  A novel approach to quantile estimation in multivariate linear regression
models with change-points is proposed: the change-point detection and the model
estimation are both performed automatically, by adopting either the quantile
fused penalty or the adaptive version of the quantile fused penalty. These two
methods combine the idea of the check function used for the quantile estimation
and the $L_1$ penalization principle known from the signal processing and,
unlike some standard approaches, the presented methods go beyond typical
assumptions usually required for the model errors, such as sub-Gaussian or
Normal distribution. They can effectively handle heavy-tailed random error
distributions, and, in general, they offer a more complex view on the data as
one can obtain any conditional quantile of the target distribution, not just
the conditional mean. The consistency of detection is proved and proper
convergence rates for the parameter estimates are derived. The empirical
performance is investigated via an extensive comparative simulation study and
practical utilization is demonstrated using a real data example.
