The weak-feature-impact effect on the NPMLE in monotone binary regression

The nonparametric maximum likelihood estimator (NPMLE) in monotone binary regression models is studied when the impact of the features on the labels is weak. Here, weakness is colloquially understood as "close to flatness" of the feature-label relationship $x \mapsto \mathbb{P}(Y=1 | X=x)$. Statistical literature provides limit distributions of the NPMLE for the two extremal cases: If the feature-label relation is strictly monotone and sufficiently smooth, then it converges at a nonparametric rate pointwise and in $L^1$ with scaled Chernoff-type and Gaussian limit distribution, respectively, and it converges at the parametric $\sqrt{n}$-rate if the underlying relation is flat. To explore the distributional transition of the NPMLE from the nonparametric to the parametric regime, we introduce a novel mathematical scenario. New restricted minimax lower bounds and matching pointwise and $L^1$-rates of convergence of the NPMLE in the weak-feature-impact scenario together with corresponding limit distributions are derived. They are shown to exhibit an elbow and a phase transition respectively, solely characterized by the level of feature impact.