K., and J. C. Monahan. 1992. “An improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator.” Econometrica 60 (4): 953-66, doi:10.2307/2951574. · Zbl 0778.62103 · doi:10.2307/2951574 |
| [3] | Aue, A., and J. Klepsch. 2017. Estimating Functional Time Series by Moving Average Model Fitting, Technical Report. Davies: University of California. https://arxiv.org/abs/1701.00770. |
| [4] | Aue, A., D. D. Norinho, and S. Hörmann. 2015. “On the Prediction of Stationary Functional Time Series.” Journal of the American Statistical Association 110 (509): 378-92, doi:10.1080/01621459.2014.909317. · Zbl 1373.62462 · doi:10.1080/01621459.2014.909317 |
| [5] | Beran, J. 1994. Statistics for Long Memory Processes. New York: Chapman & Hall/CRC Press. · Zbl 0869.60045 |
| [6] | Beran, J., Y. Feng, S. Ghosh, and R. Kulik. 2013. Long-Memory Processes: Probabilistic Properties and Statistical Methods. Berlin, Heidelberg: Springer. · Zbl 1282.62187 |
| [7] | Berkes, I., L. Horváth, and G. Rice. 2016. “On the Asymptotic Normality of Kernel Estimators of the Long Run Covariance of Functional Time Series.” Journal of Multivariate Analysis 144: 150-75, doi:10.1016/j.jmva.2015.11.005. · Zbl 1360.62450 · doi:10.1016/j.jmva.2015.11.005 |
| [8] | Bosq, D. 2000. Linear Processes in Function Spaces. New York: Lecture notes in Statistics. · Zbl 0971.62023 |
| [9] | Bosq, D., and D. Blanke. 2007. Inference and Prediction in Large Dimensions. Chichester: John Wiley & Sons. · Zbl 1183.62157 |
| [10] | Chen, S. X., L. Lei, and Y. Tu. 2016. “Functional Coefficient Moving Average Model with Applications to Forecasting Chinese CPI.” Statistica Sinica 26 (4): 1649-72, doi:10.5705/ss.2014.196t. · Zbl 1356.62136 · doi:10.5705/ss.2014.196t |
| [11] | Chiou, J.-M., and H.-G. Müller. 2009. “Modeling Hazard Rates as Functional Data for the Analysis of Cohort Lifetables and Mortality Forecasting.” Journal of the American Statistical Association 104 (486): 572-85, doi:10.1198/jasa.2009.0023. · doi:10.1198/jasa.2009.0023 |
| [12] | Dahlhaus, R. 1989. “Efficient Parameter Estimation for Self-Similar Processes.” The Annals of Statistics 17 (4): 1749-66, doi:10.1214/aos/1176347393. · Zbl 0703.62091 · doi:10.1214/aos/1176347393 |
| [13] | Doukhan, P., G. Oppenheim, and M. Taqqu, eds (2003). Theory and Applications of Long-Range Dependence. Boston, MA: Birkhäuser. · Zbl 1005.00017 |
| [14] | Embrechts, P., and M. Maejima. 2002. Selfsimiliar Processes. Princeton: Princeton University Press. · Zbl 1008.60003 |
| [15] | Fox, R., and M. S. Taqqu. 1986. “Large-sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series.” The Annals of Statistics 14 (2): 517-32. · Zbl 0606.62096 |
| [16] | Geweke, J., and S. Porter-Hudak. 1983. “The Estimation and Application of Long Memory Time Series Models.” Journal of Time Series Analysis 4 (4): 221-37, doi:10.1111/j.1467-9892.1983.tb00371.x. · Zbl 0534.62062 · doi:10.1111/j.1467-9892.1983.tb00371.x |
| [17] | Higuchi, T. 1988. “Approach to an Irregular Time Series on the Basis of the Fractal Theory.” Physica D 31 (2): 277-83, doi:10.1016/0167-2789(88)90081-4. · Zbl 0649.58046 · doi:10.1016/0167-2789(88)90081-4 |
| [18] | Hörmann, S., L. Kidziński, and M. Hallin. 2015. “Dynamic Functional Principal Components.” Journal of the Royal Statistical Society: Series B 77 (2): 319-48. · Zbl 1414.62133 |
