Sensitivity analysis of constrained linear L1L1 regression: Perturbations to constraints, addition and deletion of observations

This paper extends the direct sensitivity analysis of Shi and Lukas [2005, Sensitivity analysis of constrained linear L1L1 regression: perturbations to response and predictor variables. Comput. Statist. Data Anal. 48, 779–802] of linear L1L1 (least absolute deviations) regression with linear equality and inequality constraints on the parameters. Using the same active set framework of the reduced gradient algorithm (RGA), we investigate the effect on the L1L1 regression estimate of small perturbations to the constraints (constants and coefficients). It is shown that the constrained estimate is stable, but not uniformly stable, and in certain cases it is unchanged. We also consider the effect of addition and deletion of observations and determine conditions under which the estimate is unchanged. The results demonstrate the robustness of L1L1 regression and provide useful diagnostic information about the influence of observations. Results characterizing the (possibly non-unique) solution set are also given. The sensitivity results are illustrated with numerical simulations on the problem of derivative estimation under a concavity constraint.