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Adaptive semiparametric wavelet estimator and goodness-of-fit test for long-memory linear processes. (English) Zbl 1295.62082

Summary: This paper is first devoted to the study of an adaptive wavelet-based estimator of the long-memory parameter for linear processes in a general semiparametric frame. As such this is an extension of the previous contribution of J.-M. Bardet et al. [Bernoulli 14, No. 3, 691–724 (2008; Zbl 1155.62060)] which only concerned Gaussian processes. Moreover, the definition of the long-memory parameter estimator has been modified and the asymptotic results are improved even in the Gaussian case. Finally an adaptive goodness-of-fit test is also built and easy to be employed: it is a chi-square type test. Simulations confirm the interesting properties of consistency and robustness of the adaptive estimator and test.

MSC:

62M07 Non-Markovian processes: hypothesis testing
62M09 Non-Markovian processes: estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
60F05 Central limit and other weak theorems

Citations:

Zbl 1155.62060
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Full Text: DOI arXiv Euclid

References:

[1] Abry, P., Flandrin, P., Taqqu, M.S. and Veitch, D. (2003). Self-similarity and long-range dependence through the wavelet lens, In Doukhan, P., Oppenheim, G. and Taqqu, M.S. (Editors), Theory and applications of long-range dependence , 527-556, Birkhäuser, Boston. · Zbl 1029.60028
[2] Andrews, D.W.K. and Sun, Y. (2004). Adaptive local polynomial Whittle estimation of long-range dependence, Econometrica , 72, 569-614. · Zbl 1131.62317 · doi:10.1111/j.1468-0262.2004.00501.x
[3] Bardet, J.M. and Bertrand, P. (2007) Definition, properties and wavelet analysis of multiscale fractional Brownian motion, Fractals 15, 73-87. · Zbl 1142.60329 · doi:10.1142/S0218348X07003356
[4] Bardet, J.M., Bibi, H. and Jouini, A. (2008). Adaptive wavelet-based estimator of the memory parameter for stationary Gaussian processes, Bernoulli , 14, 691-724. · Zbl 1155.62060 · doi:10.3150/07-BEJ6151
[5] Bardet, J.M., Billat V. and Kammoun, I. (2012). A new process for modeling heartbeat signals during exhaustive run with an adaptive estimator of its fractal parameters, J. of Applied Statistics , 39, 1331-1351. · doi:10.1080/02664763.2011.646962
[6] Bardet, J.M., Lang, G., Moulines, E. and Soulier, P. (2000). Wavelet estimator of long range-dependant processes, Statist. Infer. Stochast. Processes , 3, 85-99. · Zbl 1054.62579 · doi:10.1023/A:1009953000763
[7] Bardet, J.M., Lang, G., Oppenheim, G., Philippe, A., Stoev, S. and Taqqu, M.S. (2003). Semiparametric estimation of the long-range dependence parameter: a survey. In, Theory and applications of long-range dependence , Birkhäuser Boston, 557-577. · Zbl 1032.62077
[8] Bhansali, R. Giraitis L. and Kokoszka, P.S. (2006). Estimation of the memory parameter by fitting fractionally-differenced autoregressive models, J. Multivariate Analysis , 97, 2101-2130. · Zbl 1101.62073 · doi:10.1016/j.jmva.2006.01.003
[9] Doukhan, P., Oppenheim, G. and Taqqu M.S. (Editors) (2003)., Theory and applications of long-range dependence , Birkhäuser. · Zbl 1005.00017
[10] Fefferman, C. (1973). Pointwise convergence of Fourier series., Ann. of Math. , 98, 551-571. · Zbl 0268.42009 · doi:10.2307/1970917
[11] Furmanczyk, K. (2007). Some remarks on the central limit theorem for functionals of linear processes under short-range dependence., Probab. Math. Statist. , 27, 235-245. · Zbl 1154.60019
[12] Giraitis, L. (1985). The central limit theorem for functionals of a linear process., Lith. Mat. J. , 25, 43-57. · Zbl 0568.60020 · doi:10.1007/BF00966294
[13] 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, 49-61. · Zbl 0870.62073 · doi:10.1111/1467-9892.00038
[14] Giraitis, L., Robinson P.M., and Samarov, A. (2000). Adaptive semiparametric estimation of the memory parameter., J. Multivariate Anal. , 72, 183-207. · Zbl 1065.62513 · doi:10.1006/jmva.1999.1865
[15] Henry, M. and Robinson, P.M. (1996). Bandwidth choice in Gaussian semiparametric estimation of long range dependence. In: Athens Conference on Applied Probability and Time Series Analysis, Vol. II, 220-232, Springer, New, York.
[16] Hurvich, C.M., Moulines, E. and Soulier, P. (2002). The FEXP estimator for potentially non-stationary linear time series., Stochastic Process. Appl. , 97, 307-340. · Zbl 1057.62074 · doi:10.1016/S0304-4149(01)00136-3
[17] Iouditsky, A., Moulines, E. and Soulier, P. (2001). Adaptive estimation of the fractional differencing coefficient., Bernoulli , 7, 699-731. · Zbl 1006.62082 · doi:10.2307/3318538
[18] Künsch, H. (1987). Statistical aspects of self-similar processes. Proceedings of the 1st World Congress of the Bernoulli Society, 67-74, VNU Sci. Press, Utrecht. · Zbl 0673.62073
[19] Moulines, E., Roueff, F. and Taqqu, M.S. (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, J. Time Ser. Anal. , 28, 155-187. · Zbl 1150.62058 · doi:10.1111/j.1467-9892.2006.00502.x
[20] Moulines, E. and Soulier, P. (1999). Broadband log-periodogram regression of time series with long-range dependence, Ann. Statist. , 27, 1415-1439. · Zbl 0962.62085 · doi:10.1214/aos/1017938932
[21] Moulines, E. and Soulier, P. (2003). Semiparametric spectral estimation for fractionnal processes, In Doukhan, P., Oppenheim, G. and Taqqu, M.S. (Editors), Theory and applications of long-range dependence , 251-301, Birkhäuser, Boston. · Zbl 1109.62353
[22] Peltier, R. and Lévy Véhel, J. (1995) Multifractional Brownian Motion: definition and preliminary results. Preprint INRIA, available on, . · Zbl 0867.94005 · doi:10.1142/S0218348X95000679
[23] 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
[24] Robinson, P.M. (1995b). Gaussian semiparametric estimation of long range dependence, Ann. Statist. , 23, 1630-1661. · Zbl 0843.62092 · doi:10.1214/aos/1176324317
[25] Roueff, F. and Taqqu, M.S. (2009a). Asymptotic normality of wavelet estimators of the memory parameter for linear processes., J. Time Ser. Anal. , 30, 534-558. · Zbl 1224.62068 · doi:10.1111/j.1467-9892.2009.00627.x
[26] Roueff, F. and Taqqu, M.S. (2009b). Central limit theorems for arrays of decimated linear processes., Stochastic Process. Appl. , 119, 3006-3041 · Zbl 1173.60311 · doi:10.1016/j.spa.2009.03.009
[27] Samorodnitsky G. and Taqqu, M.S. (1994). Stable non-Gaussian random processes, London, U.K., Chapman &, Hall. · Zbl 0925.60027
[28] Veitch, D., Abry, P. and Taqqu, M.S. (2003). On the Automatic Selection of the Onset of Scaling, Fractals , 11, 377-390. · Zbl 1056.62102 · doi:10.1142/S0218348X03002099
[29] Wu, W.B. (2002). Central limit theorems for functionals of linear processes and their applications., Statist. Sinica , 12, 635-649. · Zbl 1018.60022
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