Higher-order infinitesimal robustness

Using the von Mises expansion, we study the higher-order infinitesimal robustness of a general M-functional and characterize its second-order properties. We show that second-order robustness is equivalent to the boundedness of both the estimator's estimating function and its derivative with respect to the parameter. It implies, at the same time, (i) variance-robustness and (ii) robustness of higher-order saddlepoint approximations to the estimator's finite sam- ple density. The proposed construction of second-order robust M-estimators is fairly general and potentially useful in a variety of relevant settings. Besides the theoretical contributions, we discuss the main computational issues and provide an algorithm for the implementation of second-order robust M-estimators. Finally, we illustrate our theory by Monte Carlo simulation and in a real-data estimation of the maximal losses of Nikkei 225 index returns. Our findings indicate that second-order robust estimators can improve on other widely-applied robust esti- mators, in terms of efficiency and robustness, for moderate to small sample sizes and in the presence of deviations from ideal parametric models.