Issue ESAIM: PS Volume 13, 2009 Page(s) 135 - 151 DOI 10.1051/ps:2008001 Published online 26 March 2009

ESAIM: PS, March 2009, Vol. 13, p. 135-151
DOI: 10.1051/ps:2008001

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

Anton Schick¹ and Wolfgang Wefelmeyer<sup>2

1</sup>  Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA.
²  Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany; wefelm@math.uni.koeln.de

Received July 24, 2007. Revised November 6, 2007. Published online 26 March 2009

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.

Mathematics Subject Classification. 62M05, 62M10

Key words: Empirical likelihood, Owen estimator, least dispersed regular estimator, efficient influence function, stochastic expansion of residual-based kernel density estimator


© EDP Sciences, SMAI 2009

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