学习平滑参数总体的极小极大速率

中文摘要 AI 辅助

我们研究平滑二项混合密度的逐点估计，并刻画其极小极大速率。假设混合密度是$s$-Hölder光滑的，我们首先考虑具有共同二项试验次数$t$的同质设定。我们推导出匹配的上下界，揭示了取决于样本量$n$与试验次数$t$相对大小的三个区间。当$t$较小时，有限的试验次数造成识别障碍，该障碍不随样本量增大而消失；在中等$t$时，更多数据减少抽样不确定性，而试验次数限制了混合密度可被恢复的精细程度；当$t$足够大时，恢复通常的非参数密度估计速率。我们的下界利用二项混合模型的多项式结构，而可达性通过适当正则化的正交级数估计量实现。随后我们将分析扩展到异质试验参数，其中极小极大速率依赖于通过每个多项式次数处有效样本量体现的完整试验分布。这些结果刻画了在同质和异质试验下学习二项概率平滑总体的基本统计极限。

英文摘要

We study pointwise estimation of a smooth binomial mixing density and characterize its minimax rates. Assuming that the mixing density is $s$-Hölder smooth, we first consider the homogeneous setting with a common number of binomial trials $t$. We derive matching lower and upper bounds that reveal three regimes depending on the relative sizes of the sample size $n$ and the number of trials $t$. When $t$ is small, the finite number of trials creates an identification barrier that persists regardless of sample size; at intermediate $t$, more data reduce sampling uncertainty, while the number of trials limits how finely the mixing density can be recovered; and when $t$ is sufficiently large, the usual nonparametric density estimation rate is recovered. Our lower bounds exploit the polynomial structure of the binomial mixture model, while attainability is achieved by a suitably regularized orthogonal series estimator. We then extend the analysis to heterogeneous trial parameters, where the minimax rate depends on the full trial profile through the effective sample size available at each polynomial degree. These results characterize the fundamental statistical limits of learning smooth populations of binomial probabilities under both homogeneous and heterogeneous trials.

发表机构

• Carnegie Mellon University（卡内基梅隆大学）

机构由 AI 辅助整理，请以论文原文为准。