  We propose regularization methods for linear models based on the
$L_q$-likelihood, which is a generalization of the log-likelihood using a power
function. Some heavy-tailed distributions are known as $q$-normal
distributions. We find that the proposed methods for linear models with
$q$-normal errors coincide with the regularization methods that are applied to
the normal linear model. The proposed methods work well and efficiently, and
can be computed using existing packages. We examine the proposed methods using
numerical experiments, showing that the methods perform well, even when the
error is heavy-tailed.
