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Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

Author

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  • Kaiyu Liang

    (School of Mathematic, Jilin University, Changchun 130012, China)

  • Yong Zhang

    (School of Mathematic, Jilin University, Changchun 130012, China)

Abstract

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Suggested Citation

  • Kaiyu Liang & Yong Zhang, 2024. "Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes," Mathematics, MDPI, vol. 12(10), pages 1-16, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1482-:d:1391826
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    References listed on IDEAS

    as
    1. Cheng, Fuxia, 2015. "Strong consistency of the distribution estimator in the nonlinear autoregressive time series," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 41-47.
    2. K. S. Chan & H. Tong, 1986. "On Estimating Thresholds In Autoregressive Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 7(3), pages 179-190, May.
    3. Yan Wang & Mingzhi Mao & Xiaohua Hu & Tingting He, 2014. "The Law of Iterated Logarithm for Autoregressive Processes," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-8, November.
    4. Yong Zhang, 2016. "An extension of almost sure central limit theorem for self-normalized products of sums for mixing sequences," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(22), pages 6625-6640, November.
    5. Jian-Feng Yao, 2000. "On Least Squares Estimation for Stable Nonlinear AR Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 316-331, June.
    6. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
    7. Yong Zhang, 2020. "Further research on limit theorems for self-normalized sums," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(2), pages 385-402, January.
    8. Jie Li, 2014. "Asymptotics of the Lp-Norms of Density Estimators in the Nonlinear Autoregressive Models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(22), pages 4845-4855, November.
    9. Cheng, Fuxia & Sun, Shuxia, 2008. "A goodness-of-fit test of the errors in nonlinear autoregressive time series models," Statistics & Probability Letters, Elsevier, vol. 78(1), pages 50-59, January.
    10. Liebscher, Eckhard, 2003. "Strong convergence of estimators in nonlinear autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 247-261, February.
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