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Plug-in estimators for higher-order transition densities in autoregression

Published online by Cambridge University Press:  26 March 2009

Anton Schick  and
Wolfgang Wefelmeyer
Anton Schick
Affiliation:
Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA.
Wolfgang Wefelmeyer
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany; wefelm@math.uni.koeln.de
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Abstract

In this paper we obtain root-n consistency and functional central limit theorems in weighted L1-spaces for plug-in estimators of the two-step transition density in the classical stationary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-based kernel estimator for the unknown innovation density improves on plugging in an unweighted residual-based kernel estimator. These weights are chosen to exploit the fact that the innovations have mean zero. If an efficient estimator for the autoregression parameter is used, then the weighted plug-in estimator for the two-step transition density is efficient. Our approach generalizes to invertible linear processes.

Keywords

Empirical likelihoodOwen estimatorleast dispersed regular estimatorefficient influence functionstochastic expansion of residual-based kernel density estimator
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 135 - 151
Copyright
© EDP Sciences, SMAI, 2009

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