Higher-Order Infinitesimal Robustness

Abstract

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 sample 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 estimators, in terms of efficiency and robustness, for moderate to small sample sizes and in the presence of deviations from ideal parametric models. Supplementary materials for this article are available online.

Keywords:

• Generalized extreme value distribution
• M-estimator
• Robustness
• Saddlepoint
• von Mises Expansion

Acknowledgments

The authors thank the Editor, the Associate Editor, an anonymous Referee, M. Hallin, F. Peracchi, J. Fan, J. Kolassa, and seminar participants at Princeton University, Rutgers University, HEC Paris, Monash University, and at the Winter Meeting of the Econometric Society in Rome, for useful comments that improved an earlier version of the article. Financial support by the Swiss National Science Foundation Pro*Doc Program 114533 and NCCR-FinRisk (National Centers for Competence in Research-Financial Evaluation and Risk Management) is gratefully acknowledged.