  In this paper, we propose a novel conceptual framework to detect outliers
using optimal transport with a concave cost function. Conventional outlier
detection approaches typically use a two-stage procedure: first, outliers are
detected and removed, and then estimation is performed on the cleaned data.
However, this approach does not inform outlier removal with the estimation
task, leaving room for improvement. To address this limitation, we propose an
automatic outlier rectification mechanism that integrates rectification and
estimation within a joint optimization framework. We take the first step to
utilize the optimal transport distance with a concave cost function to
construct a rectification set in the space of probability distributions. Then,
we select the best distribution within the rectification set to perform the
estimation task. Notably, the concave cost function we introduced in this paper
is the key to making our estimator effectively identify the outlier during the
optimization process. We demonstrate the effectiveness of our approach over
conventional approaches in simulations and empirical analyses for mean
estimation, least absolute regression, and the fitting of option implied
volatility surfaces.
