Asymptotic variance of M-estimators for dependent Gaussian random variables

This paper discusses the asymptotic behavior of M-estimators for dependent Gaussian random variables. We show that for a Gaussian distribution, the asymptotic variance of an M-estimator of scale is minimal in the independent case and must necessarily increase for dependent data. This is not true for location estimation where the asymptotic variance can increase or decrease for dependent observations, depending on the sign of the correlation. Several examples are analyzed, showing that the asymptotic variance of the maximum likelihood estimator varies widely under dependencies.