  We consider the problem of linear fitting of noisy data in the case of broad
(say $\alpha$-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.
