Min-max asymptotic variance of M-estimates of location when scale is unknown

This paper extends Huber's (1964) min-max result to the case when the scale parameter is unknown and must be estimated along with the location parameter. A min-max problem in which nature chooses F from a family F of symmetric distribution functions around a given location-scale central model, the statistician chooses an M-estimate of location, that is, specifies the influence curve or score function ψ and the auxiliary scale estimate s_n, is solved. The optimal choise for s_n is an M-estimate of scale applied to the residuals about the median. The optimal choice for the score function ψ is a truncated and rescaled maximum likelihood score function for the central model. In he Gaussian case rescaling is not necessary and so, except for the truncation point which is now smaller, Huber's (1964) classical result obtains.