| [19] | Horváth, L., P. Kokoszka, and R. Reeder. 2013. “Estimation of the Mean of Functional Time Series and A Two Sample Problem.” Journal of the Royal Statistical Society: Series B 75 (1): 103-22, doi:10.1111/j.1467-9868.2012.01032.x. · Zbl 07555440 · doi:10.1111/j.1467-9868.2012.01032.x |
| [20] | Hou, J., and P. Perron. 2014. “Modified Local Whittle Estimator for Long Memoy Processes in the Presence of Low Frequency (and other) Contaminations.” Journal of Econometrics 182: 309-28, doi:10.1016/j.jeconom.2014.05.004. · Zbl 1311.62137 · doi:10.1016/j.jeconom.2014.05.004 |
| [21] | Hurst, H. E. 1951. “Long-term Storage Capacity of Reservoirs.” Transactions of the American Society of Civil Engineers 116 (1): 770-99. |
| [22] | Hyndman, R. J., and H. L. Shang. 2009. “Forecasting Functional Time Series (with Discussion).” Journal of the Korean Statistical Society 38 (3): 199-221, doi:10.1016/j.jkss.2009.06.002. · Zbl 1293.62267 · doi:10.1016/j.jkss.2009.06.002 |
| [23] | Hyndman, R. J., and M. Ullah. 2007. “Robust Forecasting of Mortality and Fertility Rates: A Functional Data Approach.” Computational Statistics and Data Analysis 51 (10): 4942-56, doi:10.1016/j.csda.2006.07.028. · Zbl 1162.62434 · doi:10.1016/j.csda.2006.07.028 |
| [24] | Jensen, M. J. 1999. “Using Wavelets to Obtain a Consistent Ordinary Least Squares Estimator of the Long-Memory Parameter.” Journal of Forecasting 18 (1): 17-32. |
| [25] | Jeong, H.-D. J., J.-S. R. Lee, D. McNickle, and K. Pawlikowski. 2007. “Comparison of Various Estimators in Simulated FGN.” Simulation and Modelling Practice and Theory 15 (9): 1173-91, doi:10.1016/j.simpat.2007.08.004. · doi:10.1016/j.simpat.2007.08.004 |
| [26] | Klepsch, J., and C. Klüppelberg. 2017. “An Innovations Algorithm for the Prediction of Functional Linear Processes.” Journal of Multivariate Analysis 155: 252-71, doi:10.1016/j.jmva.2017.01.005. · Zbl 1397.62346 · doi:10.1016/j.jmva.2017.01.005 |
| [27] | Klepsch, J., C. Klüppelberg, and T. Wei. 2017. “Prediction of Functional ARMA Processes with an Application to Traffic Data.” Econometrics and Statistics 1: 128-49, doi:10.1016/j.ecosta.2016.10.009. · doi:10.1016/j.ecosta.2016.10.009 |
| [28] | Kokoszka, P., and M. Reimherr. 2013. “Determining the Order of the Functional Autoregressive Model.” Journal of Time Series Analysis 34 (1): 116-29, doi:10.1111/j.1467-9892.2012.00816.x. · Zbl 1274.62600 · doi:10.1111/j.1467-9892.2012.00816.x |
| [29] | Kokoszka, P., and M. Reimherr. 2017. Introduction to Functional Data Analysis. Boca Raton: Chapman & Hall/CRC Press. · Zbl 1411.62004 |
| [30] | Künsch, H. R. 1987. “Statistical Aspects of Self-Similar Processes.” In Proceedings of the World Congress of the Bernoulli Society, Vol. 1, Tashkent, 67-74. · Zbl 0673.62073 |
| [31] | Li, D., P. M. Robinson, and H. L. Shang. 2019. “Long-range Dependent Curve Time Series.” Journal of the American Statistical Association: Theory and Methods. doi:10.1080/01621459.2019.1604362. · Zbl 1445.62231 · doi:10.1080/01621459.2019.1604362 |
| [32] | Liu, X., H. Xiao, and R. Chen. 2016. “Convolutional Autoregressive Models for Functional Time Series.” Journal of Econometrics 194 (2): 263-82, doi:10.1016/j.jeconom.2016.05.006. · Zbl 1443.62277 · doi:10.1016/j.jeconom.2016.05.006 |
| [33] | Mandelbrot, B. B. 1963. “The Variation of Certain Speculative Prices.” The Journal of Business 36 (4): 394-419. |
