当真值为平坦时估计量单调重排的极限性质
Limiting properties of monotone rearrangements of estimators when the truth is flat
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中文总结 AI 辅助
本文针对真值为平坦的情形，研究单调重排估计量的极限性质，推导了均匀密度直方图重排估计量及两类重排copula估计量的弱收敛结果，并证明了相关相依测度的渐近正态性。
中文摘要 AI 辅助
单调重排为估计量施加形状约束提供了一种简单方法，但现有分布理论未覆盖平坦区域——目标在该区域无局部序关系。我们研究两类典型平坦设定下的重排估计量：其一，针对均匀密度的直方图估计量，经额外确定性中心化后，在区间(0,1)的紧子集上，其非递减重排以参数速率成立函数弱收敛，该结果与严格单调密度的已知结果显著不同；其二，基于经验copula增量和棋盘近似，考虑独立性下重排copula的两类重排估计量，经适当中心化与缩放后，二者在[0,1]^2上弱收敛至一个此前未知的积分高斯过程。我们进一步利用这些结果，证明了Strothmann等人（2024）新近讨论的一类广泛的基于重排copula的相依测度的渐近正态性。
英文摘要
Monotone rearrangements provide a simple way to enforce shape constraints of an estimator, but existing distributional theory does not cover flat regions, where the target induces no local ordering. We study rearranged estimators in two canonical flat settings. First, for a histogram estimator of the uniform density, we establish functional weak convergence of its non-decreasing rearrangement at the parametric rate on compact subsets of the interval $(0,1)$ after an additional deterministic centering. This result is strikingly different from what is known for strictly monotone densities. Second, we consider two rearranged estimators of rearranged copulas under independence, based on empirical-copula increments and on a checkerboard approximation. After appropriate centering and rescaling, both estimators converge weakly on $[0,1]^2$ to an integrated Gaussian process which was not known before. We further use these results to prove asymptotic normality for a broad class of rearranged copula-based dependence measures, which were recently discussed in Strothmann et al. (2024).
