Higher-order robustness

The higher–order robustness for M–estimators is introduced and defined. The conditions needed to ensure higher stability of the asymptotic bias are provided by refining the Von Mises bias expansion. Admissible M–estimators featuring second–order robustness are thus introduced. Then, a saddle-point argument is applied in order to approximate the finite sample distribution of second–order robust M–estimators. The link between the stability of this approximation and the second–order robustness is explored. Monte Carlo simulation provides evidence that second–order robust M–estimators perform better than the MLE and Huber–type estimators, even in moderate to small sample sizes and/or for large amounts of contamination.