# Non-penalized variable selection in high-dimensional linear model settings via generalized fiducial inference

## TL;DR

This paper introduces a novel variable selection method for high-dimensional linear models using generalized fiducial inference, effectively handling collinearity and dependencies among covariates without penalization.

## Contribution

It presents a new non-penalized variable selection approach based on generalized fiducial inference that is consistent even with highly collinear and high-dimensional data.

## Key findings

- The method assigns small probabilities to redundant covariate subsets.
- It is consistent in selecting the true sparse subset as sample size grows.
- Applicable in both classical and ultra-high-dimensional settings.

## 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.