Speaker: Karl Gregory, Associate Professor of Statistics, University of South Carolina

Title: Least angle regression inference

Abstract: Least angle regression (LAR) was introduced by Efron et al. (2004) as an algorithm for linear predictions, intended as an alternative to forward selection with connections to penalized regression. However, LAR has remained somewhat of a "black box," where basic inference properties are unknown, including sampling distributions for LAR estimates. We provide a formal and novel framework for inference with LAR, which also allows LAR to be interpreted from new perspectives with several newly developed statistical properties. Furthermore, the LAR algorithm at a data level approximates a population counterpart that aims to explain a response mean along regressor variables that are ordered according to a decreasing series of population "correlation" parameters, whereby zero correlations denote unimportant variables. Despite the inter-connected steps in the LAR algorithm, estimates of all non-zero population correlations turn out to be asymptotically independent and normal, while estimates of zero-valued population correlations have a complicated, non-normal joint distribution. Due to this dichotomy, bootstrap becomes useful for approximating sampling distributions, though caution is needed as the standard (residual) bootstrap for regression generally fails for LAR inference. The bootstrap, though, can be modified to capture the entire joint distribution of all correlation estimates, which provides a practical tool for studying and interpreting the entrance of variables. The LAR inference method is studied through simulation and illustrated with data examples.