Unified Robust Estimation

Robust estimation is primarily concerned with how to provide reliable parameter estimates in the presence of outliers. Numerous robust loss functions have been proposed in regression and classification, along with various computing algorithms. In modern penalized generalized linear model (GLM), however, there is limited research on robust estimation that can provide weights to determine outlier status of the observations. This article proposes a unified framework based on a large family of loss functions, a composite of concave and convex functions (CC-family). Properties of the CC-family are investigated, and CC-estimation is innovatively conducted via the iteratively reweighted convex optimization (IRCO), a generalization of the iteratively reweighted least squares in robust linear regression. For robust GLM, the IRCO becomes iteratively reweighted GLM. The unified framework contains penalized estimation and robust support vector machine and is demonstrated with a variety of data applications.