# Inference Without Compatibility

## TL;DR

This paper develops new estimators for high-dimensional linear models that are consistent at the root-n rate without requiring compatibility conditions, broadening the scope of hypothesis testing in such models.

## Contribution

It introduces the first root-n consistent estimators for coefficients, signal strength, and noise level under minimal sparsity assumptions and no compatibility conditions.

## Key findings

- Estimators are supported by numerical simulations.
- Comparisons show advantages over existing methods.
- Establishes theoretical guarantees for the estimators.

## 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 $\sqrt{n}$-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.