Finite-sample lower bound for the MLE norm bias

Establish that the maximum likelihood estimator in logistic regression with Gaussian design satisfies the finite-sample lower bound \(\widehat R_{\mathrm{MLE}}-R \gtrsim R^2d/n\) under the paper’s relevant parameter and sample-size conditions, thereby proving that the dimension-dependent term in the MLE norm-estimation error is intrinsic rather than a proof artifact.

Background

The paper improves the known finite-sample upper bound for the MLE’s norm error from $\widetilde O(\sqrt{R^3d/n})$ to $\widetilde O(\sqrt{R^3/n}+R^2d/n)$. The authors construct a separate debiased estimator achieving the minimax-optimal norm rate $O(\sqrt{R^3/n})$, which shows that the additional $R^2d/n$ term is not information-theoretically necessary.

High-dimensional asymptotic calculations and numerical experiments indicate that the MLE has a positive radial bias of order $R^2d/n$. However, these results do not establish the corresponding lower bound in finite samples. Proving it would show that the MLE is statistically suboptimal for parameter estimation in the regime identified by the authors.