  In practical data analysis under noisy environment, it is common to first use
robust methods to identify outliers, and then to conduct further analysis after
removing the outliers. In this paper, we consider statistical inference of the
model estimated after outliers are removed, which can be interpreted as a
selective inference (SI) problem. To use conditional SI framework, it is
necessary to characterize the events of how the robust method identifies
outliers. Unfortunately, the existing methods cannot be directly used here
because they are applicable to the case where the selection events can be
represented by linear/quadratic constraints. In this paper, we propose a
conditional SI method for popular robust regressions by using homotopy method.
We show that the proposed conditional SI method is applicable to a wide class
of robust regression and outlier detection methods and has good empirical
performance on both synthetic data and real data experiments.
