Weakly adaptive estimators in explosive autoregression
type: Item
instance of: scholarly article
title: Weakly adaptive estimators in explosive autoregression
zbMATH Open document ID: 0709.62083
author: Hira L. Koul
author: G. Ch. Pflug
published in: The Annals of Statistics
publication date: 1990-01-01T00:00:00Z
review text: Let \(X(t)=\alpha \cdot X(t-1)+\epsilon (t)\), \(t\geq 1\), be an AR[1]- process with \(| \alpha | >1\). The \(\epsilon\) (t) are independent and have a common unknown density \(\psi\). The paper establishes a set of sufficient conditions on \(\psi\) under which the log-likelihood process \[ \Lambda_ n(t| \psi)=\log (dP^ n_{\alpha +\delta (n)\cdot t,\psi}/dP^ n_{\alpha,\psi}) \] has a nondegenerate limit \(\Lambda\) (t\(| \psi)\) when the shrinking factor \(\delta\) (n) is suitably choosen. This limit is neither LAN nor LAMN in general. Under some additional moment conditions the sequence of weakly adaptive Pitman-type estimators \[ {\hat \alpha}_ n=\int u\prod^{n}_{i=k+1}{\hat \psi}_ n(X_ i-uX_{i-1})du/\int \prod^{n}_{i=k+1}{\hat \psi}_ n(X_ i-uX_{i-1})du, \] where \({\hat \psi}\) is a kernel type density estimate, is shown to be asymptotically minimax for the limit \(\Lambda\) (t\(| \psi)\) under a weighted squared error loss function.
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62F12
zbMATH DE Number: 4166410
zbMATH Keywords: weakly adaptive estimators
zbMATH Keywords: explosive autoregression
zbMATH Keywords: minimax estimators
zbMATH Keywords: nonstationary
zbMATH Keywords: translation invariant
zbMATH Keywords: log-likelihood process
zbMATH Keywords: moment conditions
zbMATH Keywords: sequence of weakly adaptive Pitman-type estimators
zbMATH Keywords: asymptotically minimax
zbMATH Keywords: weighted squared error loss
reviewed by: Rainer Schlittgen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176347634
OpenAlex ID: W1981273527
DOI: 10.1214/AOS/1176347634
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description: scientific article; zbMATH DE number 4166410