  In this paper, a linear model with grouped explanatory variables is
considered. The idea is to perform an automatic detection of different
successive groups of the unknown coefficients under the assumption that the
number of groups is of the same order as the sample size. The standard least
squares loss function and the quantile loss function are both used together
with the fused and adaptive fused penalty to simultaneously estimate and group
the unknown parameters. The proper convergence rate is given for the obtained
estimators and the upper bound for the number of different successive group is
derived. A simulation study is used to compare the empirical performance of the
proposed fused and adaptive fused estimators and a real application on the air
quality data demonstrates the practical applicability of the proposed methods.
