New neighborhoods in robustness and least favorable pairs for special capacities

Abstract

We propose a new neighborhood of distributions for robust inference, which is generated by a certain special capacity with three constants. It covers not only the familiar neighborhoods defined in terms of ε-contamination, total variation and their combination, but also their intersections. It is shown that the neighborhood is also obtained from contaminating some neighborhood which consists of absolutely continuous distributions. We derive the explicit forms of Radon-Nikodym derivatives and the least favorable pairs for such special capacities and neighborhoods on the real line, which generalize the earlier results of Huber and Rieder for minimax testing, and reinforce the general results of Bednarski.

Keywords:

• Robustness
• special capacity
• Radon-Nikodym derivative
• least favorable pair
• minimax test

MSC 2020 Subject classifications:

• Primary 62G35
• Primary Secondary 62C20
• Secondary 62F35

Acknowledgments

The authors thank the anonymous reviewers for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).