Title: Pathwise Least Angle Regression and a Significance Test for the Elastic Net,
Abstract: Least angle regression (LARS) by Efron et al. (2004) is a novel method for
constructing the piece-wise linear path of Lasso solutions. For several years,
it remained also as the de facto method for computing the Lasso solution before
more sophisticated optimization algorithms preceded it. LARS method has
recently again increased its popularity due to its ability to find the values
of the penalty parameters, called knots, at which a new parameter enters the
active set of non-zero coefficients. Significance test for the Lasso by
Lockhart et al. (2014), for example, requires solving the knots via the LARS
algorithm. Elastic net (EN), on the other hand, is a highly popular extension
of Lasso that uses a linear combination of Lasso and ridge regression
penalties. In this paper, we propose a new novel algorithm, called pathwise
(PW-)LARS-EN, that is able to compute the EN knots over a grid of EN tuning
parameter {\alpha} values. The developed PW-LARS-EN algorithm decreases the EN
tuning parameter and exploits the previously found knot values and the original
LARS algorithm. A covariance test statistic for the Lasso is then generalized
to the EN for testing the significance of the predictors. Our simulation
studies validate the fact that the test statistic has an asymptotic Exp(1)
distribution.