Sensitivity analysis of constrained linear L1 regression: perturbations to response and predictor variables

The active set framework of the reduced gradient algorithm is used to develop a direct sensitivity analysis of linear L1 (least absolute deviations) regression with linear equality and inequality constraints on the parameters. We investigate the effect on the L1 regression estimate of a perturbation to the values of the response or predictor variables. For observations with nonzero residuals, we find intervals for the values of the variables for which the estimate is unchanged. For observations with zero residuals, we find the change in the estimate due to a small perturbation to the variable value. The results provide practical diagnostic formulae. They quantify some robustness properties of constrained L1 regression and show that it is stable, but not uniformly stable. The level of sensitivity to perturbations depends on the degree of collinearity in the model and, for predictor variables, also on how close the estimate is to being nonunique. The results are illustrated with numerical simulations on examples including curve fitting and derivative estimation using trigonometric series.