On approximate robust confidence distributions

A confidence distribution is a complete tool for making frequentist inference for a parameter of interest based on an assumed parametric model. Indeed, it provides point estimates, along with confidence intervals, allows to define rejection regions for testing unilateral and bilateral hypotheses, to assign measures of evidence or levels of confidence to prespecified regions of the parameter space, and to compare the parameter of interest with other parameters from other studies. The aim is to discuss robust confidence distributions derived from unbiased M-estimating functions, which provide bounded-influence inference for a parameter of interest, when the assumed central model is just an approximate parametric model or in the presence of deviant values in the observed data. Paralleling likelihood-based results and extending results available for robust scoring rules, two methods are proposed for deriving robust confidence distributions: the first one uses the asymptotic theory of robust pivotal quantities and the second one is based on simulation methods. An application and a simulation study are illustrated in the context of non-inferiority and superiority testing.