Beyond Modern Asymptotics for Log-Likelihood Ratios in Logistic Regression
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 $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-δ)$ quantile over all fixed collections of design vectors and all target parameters: \[ d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}δ\right). \] This is a nonasymptotic analogue of the Wilks $χ^2_d$ 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 \[ \log\log\log n+\log\left(\frac{1}δ\right). \] The worst cas...
| Authors | Hugo Chardon, Reese Pathak, Nikita Zhivotovskiy |
|---|---|
| Primary category | math.ST |
| Published | 3 August 2026 |
| Last updated at source | 3 August 2026 |
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