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Spectral estimation for non-linear long range dependent discrete time trawl processes. (English) Zbl 1451.62099

A discrete time trawl process \(X = {X_k}\), \(k \in \mathbb Z\), is defined by \[ X_k \sum^\infty_{j=0} \gamma_{k-j} (a_j ), \, k \in \mathbb{Z}. \] Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence \((a_j )_j \in \mathbb N\) and defined through a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process \((\gamma(t))\), \( t \in \mathbb R\), called the seed process. They provide various examples of trawl processes. Although the usual causal linear models for long range dependence (such as ARFIMA processes) constitute specific examples of trawl processes, they here focus on the nonlinear models, and specify simple sufficient conditions implying the assumptions used in the general results. The proofs of the results are presented. Finally, they present numerical experiments focusing on the estimation of the long memory parameter \(d^{*}\) comparing their approach to the more classical local Whittle estimator, which is known to perform well for standard linear models.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
62F12 Asymptotic properties of parametric estimators
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:

[1] Barndorff-Nielsen, O. E., Lunde, A., Shepard, N., and Veraart, A. E. D. (2014). Integer-valued trawl processes: a class of stationary infinitely divisible processes., Scand. J. Statist., 41:693-724. · Zbl 1309.60033
[2] Dalla, V., Giraitis, L., and Hidalgo, J. (2005). Consistent estimation of the memory parameter for nonlinear time series., J. of Time Series Analysis, 211-251:87-104. · Zbl 1115.62084
[3] Dedecker, J., Doukhan, P., Lang, G., León, J. R., Louhichi, S., and Prieur, C. (2007)., Weak dependence: With Examples and Applications, volume 190 of Lecture Notes in Statistics. Springer. · Zbl 1165.62001
[4] Doukhan, P., Jakubowski, A., Lopes, S. R. C., and Surgailis, D. (2019). Discrete-time trawl processes., Stochastic Process. Appl., 129(4):1326-1348. · Zbl 1488.60243
[5] Doukhan, P., Oppenheim, G., and Taqqu, M. S. (2002)., Theory and Applications of Long-Range Dependence. Birkhäuser, Boston.
[6] Fay, G., Roueff, F., and Soulier, P. (2007). Estimation of the memory parameter of the infinite-source Poisson process., Bernoulli, 13(2):473-491. · Zbl 1127.62070
[7] Giraitis, L., Koul, H. L., and Surgailis, D. (2012)., Large Sample Inference for Long Memory Processes. Imperial College Press, London. · Zbl 1279.62016
[8] Hannan, E. J. (1973). The asymptotic theory of linear time-series models., J. Appl. Probability, 10:130-145, corrections, ibid. 10 (1973), 913. · Zbl 0269.62075
[9] Hurvich, C. M., Moulines, E., and Soulier, P. (2005). Estimating long memory in volatility., Econometrica, 73(4):1283-1328. · Zbl 1151.91702
[10] Petrov, V. V. (1995)., Limit Theorems of Probability Theory, volume 4 of Oxford Studies in Probability. The Clarendon Press, Oxford University Press, New York. Sequences of independent random variables, Oxford Science Publications. · Zbl 0826.60001
[11] Pipiras, V. and Taqqu, M. S. (2017)., Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. · Zbl 1377.60005
[12] Robinson, P. · Zbl 0843.62092
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