  Coefficient estimation and variable selection in multiple linear regression
is routinely done in the (penalized) least squares (LS) framework. The concept
of model selection oracle introduced by Fan and Li [J. Amer. Statist. Assoc. 96
(2001) 1348--1360] characterizes the optimal behavior of a model selection
procedure. However, the least-squares oracle theory breaks down if the error
variance is infinite. In the current paper we propose a new regression method
called composite quantile regression (CQR). We show that the oracle model
selection theory using the CQR oracle works beautifully even when the error
variance is infinite. We develop a new oracular procedure to achieve the
optimal properties of the CQR oracle. When the error variance is finite, CQR
still enjoys great advantages in terms of estimation efficiency. We show that
the relative efficiency of CQR compared to the least squares is greater than
70% regardless the error distribution. Moreover, CQR could be much more
efficient and sometimes arbitrarily more efficient than the least squares. The
same conclusions hold when comparing a CQR-oracular estimator with a
LS-oracular estimator.
