Local Influence in LASSO Regression

The least absolute shrinkage and selection operator (LASSO) regression is very popular in these days, because we can get the high dimensional data cheaply. However, there are few research results on the influence analysis of the LASSO regression. We study the local influence on the LASSO estimator in linear regression. We derive a generalized Cook s distance of LASSO estimator, which is equivalent to the Cook s approach under simultaneous perturbation of the assumed model. We consider the perturbations of variance and explanatory variables. In particular we can get the influence information under an explanatory variables, which can not be obtained by the case deletion method. Since the closed form of the LASSO estimator can not be obtained, we can get the influence information based on the estimation of ridge regression. Furthermore, the final results on the local influence can be represented in a closed form. Numerical examples, for which we apply the proposed diagnostic methods, are given for illustration. The proposed method gives important influence information on the LASSO estimate by a simple graphical method.