When a unique $M$-estimate exists, its density is obtained as a corollary to a more general theorem which asserts that under mild conditions the intensity function of the point process of solutions of the estimating equations exists and is given by the density of the estimating function standardized by multiplying it by the inverse of its derivative. We apply the results to give a result for Huber’s proposal 2 applied to regression and scale estimates. We also give a saddlepoint approximation for the density and use this to give approximations for tail areas for smooth functions of the $M$-estimates.
"The density of multivariate $M$-estimates." Ann. Statist. 28 (1) 275 - 297, February 2000. https://doi.org/10.1214/aos/1016120373