  The investigation of the non-linearity of the Leavitt law is a topic that
began more than seven decades ago, when some of the studies in this field found
that the Leavitt law has a break at about ten days. The goal of this work is to
investigate a possible statistical cause of this non-linearity. By applying
linear regressions to OGLE-II and OGLE-IV data, we find that, in order to
obtain the Leavitt law by using linear regression, robust techniques to deal
with influential points and/or outliers are needed instead of the ordinary
least-squares regression traditionally used. In particular, by using $M$- and
$MM$-regressions we establish firmly and without doubts the linearity of the
Leavitt law in the Large Magellanic Cloud, without rejecting or excluding
Cepheid data from the analysis. This implies that light curves of Cepheids
suggesting blending, bumps, eclipses or period changes, do not affect the
Leavitt law for this galaxy. For the SMC, including this kind of Cepheids, it
is not possible to find an adequate model, probably due to the geometry of the
galaxy. In that case, a possible influence of these stars could exist.
