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Joint Asymptotic Properties of Stopping Times and Sequential Estimators for Stationary First-order Autoregressive Models

Author

Listed:
  • Kohtaro Hitomi

    (Kyoto Institute of Technology)

  • Keiji Nagai

    (Yokohama National University)

  • Yoshihiko Nishiyama

    (Institute of Economic Research, Kyoto University)

  • Junfan Tao

    (JSPS International Research Fellow (Kyoto University), Institute of Economic Research, Kyoto University)

Abstract

Currently, because online data is abundant and can be collected more easily , people often face the problem of making correct statistical decisions as soon as possible. If the online data is sequentially available, sequential analysis is appropriate for handling such a problem. We consider the joint asymptotic properties of stopping times and sequential estimators for stationary first-order autoregressive (AR(1)) processes under independent and identically distributed errors with zero mean and finite variance. Using the stopping times introduced by Lai and Siegmund (1983) for AR(1), we investigate the joint asymptotic properties of the stopping times, the sequential least square estimator (LSE), and the estimator of σ². The functional central limit theorem for nonlinear ergodic stationary processes is crucial for obtaining our main results with respect to their asymptotic properties. We found that the sequential least square estimator and stopping times exhibit joint asymptotic normality. When σ² is estimated, the joint limiting distribution degenerates and the asymptotic variance of the stopping time is strictly smaller than that of the stopping time with a known σ².

Suggested Citation

  • Kohtaro Hitomi & Keiji Nagai & Yoshihiko Nishiyama & Junfan Tao, 2021. "Joint Asymptotic Properties of Stopping Times and Sequential Estimators for Stationary First-order Autoregressive Models," KIER Working Papers 1060, Kyoto University, Institute of Economic Research.
    • Handle: RePEc:kyo:wpaper:1060
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    References listed on IDEAS

    as
    1. Shete, Sanjay & Sriram, T. N., 1998. "Fixed precision estimator of the offspring mean in branching processes," Stochastic Processes and their Applications, Elsevier, vol. 77(1), pages 17-33, September.
    2. Keiji Nagai & Yoshihiko Nishiyama & Kohtaro Hitomi, 2018. "Sequential test for unit root in AR(1) model," KIER Working Papers 1003, Kyoto University, Institute of Economic Research.
    3. Galtchouk, L. & Konev, V., 2004. "On uniform asymptotic normality of sequential least squares estimators for the parameters in a stable AR(p)," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 119-142, November.
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    Cited by:

    1. Jianwei Jin & Keiji Nagai, 2022. "Sequential unit root test for first-order autoregressive processes with initial values," KIER Working Papers 1085, Kyoto University, Institute of Economic Research.
    2. Kohtaro Hitomi & Jianwei Jin & Keiji Nagai & Yoshihiko Nishiyama & Junfan Tao, 2022. "Unit root tests considering initial values and a concise method for computing powers," KIER Working Papers 1084, Kyoto University, Institute of Economic Research.

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    More about this item

    Keywords

    Observed Fisher information; joint asymptotic normality; functional central limit theorem in D[0; ∞); Anscombebe's Theorem;
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