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Title: Joint Asymptotic Properties of Stopping Times and Sequential Estimators for Stationary First-order Autoregressive Models
Authors: Hitomi, Kohtaro
Nagai, Keiji
Nishiyama, Yoshihiko
Tao, Junfan
Keywords: 62M10
Observed Fisher information
joint asymptotic normality
functional central limit theorem in D[0, ∞)
Anscombe's Theorem
Issue Date: Jun-2021
Publisher: Institute of Economic Research, Kyoto University
Journal title: KIER Discussion Paper
Volume: 1060
Start page: 1
End page: 17
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.
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Appears in Collections:KIER Discussion Paper (English)

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