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Media type:
E-Article
Title:
Weakly Adaptive Estimators in Explosive Autoregression
Contributor:
Koul, Hira L.
imprint:
Institute of Mathematical Statistics, 1990
Published in:The Annals of Statistics
Language:
English
ISSN:
0090-5364
Origination:
Footnote:
Description:
<p>Consider the model<tex-math>$X_i = \rho X_{i - 1} + \varepsilon_i, |\rho| > 1$</tex-math>, where X<sub>0</sub>, ε<sub>1</sub>, ε<sub>2</sub>, ⋯ are independent random variables with ε<sub>1</sub>, ε<sub>2</sub>, ⋯ having common density ψ. This paper gives sufficient conditions under which the sequence of experiments induced by {X<sub>0</sub>, X<sub>1</sub>, ⋯, X<sub>n</sub>} 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 ρ at a given ψ converges weakly to T, where T is minimax for the limiting experiment under a weighted squared error loss function. Finally, for an unknown ψ, 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.</p>