分层Pitman-Yor模型中随机索引簇计数的中偏差

中文摘要 AI 辅助

考虑来自两层分层Pitman-Yor模型的大小为$N$的样本，设$\xi(N)$表示层级第一层生成的随机簇数。我们研究在随机样本大小$\xi(N)$下评估的划分统计量的中偏差原理。特别地，我们建立了总簇数$K_{\xi(N)}$和频率计数$M_{l,\xi(N)}$（即由恰好$l$个第一层簇表示的上层簇数）的中偏差原理。我们的分析将固定样本大小的小倾斜渐近性与随机索引$\xi(N)$的中偏差原理相结合。通过一致小倾斜论证，结合指数加权尾部估计，我们能够从确定性样本大小过渡到分层随机索引设置。我们在典型增长阶$N^{\alpha_1\alpha_2}$与线性尺度$N$之间的一系列中间尺度上获得了显式的偏差速度和良好的速率函数。所得速率函数表现出由乘积$\alpha_1\alpha_2$控制的共同幂律结构，揭示了层级两层的乘法效应。

英文摘要

Consider a sample of size $N$ from a two-layer hierarchical Pitman--Yor model, and let $ξ(N)$ denote the random number of clusters generated at the first layer of the hierarchy. We study moderate deviation principles for partition statistics evaluated at the random sample size $ξ(N)$. In particular, we establish moderate deviation principles for the total number of clusters $K_{ξ(N)}$ and for the frequency count $M_{l,ξ(N)}$, the number of upper-level clusters represented by exactly $l$ first-level clusters. Our analysis combines fixed-sample-size small-tilt asymptotics with a moderate deviation principle for the random index $ξ(N)$. A uniform small-tilt argument, together with exponentially weighted tail estimates, allows us to pass from deterministic sample sizes to the hierarchical random-index setting. We obtain explicit deviation speeds and good rate functions on a family of intermediate scales between the typical growth order $N^{α_1α_2}$ and the linear scale $N$. The resulting rate functions exhibit a common power-law structure governed by the product $α_1α_2$, revealing a multiplicative effect of the two levels of the hierarchy.

发表机构

• Shenzhen University（深圳大学）

机构由 AI 辅助整理，请以论文原文为准。