Inference without compatibility: Using exponential weighting for inference on a parameter of a linear model

Abstract

We consider hypotheses testing problems for three parameters in high-dimensional linear models with minimal sparsity assumptions of their type but without any compatibility conditions. Under this framework, we construct the first $\sqrt{\mathit{n}}$-consistent estimators for low-dimensional coefficients, the signal strength, and the noise level. We support our results using numerical simulations and provide comparisons with other estimators.