Beyond Modern Asymptotics for Log-Likelihood Ratios in Logistic Regression

Abstract

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 $(1-\delta)$ quantile over all fixed collections of design vectors and all target parameters: \[ d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}{\delta}\right). \] This is a nonasymptotic analogue of the Wilks $\chi^2_d$ 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 \[ \log\log\log n+\log\left(\frac{1}{\delta}\right). \] The worst case quantile in dimension $d=1$ is of order $\log(1/\delta)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/\delta)$, we prove the sharp bound \[ d+\log\left(\frac{1}{\delta}\right). \] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $\delta$.