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Asymptotic normality of wavelet estimators of the memory parameter for linear processes

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  • F. Roueff
  • M. S. Taqqu

Abstract

. We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi‐parametrically using wavelets from a sample X1,…, Xn of the process. We treat both the log‐regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size n → ∞ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the scalogram for linear processes, conveniently centred and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast to quadratic forms computed on the basis of Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size n → ∞.

Suggested Citation

  • F. Roueff & M. S. Taqqu, 2009. "Asymptotic normality of wavelet estimators of the memory parameter for linear processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(5), pages 534-558, September.
  • Handle: RePEc:bla:jtsera:v:30:y:2009:i:5:p:534-558
    DOI: 10.1111/j.1467-9892.2009.00627.x
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    References listed on IDEAS

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    1. Stoev, Stilian & Taqqu, Murad S. & Park, Cheolwoo & Michailidis, George & Marron, J.S., 2006. "LASS: a tool for the local analysis of self-similarity," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2447-2471, May.
    2. E. Moulines & F. Roueff & M. S. Taqqu, 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, Wiley Blackwell, vol. 28(2), pages 155-187, March.
    3. Roueff, F. & Taqqu, M.S., 2009. "Central limit theorems for arrays of decimated linear processes," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 3006-3041, September.
    4. Faÿ, Gilles & Moulines, Eric & Roueff, François & Taqqu, Murad S., 2009. "Estimators of long-memory: Fourier versus wavelets," Journal of Econometrics, Elsevier, vol. 151(2), pages 159-177, August.
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    Cited by:

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    2. Gannaz, Irène, 2023. "Asymptotic normality of wavelet covariances and multivariate wavelet Whittle estimators," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 485-534.
    3. Roueff, François & von Sachs, Rainer, 2011. "Locally stationary long memory estimation," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 813-844, April.
    4. Kei Nanamiya, 2014. "Modelling For The Wavelet Coefficients Of Arfima Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(4), pages 341-356, July.
    5. Patrice Abry & B. Cooper Boniece & Gustavo Didier & Herwig Wendt, 2023. "Wavelet eigenvalue regression in high dimensions," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 1-32, April.
    6. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.

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