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 (β₁ , . . . , β_p) when the scale and the intercept parameters are unknown. The minimax-variance estimates of (β₁ , . . . , β_p) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ∈-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.