Abstract:Statistical literature provides pointwise limiting distributions of the nonparametric maximum likelihood estimator (NPMLE) in monotone binary regression for the two extremal cases: If the feature-label relation is strictly monotone and sufficiently smooth, it converges at a nonparametric rate with scaled Chernoff-type limiting distribution, and it converges at the parametric $\sqrt{n}$-rate if the underlying relation is flat. In this article, we provide the complete picture of the distributional metamorphosis of the NPMLE, revealing a new limiting distribution which provides a significantly better distributional approximation for small samples in case of a weak feature-label relationship. It is shown to continuously interpolate between the two extremal cases. The innovative way to determine this distribution is to generate it as a limit of the NPMLE in the newly introduced weak-feature-impact triangular array for a particular parameter-sample-size constellation. Moreover, a phase transition is likewise observed for the suitably rescaled $L^{1}$-error in this weak-feature-impact scenario. As a by-product, its limiting distribution for flat regression functions is obtained, which was unknown before. The proof develops a completely new strategy, notably not based on the switch relation. A novel type of local minimax lower bounds accompanies these results.
From: Dario Kieffer [view email]
[v1]
Sun, 13 Apr 2025 13:33:15 UTC (112 KB)
[v2]
Wed, 4 Jun 2025 12:16:40 UTC (125 KB)
[v3]
Thu, 18 Jun 2026 15:28:07 UTC (449 KB)
[v4]
Tue, 8 Sep 2026 13:31:21 UTC (503 KB)
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