Robust Estimation of Location in a Gaussian Parametric Model: II

This paper extends the results of Andrews (1984) which considers the problem of robust estimation of location in a model with stationary strong mixing Gaussian parametric distributions. Three neighbourhood systems are considered, each of which contains the Hellinger neighbourhoods used in Andrews (1984). Optimal robust estimators for this dependent random variable model are found to be bounded influence estimators with optimal psi functions which are very nearly of Huber shape. These estimators are quite robust against different “amounts” of dependence, and against lack of dependence. To generate the optimal estimators a minimax asymptotic risk criterion is used, where minimaxing is done over neighbourhoods of the parametric Gaussian distributions. The neighbourhood systems include distributions of strong mixing processes. They allow for deviations from stationarity and from the Gaussian structure of dependence. In addition, deviations from the normal univariate parametric distributions are allowed within the neighbourhoods defined by (i) εn-contamination, (ii) variational metric distance, and (iii) Kolmogorov metric distance.