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 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 $n\gtrsim d+\log(1/δ)$, we prove the sharp bound \[ d+\log\left(\frac{1}δ\right). \] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $δ$.

Finite-sample performance of the maximum likelihood estimator in logistic regression

Logistic regression is a classical model for describing the probabilistic dependence of binary responses to multivariate covariates. We consider the predictive performance of the maximum likelihood estimator (MLE) for logistic regression, assessed in terms of logistic risk. We consider two questions: first, that of the existence of the MLE (which occurs when the dataset is not linearly separated), and second that of its accuracy when it exists. These properties depend on both the dimension of covariates and on the signal strength. In the case of Gaussian covariates and a well-specified logistic model, we obtain sharp non-asymptotic guarantees for the existence and excess logistic risk of the MLE. We then generalize these results in two ways: first, to non-Gaussian covariates satisfying a certain two-dimensional margin condition, and second to the general case of statistical learning with a possibly misspecified logistic model. Finally, we consider the case of a Bernoulli design, where the behavior of the MLE is highly sensitive to the parameter direction.