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Is the regression using the mean squared error any different if the absolute error is used? Assuming the formula is $Y = ax + b$ and both a and b are found using the mean squared error, would they be different than with the other method? Should I take the formula of the mean abs error and minimize it?

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  • $\begingroup$ I do not know , how $a$ and $b$ can be calculated if we want to minimize the maximal absolut error. The values $a$ and $b$ will be different in general from the values we get using the minimal mean squared error. $\endgroup$ – Peter Nov 9 '16 at 16:00
  • $\begingroup$ You can also take a look at this: stats.stackexchange.com/questions/118/… or this: stats.stackexchange.com/questions/46019/… "..squaring makes the algebra much easier to work with and offers properties that the absolute method does not.." $\endgroup$ – Ev. Kounis Nov 9 '16 at 16:03
  • $\begingroup$ One of the reasons to use mean squared error instead of absolute error is that the former is differentiable but the latter needn't be (basically because the absolute value function has a point of non-differentiability). Differentiability is extremely useful in finding extrema, and in particular in minimizing the error. $\endgroup$ – MPW Nov 9 '16 at 16:05
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    $\begingroup$ You could define any objective function of your choice. It will become a minimization problem for which I should use the results of a classical linear regression as a starting point. $\endgroup$ – Claude Leibovici Nov 9 '16 at 16:14

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