Abstract

Robust estimation is primarily concerned with providing 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 models (GLM), however, there is limited research on robust estimation
that can provide weights to determine the 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), which is a generalization of the iteratively reweighted least squares in
robust linear regression. For robust GLM, the IRCO becomes the iteratively reweighted
GLM. The unified framework contains penalized estimation and robust support vector
machine and is demonstrated with a variety of data applications.