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≥ 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.