The Cramér-Rao bound and robust M-estimates for autoregressions

Abstract

SUMMARY A Cramér-Rao lower bound is computed for estimates of the location, innovations scale and autoregressive parameters for a finite-variance p th-order autoregression. The implication of the bound is that the usual least-squares estimates of all of these parameters have asymptotically the same lack of efficieney robustness toward heavy-tailed innovations distributions as does the sample mean for estimating location. On the other hand, autoregression analogues of Huber's regression M -estimates, with the location estimate obtained from the intercept and autoregressive parameter estimates, are shown to be efficiency robust. The location estimate is also shown to be minimax robust.