Robust confidence distributions from proper scoring rules

Abstract

A confidence distribution is a distribution for a parameter of interest based on a parametric statistical model. As such, it serves the same purpose for frequentist statisticians as a posterior distribution for Bayesians, since it allows to reach point estimates, to assess their precision, to set up tests along with measures of evidence, to derive confidence intervals, comparing the parameter of interest with other parameters from other studies, etc. A general recipe for deriving confidence distributions is based on classical pivotal quantities and their exact or approximate distributions. However, in the presence of model misspecifications or outlying values in the observed data, classical pivotal quantities, and thus confidence distributions, may be inaccurate. The aim of this paper is to discuss the derivation and application of robust confidence distributions. In particular, we discuss a general approach based on the Tsallis scoring rule in order to compute a robust confidence distribution. Examples and simulation results are discussed for some problems often encountered in practice, such as the two-sample heteroschedastic comparison, the receiver operating characteristic curves and regression models.

Keywords:

• M-estimators
• pivotal quantity
• robustness
• scoring rule
• Tsallis score

Acknowledgments

The authors acknowledge the associate editor and the anonymous referee for the useful comments that greatly improved the paper.

Disclosure statement

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

Additional information

Funding

This research work was partially supported by the University of Padova (BIRD197903) and by MIUR (PRIN 2015, grant 2015EASZFS_003).