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Biometrika 1982 69(2):437-442; doi:10.1093/biomet/69.2.437
© 1982 by Biometrika Trust
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The Cramér-Rao bound and robust M-estimates for autoregressions

R. DOUGLAS MARTIN

Department of Statistics, University of Washington Seattle, Washington, U.S.A.

A Cramér-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 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.

Key Words: Austoregression • Cramér-Rao bound • Efficiency robustness • Information matrix • Innovations • M-estimate • Minimax robustness • Qualitative robustness • Robust estimation • Time series


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