Title: The weak-feature-impact effect on the NPMLE in monotone binary regression Show Chinese title

Title: 单调二元回归中弱特征影响对NPMLE的影响

Abstract: 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. Show Chinese abstract

Abstract: 研究了特征对标签影响较弱时单调二元回归模型中的非参数最大似然估计量（NPMLE）。这里，弱性被通俗地理解为“接近平坦性”的特征-标签关系$x \mapsto \mathbb{P}(Y=1 | X=x)$。统计文献提供了两种极端情况下的NPMLE的极限分布：如果特征-标签关系严格单调且足够光滑，则它在各点收敛于非参数速率，并分别以按比例缩放的Chernoff型和高斯极限分布，在$L^1$收敛；如果底层关系是平坦的，则它以参数$\sqrt{n}$-速率收敛。为了探索NPMLE从非参数到参数区域的分布转换，我们引入了一种新的数学场景。推导出了弱特征影响场景下NPMLE的新限制最小下界以及点态和$L^1$-收敛速度，并得到了相应的极限分布。它们分别显示出肘部效应和相变，仅由特征影响的程度决定。