M estimators of location for Gaussian and related processes with slowly decaying serial correlations

Abstract

We investigate the behavior of M estimators of the location parameter for stochastic processes with long-range dependence. The processes considered are Gaussian or one-dimensional transformations of Gaussian processes. It turns out that, up to a constant, all M estimators are asymptotically equivalent to the arithmetic mean. For Gaussian processes this constant is always equal to one, independently of the ψ function. In view of the case of iid observations, the results are surprising. They are related to earlier work by Gastwirth and Rubin. Some simulations illustrate the results.