Robust linear regression with broad distributions of errors

• Correct estimating of the linear fit parameters in the presence of large outliers.
• The median of the empirical distribution of the residues determines line’s shift.
• The minimum of interquantile width determines line’s slope (1st method).
• The maximum of characteristic function’s residues determines line’s slope (2nd method).

We consider the problem of linear fitting of noisy data in the case of broad (say αα-stable) distributions of random impacts (“noise”), which can lack even the first moment. This situation, common in statistical physics of small systems, in Earth sciences, in network science or in econophysics, does not allow for application of conventional Gaussian maximum-likelihood estimators resulting in usual least-squares fits. Such fits lead to large deviations of fitted parameters from their true values due to the presence of outliers. The approaches discussed here aim onto the minimization of the width of the distribution of residua. The corresponding width of the distribution can either be defined via the interquantile distance of the corresponding distributions or via the scale parameter in its characteristic function. The methods provide the robust regression even in the case of short samples with large outliers, and are equivalent to the normal least squares fit for the Gaussian noises. Our discussion is illustrated by numerical examples.