  A simultaneous change-point detection and estimation in a piece-wise constant
model is a common task in modern statistics. If, in addition, the whole
estimation can be performed automatically, in just one single step without
going through any hypothesis tests for non-identifiable models, or unwieldy
classical a-posterior methods, it becomes an interesting, but also challenging
idea. In this paper we introduce the estimation method based on the quantile
LASSO approach. Unlike standard LASSO approaches, our method does not rely on
typical assumptions usually required for the model errors, such as sub-Gaussian
or Normal distribution. The proposed quantile LASSO method can effectively
handle heavy-tailed random error distributions, and, in general, it offers a
more complex view of the data as one can obtain any conditional quantile of the
target distribution, not just the conditional mean. It is proved that under
some reasonable assumptions the number of change-points is not underestimated
with probability tenting to one, and, in addition, when the number of
change-points is estimated correctly, the change-point estimates provided by
the quantile LASSO are consistent. Numerical simulations are used to
demonstrate these results and to illustrate the empirical performance robust
favor of the proposed quantile LASSO method.
