If you assume a Gaussian distribution for the "errors" of your regression model, you can express the maximum likelihood, so the coefficients giving the highest likelihood for the observed data, with the sum of squared "errors" of your regression model.
I am trying to calculate the AIC for my quantile regression models, and in order to do that I need to get the likelihood assuming some distribution. The distribution "associated" with the quantile loss function is the asymmetric Laplace distribution.
Of course it is possible to code the pdf and use optimization algorithms to get the maximum likelihood, but I would like to know if something similar to the Gaussian case is available. So a formula linking the maximum likelihood of the asymmetric Laplace and the "errors", or residuals rather than errors since I will be working with predicted and observed values.
I would like you to point out any nonsense terminology I might be using.