Least-absolute-deviations fits for generalized linear models

The fitting by quasi-likelihoods is based on Euclidean distance and thereby related to the least-squares norm. This paper examines the consequences of replacing the L2-norm by the L1-norm in the derivation of quasi-likelihoods. Since the least-absolute-deviations centre of a distribution is its median rather than its mean, the natural models for the L1-fitting involve medians. However, even if we model the mean response rather than the median response, an L1-type criterion is applicable and leads to alternatives for maximum likelihood fits.