Non-asymptotic minimax optimality of the Firth correction

Establish non-asymptotic high-dimensional error bounds proving that the Firth correction for the Bradley–Terry–Luce model with linear-in-parameter utility functions is minimax optimal when the sample size satisfies the stated order condition, up to factors depending on the dynamic-range bound, questionnaire coherence, and logarithmic terms.

Background

The paper compares the unconstrained canonical maximum likelihood estimator with the Firth correction, which remains finite under separation and empirically exhibits more stable finite-sample behavior. The theoretical results establish minimax lower bounds and matching upper bounds for the canonical MLE under sufficiently large sample sizes, but they do not provide corresponding non-asymptotic guarantees for the Firth correction.

The numerical experiments suggest that the Firth correction may achieve minimax performance at a sample size of order d_theta{3/2}, subject to factors involving the dynamic-range parameter B, the questionnaire coherence level mu_Q, and logarithmic terms. The authors explicitly formulate this as a conjecture and note the absence of currently available high-dimensional non-asymptotic error bounds.

References

These results support the Firth correction as a more reliable default for the finite-sample, fixed-design preference elicitation problem considered here, and we conjecture that it is minimax optimal when $N\gtrsim d_\theta{3/2}$, up to $B$, $\mu_Q$, and logarithmic factors. To the best of our knowledge, however, non-asymptotic error bounds for the Firth correction in high-dimensional settings are not currently available.

— Error Bounds for Statistical Estimators in BTL Model with Parametric Multivariate Utility Functions  (2609.26326 - Li et al., 22 Sep 2026) in Section 5, subsection “Synthetic Questionnaire”