Title: Non-penalized variable selection in high-dimensional linear model settings via generalized fiducial inference,
Abstract: Standard penalized methods of variable selection and parameter estimation
rely on the magnitude of coefficient estimates to decide which variables to
include in the final model. However, coefficient estimates are unreliable when
the design matrix is collinear. To overcome this challenge an entirely new
perspective on variable selection is presented within a generalized fiducial
inference framework. This new procedure is able to effectively account for
linear dependencies among subsets of covariates in a high-dimensional setting
where $p$ can grow almost exponentially in $n$, as well as in the classical
setting where $p \le n$. It is shown that the procedure very naturally assigns
small probabilities to subsets of covariates which include redundancies by way
of explicit $L_{0}$ minimization. Furthermore, with a typical sparsity
assumption, it is shown that the proposed method is consistent in the sense
that the probability of the true sparse subset of covariates converges in
probability to 1 as $n \to \infty$, or as $n \to \infty$ and $p \to \infty$.
Very reasonable conditions are needed, and little restriction is placed on the
class of possible subsets of covariates to achieve this consistency result.