MODELLING FOR THE WAVELET COEFFICIENTS OF ARFIMA PROCESSES
We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical...
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| Vydáno v: | Journal of time series analysis Ročník 35; číslo 4; s. 341 - 356 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Blackwell Publishing Ltd
01.07.2014
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| ISSN: | 0143-9782, 1467-9892 |
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| Abstract | We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale. |
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| AbstractList | We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long-range dependent processes, which many authors have explained in semi-parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non-decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale. [PUBLICATION ABSTRACT] We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale. We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long-range dependent processes, which many authors have explained in semi-parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐ ;decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale. Reprinted by permission of Blackwell Publishers |
| Author | Nanamiya, Kei |
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| Cites_doi | 10.1109/34.192463 10.2307/1390751 10.1111/j.1467-9892.1995.tb00221.x 10.1109/18.992817 10.1016/S0165-1889(99)00010-X 10.1109/18.391246 10.1214/07-AOS527 10.1109/18.119751 10.1017/CBO9780511841040 10.1111/j.1467-9892.2009.00627.x 10.1109/18.650984 10.1111/1467-9892.00157 10.2202/1558-3708.1051 10.1023/A:1009953000763 10.1093/biomet/68.1.165 10.1109/TSP.2005.851111 10.1016/j.jeconom.2009.03.005 10.1111/j.1467-9892.1980.tb00297.x 10.1111/j.1467-9892.2006.00502.x |
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| References_xml | – reference: Anderson TW. 1971. The Statistical Analysis of Time Series. New York: Wiley. – reference: Abry P, Veitch D. 1998. Wavelet analysis of long-range-dependent traffic. IEEE Transactions on Information Theory 44: 2-15. – reference: Moulines E, Roueff F, Taqqu MS. 2008. A wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series. The Annals of Statistics 36: 1925-1956. – reference: Craigmile PF, Guttorp P, Percival DB. 2005. Wavelet-based parameter estimation for polynomial contaminated fractionally differenced processes. IEEE Transactions on Signal Processing 53: 3151-3161. – reference: Bardet JM, Lang G, Moulines E, Soulier P. 2000. Wavelet estimator of long-range dependent processes. Statistical Inference for Stochastic Processes 3: 85-99. – reference: Flandrin P. 1992. Wavelet analysis and synthesis of fractional Brownian motion. IEEE Transactions on Information Theory 38: 910-917. – reference: McCoy EJ, Walden AT. 1996. Wavelet analysis and synthesis of stationary long-memory processes. Journal of Computational and Graphical Statistics 5: 26-56. – reference: Mallat SG. 1989. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11: 647-693. – reference: Granger CWJ, Joyeux R. 1980. An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1: 15-29. – reference: Hosking JRM. 1981. Fractional differencing. Biometrika 68: 165-176. – reference: Moulines E, Roueff F, Taqqu MS. 2007. On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter. Journal of Time Series Analysis 28: 155-187. – reference: Krim H, Pesquet JC. 1995. Multiresolution analysis of a class of nonstationary processes. IEEE Transactions on Information Theory 41: 1010-1020. – reference: Hurvich CM, Ray BK. 1995. Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. Journal of Time Series Analysis 16: 17-41. – reference: Kato T, Masry E. 1999. On the spectral density of the wavelet transform of fractional Brownian motion. Journal of Time Series Analysis 20: 559-563. – reference: Percival DB, Walden AT. 2000. Wavelet Methods for Time Series Analysis. Cambridge: Cambridge University Press. – reference: Roueff F, Taqqu MS. 2009. Asymptotic normality of wavelet estimators of the memory parameter for linear processes. Journal of Time Series Analysis 30: 534-558. – reference: Faÿ G, Moulines E, Roueff F, Taqqu MS. 2009. Estimators of long-memory: Fourier versus wavelets. Journal of Econometrics 151: 159-177. – reference: Jensen MJ. 2000. An alternative maximum likelihood estimator of long-memory processes using compactly supported wavelets. Journal of Economic Dynamics and Control 24: 361-387. – reference: Yaglom AM. 1958. Correlation theory of processes with random stationary nth increments. American Mathematical Society