| [34] | Mandelbrot, B. B. 1975. “Limit Theorems on the Self-Normalized Range for Weakly and Strongly Dependent Processes.” Probability Theory and Related Fields 31 (4): 271-85, doi:10.1007/BF00534968. · Zbl 0288.60033 · doi:10.1007/BF00534968 |
| [35] | Mandelbrot, B. B., and M. S. Taqqu. 1979. “Robust R/S Analysis of Long Run Serial Correlation.” In Proceedings of the 42nd Session of the International Statistical Institute, Manila, 69-104. · Zbl 0518.62036 |
| [36] | Mandelbrot, B. B., and J. W. van Ness. 1968. “Fractional Brownian Motions, Fractional Noises and applications.” SIAM Review 10 (4): 422-37. · Zbl 0179.47801 |
| [37] | Mandelbrot, B. B., and J. R. Wallis. 1969. “Robustness of the Rescaled Range R/S in the Measurement of Noncyclic Long Run Statistical Dependence.” Water Resources Research 5 (5): 967-88, doi:10.1029/WR005i005p00967. · doi:10.1029/WR005i005p00967 |
| [38] | Palma, W. 2007. Long-Memory Time Series: Theory and Methods. Hoboken, New Jersey: Wiley. · Zbl 1183.62153 |
| [39] | Panaretos, V. M., and S. Tavakoli. 2012. “Fourier Analysis of Stationary Time Series in Function Space.” Annals of Statistics 41 (2): 568-603, doi:10.1214/13-AOS1086. · Zbl 1267.62094 · doi:10.1214/13-AOS1086 |
| [40] | Peng, C. K., S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger. 1994. “Mosaic Organization of DNA Nucleotides.” Physical Review E 49 (2): 1685-9, doi:10.1103/physreve.49.1685. · doi:10.1103/physreve.49.1685 |
| [41] | Reisen, V. A. 1994. “Estimation of the Fractional Difference Parameter in the ARIMA (p, d,q) Model Using the Smoothed Periodogram.” Journal of Time Series Analysis 15 (3): 335-0, doi:10.1111/j.1467-9892.1994.tb00198.x. · Zbl 0803.62084 · doi:10.1111/j.1467-9892.1994.tb00198.x |
| [42] | Rice, G., and H. L. Shang. 2017. “A Plug-in Bandwidth Selection Procedure for Long-Run Covariance Estimation with Stationary Functional Time Series.” Journal of Time Series Analysis 38 (4): 591-609, doi:10.1111/jtsa.12229. · Zbl 1367.62094 · doi:10.1111/jtsa.12229 |
| [43] | Robinson, P. M. 1995a. “Gaussian Semiparametric Estimation of Long Range Dependence.” The Annals of Statistics 23 (5): 1630-61. · Zbl 0843.62092 |
| [44] | Robinson, P. M. 1995b. “Log-periodogram Regression of Time Series with Long Range Dependence.” The Annals of Statistics 23 (3): 1048-72. · Zbl 0838.62085 |
| [45] | Robinson, P. M., ed. (2003). Time Series with Long Memory. Oxford: Oxford University Press. · Zbl 1113.62106 |
| [46] | Shimotsu, K. 2010. “Exact Local Whittle Estimation of Fractional Integration with Unknown Mean and Time Trend.” Econometric Theory 26: 501-40. · Zbl 1185.62163 |
| [47] | Shimotsu, K., and P. C. B. Phillips. 2005. “Exact Local Whittle Estimation of Fractional Integration.” The Annals of Statistics 33 (4): 1890-933. · Zbl 1081.62069 |
| [48] | Taqqu, M., V. Teverovsky, and W. Willinger. 1995. “Estimators for Long-Range Dependence: An Empirical Study.” Fractals 3 (4): 785-98. · Zbl 0864.62061 |
| [49] | Teverovsky, V., and M. Taqqu. 1997. “Testing for Long-Range Dependence in the Presence of Shifting Means or A Slowly Declining Trend, Using A Variance-Type Estimator.” Journal of Time Series Analysis 18 (3): 279-304, doi:10.1111/1467-9892.00050. · Zbl 0934.62094 · doi:10.1111/1467-9892.00050 |
| [50] | Velasco, C. 1999. “Gaussian Semiparametric Estimation of Non-Stationary Time Series.” Journal of Time Series Analysis 20 (1): 87-127, doi:10.1111/1467-9892.00127. · Zbl 0922.62093 · doi:10.1111/1467-9892.00127 |
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.
