The Annals of Statistics

On Adaptive Estimation in Stationary ARMA Processes

Jens-Peter Kreiss

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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

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

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


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

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


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

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