On Adaptive Estimation in Stationary ARMA Processes

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 $f$.