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For linear regression there's an assumption that error terms come from normal distribution. so that $Y = aX + b + \epsilon$, where $\epsilon$ has normal distribution with mean zero and certain variance.

Is there an analogous assumption for Huber loss (or MAE which should be easier)? So that for a variable described by relation $Y = aX + b + \epsilon$ with $\epsilon$ from this distribution huber regression (or theoretically minimization of MAE) gives a and b parameters?

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