M ‐Estimates of regression when the scale is unknown and the error distribution is possibly asymmetric: A minimax result

Huber (1964) found the minimax‐variance M ‐estimate of location under the assumption that the scale parameter is known; Li and Zamar (1991) extended this result to the case when the scale is unknown. We consider the robust estimation of the regression coefficients (β 1 ,…,β p ) when the scale and the intercept parameters are unknown. The minimax‐variance estimates of (β 1 ,…,β p ) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ϵ‐contamination neighbourhoods of a central regression model with Gaussian errors (asymmetric contamination is allowed), and the minimum is taken over a large class of generalized M ‐estimates of regression of the Mallow type. The optimal choice of estimates for the nuisance parameters (scale and intercept) is also considered.