Adaptive Estimation in Noncausal Stationary AR Processes

Abstract

We consider the estimation problem of the parameter $b$ of a stationary AR($p$) process without any of the usual causality assumptions. The aim of the paper is to derive asymptotic minimax bounds for estimators of $b$. When the distribution of the noise is known, we show LAN properties of the model and derive locally asymptotically minimax (LAM) estimators. The most important results are about the case of unknown distribution: The main result shows that, if one uses the usual parametrization, these bounds depend heavily on the causality or the noncausality of the process, so that adaptive efficient estimation is impossible in the noncausal situation: The scaling factor is shown to give the hardest one-dimensional subproblem, and an unusual scaling is exhibited that could lead to adaptive efficient estimation of the rescaled parameter even in the noncausal case.