A More General Robust Loss Function

We present a two-parameter loss function which can be viewed as a generalization of many popular loss functions used in robust statistics: the Cauchy/Lorentzian, Geman-McClure, Welsch/Leclerc, and generalized Charbonnier loss functions (and by transitivity the L2, L1, L1-L2, and pseudo-Huber/Charbonnier loss functions). If this penalty is viewed as a negative log-likelihood, it yields a general probability distribution that includes normal and Cauchy distributions as special cases. We describe and visualize this loss and its corresponding distribution, and document several of their useful properties.