Asymptotic expansions are derived for the densities of multivariate $M$-estimates. The expansion is based on a saddlepoint technique and yields good accuracy in the tails for small sample sizes. Numerical results are given for robust estimation of location and scale and these are compared with known Monte Carlo results.
Christopher Field. "Small Sample Asymptotic Expansions for Multivariate $M$-Estimates." Ann. Statist. 10 (3) 672 - 689, September, 1982. https://doi.org/10.1214/aos/1176345864