# Beyond Modern Asymptotics for Log-Likelihood Ratios in Logistic Regression

arXiv: https://arxiv.org/abs/2608.02507 · HTML: https://mlsanity.com/paper/2608.02507/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 58 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Confidence Sequences for Generalized Linear Models via Regret Analysis](https://mlsanity.com/paper/2504.16555/) (2025)
- [Finite-sample performance of the maximum likelihood estimator in logistic regression](https://mlsanity.com/paper/2411.02137/) (2024)
- [Confidence Estimation via Sequential Likelihood Mixing](https://mlsanity.com/paper/2502.14689/) (2025)
- [Volumes of logistic regression models with applications to model selection](https://mlsanity.com/paper/1408.0881/) (2014)

## Abstract (arXiv)

We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For n≥ d≥ 3, we determine, up to universal constants, its worst case (1-δ) quantile over all fixed collections of design vectors and all target parameters: [ dlogleft(frace ndright)+logleft(frac1δright). ] This is a nonasymptotic analogue of the Wilks chi2d phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension d=2 is sharply of order [ logloglog n+logleft(frac1δright). ] The worst case quantile in dimension d=1 is of order log(1/δ), with no dependence on n. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime ngtrsim d+log(1/δ), we prove the sharp bound [ d+logleft(frac1δright). ] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on n, d, and δ.