On adaptive estimation in stationary ARMA processes
type: Item
instance of: scholarly article
title: On adaptive estimation in stationary ARMA processes
zbMATH Open document ID: 0616.62042
author: Jens-Peter Kreiß
published in: The Annals of Statistics
publication date: 1987-01-01T00:00:00Z
review text: The paper deals with adaptive construction of locally asymptotically minimax (LAM) estimators for stationary ARMA processes with independent and identically, but not necessarily normally distributed innovations. First the local asymptotic normality (LAN) for this model is proved using the sufficient conditions for LAN given by \textit{G. G. Roussas} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 31-46 (1979; Zbl 0377.62011)]. Then a construction of LAM estimators is suggested if only \(\sqrt{n}\)- consistent initial estimators are available. As these estimates can depend on the distribution of the innovations strongly adaptive estimators are suggested finally which are optimal in LAM sense for a wide class of symmetric innovation distributions. The kernel estimators for the score function -f'/2f (f is the density of the innovation distribution) are used for this purpose. The suggested construction of the adaptive estimators generalizes the results of \textit{R. Beran} [Ann. Inst. Stat. Math. 28, 77-89 (1976; Zbl 0362.62093)] and \textit{P. J. Bickel} [Ann. Stat. 10, 647-671 (1982; Zbl 0489.62033)]. A comparative simulation study concludes the paper.
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3998989
zbMATH Keywords: autoregressive moving average
zbMATH Keywords: optimality criterion
zbMATH Keywords: log-likelihood ratio
zbMATH Keywords: locally asymptotically minimax
zbMATH Keywords: LAM
zbMATH Keywords: stationary ARMA processes
zbMATH Keywords: local asymptotic normality
zbMATH Keywords: LAN
zbMATH Keywords: strongly adaptive estimators
zbMATH Keywords: kernel estimators
zbMATH Keywords: score function
zbMATH Keywords: innovation distribution
zbMATH Keywords: simulation study
reviewed by: Tomáš Cipra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350256
OpenAlex ID: W2056609565
DOI: 10.1214/AOS/1176350256
description: scientific article; zbMATH DE number 3998989