The Annals of Statistics

On Adaptive Estimation in Stationary ARMA Processes

Jens-Peter Kreiss

Full-text: Open access

Abstract

We consider the estimation problem for the parameter $\vartheta_0$ of a stationary ARMA $(p, q)$ process, with independent and identically, but not necessary normally distributed errors. First we prove local asymptotic normality (LAN) for this model. Then we construct locally asymptotically minimax (LAM) estimators, which asymptotically achieve the smallest possible covariance matrix. Utilizing these, we finally obtain strongly adaptive estimators, by using usual kernel estimators for the score function $\dot{\varphi} = -f'/2 f$, where $f$ denotes the density of the error distribution. These estimates turn out to be asymptotically optimal in the LAM sense for a wide class of symmetric densities $f$.

Article information

Source
Ann. Statist., Volume 15, Number 1 (1987), 112-133.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350256

Digital Object Identifier
doi:10.1214/aos/1176350256

Mathematical Reviews number (MathSciNet)
MR885727

Zentralblatt MATH identifier
0616.62042

JSTOR
links.jstor.org

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62M09: Non-Markovian processes: estimation 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators

Keywords
ARMA process log-likelihood ratio locally asymptotically normal locally asymptotically minimax adaptive estimates kernel estimators

Citation

Kreiss, Jens-Peter. On Adaptive Estimation in Stationary ARMA Processes. Ann. Statist. 15 (1987), no. 1, 112--133. doi:10.1214/aos/1176350256. https://projecteuclid.org/euclid.aos/1176350256


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