A note on the generalized degrees of freedom under the L₁ loss function

Generalized degrees of freedom measure the complexity of a modeling procedure; a modeling procedure is a combination of model selection and model fitting. In this manuscript, we consider two definitions of generalized degrees of freedom for a modeling procedure under the L₁ loss function, and investigate the connections between those two definitions. We also propose the extended Akaike information criterion, the adaptive model selection, and the extended generalized cross-validation under the L₁ loss function. Finally, we extend the results to M-estimation.