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

SUMMARY A Cramer-Rao lower bound is computed for estimates of the location, innovations scale and autoregressive parameters for a finite-variance pth-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 efficiency robustness toward heavytailed 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.