multiple linear regression error minimization

Question

regression analysis in different statistical packages fits the best line by minimizing the error of the fit, the error term used by default is mostly MSE (mean square error), in other words, regression gives the values to model constants so the model finally generates the minimum error (MSE). regarding this basics is it possible to change this error criterion to any other user-defined criterion? it doesn't matter in which program, I'm just asking that is this possible? and if the answer is yes, how?

P.S: in non-linear regression, this is possible. for example, in SPSS there is an option named loss value which you can change it to any of your specific formulas so that the model gives the constant values that they will output the minimum error criterion of your interest (MAE instead of MSE for example)

Answer 1

There are 2 other criterion so popular over, L1 et L$_\infty$ criterion. L1 regression, say sum of abs values ! L2 minization (ie quadratic error) is fast (direct), easy computational and well established mathematically, I mean theory around fit well together (Gaussian distribution, derivative = linear, brownian motion, eig/pca, etc...). But in L1 world, mathematics are not so nice, but in some context, outperform the L2 arsenal (l1 = entropy = information minization = compression, compress sensing)

Other norm are use, like L$_\infty$ (ie minimize the max) usefull for saturation problem et 0-1 binary variable minization. More generally L$_\alpha$ (ie $||.||_\alpha$) could be use with standard IRLS scheme, but L1 is "the minimal norm" keep the formulation convex. And non-convex world is evil, so...