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June, 1990 Weakly Adaptive Estimators in Explosive Autoregression
Hira L. Koul, Georg Ch. Pflug
Ann. Statist. 18(2): 939-960 (June, 1990). DOI: 10.1214/aos/1176347634


Consider the model $X_i = \rho X_{i - 1} + \varepsilon_i, |\rho| > 1$, where $X_0, \varepsilon_1, \varepsilon_2, \cdots$ are independent random variables with $\varepsilon_1, \varepsilon_2, \cdots$ having common density $\psi$. This paper gives sufficient conditions under which the sequence of experiments induced by $\{X_0, X_1, \cdots, X_n\}$ has a weak limit in the sense of Le Cam. In general, the limiting experiment is translation invariant and neither LAN nor LAMN. The paper further shows that the sequence of Pitman-type estimators of $\rho$ at a given $\psi$ converges weakly to $T$, where $T$ is minimax for the limiting experiment under a weighted squared error loss function. Finally, for an unknown $\psi$, a sequence of Pitman-type estimators that converges weakly to $T$ is constructed. These estimators are called weakly adaptive. The class of error densities for which these results hold include some that may not have finite Fisher information.


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Hira L. Koul. Georg Ch. Pflug. "Weakly Adaptive Estimators in Explosive Autoregression." Ann. Statist. 18 (2) 939 - 960, June, 1990.


Published: June, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0709.62083
MathSciNet: MR1056345
Digital Object Identifier: 10.1214/aos/1176347634

Primary: 62G05
Secondary: 62G10, 62G20

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 2 • June, 1990
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