Weakly Adaptive Estimators in Explosive Autoregression

Abstract

Consider the model $X_i = \rho X_{i - 1} + \varepsilon_i, |\rho| > 1$, where $X_0, \varepsilon_1, \varepsilon_2, \cdots$ are independent random variables with $\varepsilon_1, \varepsilon_2, \cdots$ having common density $\psi$. This paper gives sufficient conditions under which the sequence of experiments induced by $\{X_0, X_1, \cdots, X_n\}$ has a weak limit in the sense of Le Cam. In general, the limiting experiment is translation invariant and neither LAN nor LAMN. The paper further shows that the sequence of Pitman-type estimators of $\rho$ at a given $\psi$ converges weakly to $T$, where $T$ is minimax for the limiting experiment under a weighted squared error loss function. Finally, for an unknown $\psi$, a sequence of Pitman-type estimators that converges weakly to $T$ is constructed. These estimators are called weakly adaptive. The class of error densities for which these results hold include some that may not have finite Fisher information.