Title: Penalized Maximum Tangent Likelihood Estimation and Robust Variable Selection,
Abstract: We introduce a new class of mean regression estimators -- penalized maximum
tangent likelihood estimation -- for high-dimensional regression estimation and
variable selection. We first explain the motivations for the key ingredient,
maximum tangent likelihood estimation (MTE), and establish its asymptotic
properties. We further propose a penalized MTE for variable selection and show
that it is $\sqrt{n}$-consistent, enjoys the oracle property. The proposed
class of estimators consists penalized $\ell_2$ distance, penalized exponential
squared loss, penalized least trimmed square and penalized least square as
special cases and can be regarded as a mixture of minimum Kullback-Leibler
distance estimation and minimum $\ell_2$ distance estimation. Furthermore, we
consider the proposed class of estimators under the high-dimensional setting
when the number of variables $d$ can grow exponentially with the sample size
$n$, and show that the entire class of estimators (including the aforementioned
special cases) can achieve the optimal rate of convergence in the order of
$\sqrt{\ln(d)/n}$. Finally, simulation studies and real data analysis
demonstrate the advantages of the penalized MTE.