An Adaptively Resized Parametric Bootstrap for Inference in High-dimensional Generalized Linear Models

TL;DR

This paper proposes a resized bootstrap method for reliable inference in high-dimensional generalized linear models, addressing inaccuracies of classical methods when the parameter-to-sample size ratio is large.

Contribution

It introduces a novel resized bootstrap approach that shrinks the MLE to improve inference accuracy in high-dimensional settings.

Findings

Provides valid confidence intervals in simulations

Demonstrates effectiveness on real data

Extends to various high-dimensional GLMs

Abstract

Accurate statistical inference in logistic regression models remains a critical challenge when the ratio between the number of parameters and sample size is not negligible. This is because approximations based on either classical asymptotic theory or bootstrap calculations are grossly off the mark. This paper introduces a resized bootstrap method to infer model parameters in arbitrary dimensions. As in the parametric bootstrap, we resample observations from a distribution, which depends on an estimated regression coefficient sequence. The novelty is that this estimate is actually far from the maximum likelihood estimate (MLE). This estimate is informed by recent theory studying properties of the MLE in high dimensions, and is obtained by appropriately shrinking the MLE towards the origin. We demonstrate that the resized bootstrap method yields valid confidence intervals in both simulated and real data examples. Our methods extend to other high-dimensional generalized linear models.