Translations 8: 87-141. – reference: Bardet JM. 2002. Statistical study of the wavelet analysis of fractional Brownian motion. IEEE Transactions on Information Theory 48: 991-999. – reference: Jensen MJ. 1999. An approximate wavelet MLE of short- and long-memory parameters. Studies in Nonlinear Dynamics & Econometrics 3: 239-253. – volume: 151 start-page: 159 year: 2009 end-page: 177 article-title: Estimators of long‐memory: Fourier versus wavelets publication-title: Journal of Econometrics – volume: 38 start-page: 910 year: 1992 end-page: 917 article-title: Wavelet analysis and synthesis of fractional Brownian motion publication-title: IEEE Transactions on Information Theory – volume: 3 start-page: 239 year: 1999 end-page: 253 article-title: An approximate wavelet MLE of short‐ and long‐memory parameters publication-title: Studies in Nonlinear Dynamics & Econometrics – volume: 28 start-page: 155 year: 2007 end-page: 187 article-title: On the spectral density of the wavelet coefficients of long memory time series with application to the log‐regression estimation of the memory parameter publication-title: Journal of Time Series Analysis – volume: 20 start-page: 559 year: 1999 end-page: 563 article-title: On the spectral density of the wavelet transform of fractional Brownian motion publication-title: Journal of Time Series Analysis – volume: 8 start-page: 87 year: 1958 end-page: 141 article-title: Correlation theory of processes with random stationary th increments publication-title: American Mathematical Society Translations – volume: 44 start-page: 2 year: 1998 end-page: 15 article-title: Wavelet analysis of long‐range‐dependent traffic publication-title: IEEE Transactions on Information Theory – volume: 36 start-page: 1925 year: 2008 end-page: 1956 article-title: A wavelet Whittle estimator of the memory parameter of a non‐stationary Gaussian time series publication-title: The Annals of Statistics – volume: 68 start-page: 165 year: 1981 end-page: 176 article-title: Fractional differencing publication-title: Biometrika – volume: 3 start-page: 85 year: 2000 end-page: 99 article-title: Wavelet estimator of long‐range dependent processes publication-title: Statistical Inference for Stochastic Processes – volume: 16 start-page: 17 year: 1995 end-page: 41 article-title: Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes publication-title: Journal of Time Series Analysis – volume: 5 start-page: 26 year: 1996 end-page: 56 article-title: Wavelet analysis and synthesis of stationary long‐memory processes publication-title: Journal of Computational and Graphical Statistics – year: 2000 – volume: 11 start-page: 647 year: 1989 end-page: 693 article-title: A theory for multiresolution signal decomposition: the wavelet representation publication-title: IEEE Transactions on Pattern Analysis and Machine Intelligence – volume: 48 start-page: 991 year: 2002 end-page: 999 article-title: Statistical study of the wavelet analysis of fractional Brownian motion publication-title: IEEE Transactions on Information Theory – volume: 53 start-page: 3151 year: 2005 end-page: 3161 article-title: Wavelet‐based parameter estimation for polynomial contaminated fractionally differenced processes publication-title: IEEE Transactions on Signal Processing – year: 1971 – volume: 1 start-page: 15 year: 1980 end-page: 29 article-title: An introduction to long‐memory time series models and fractional differencing publication-title: Journal of Time Series Analysis – volume: 24 start-page: 361 year: 2000 end-page: 387 article-title: An alternative maximum likelihood estimator of long‐memory processes using compactly supported wavelets publication-title: Journal of Economic Dynamics and Control – volume: 41 start-page: 1010 year: 1995 end-page: 1020 article-title: Multiresolution analysis of a class of nonstationary processes publication-title: IEEE Transactions on Information Theory – volume: 30 start-page: 534 year: 2009 end-page: 558 article-title: Asymptotic normality of wavelet estimators of the memory parameter for linear processes publication-title: Journal of Time Series Analysis – ident: e_1_2_6_16_1 doi: 10.1109/34.192463 – ident: e_1_2_6_17_1 doi: 10.2307/1390751 – ident: 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e_1_2_6_7_1 doi: 10.1016/j.jeconom.2009.03.005 – ident: e_1_2_6_9_1 doi: 10.1111/j.1467-9892.1980.tb00297.x – ident: e_1_2_6_18_1 doi: 10.1111/j.1467-9892.2006.00502.x |
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| Title | MODELLING FOR THE WAVELET COEFFICIENTS OF ARFIMA PROCESSES |
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