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

Conditional Selective Inference for Robust Regression and Outlier Detection using Piecewise-Linear Homotopy Continuation