有限总体中多集合分配占用（MAO）的统一精确阶乘矩理论
A Unified Exact Factorial-Moment Theory for Multi-set Allocation Occupancy (MAO) in Finite Populations
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中文总结 AI 辅助
本文针对有限总体的多集合分配占用问题，建立统一精确阶乘矩理论，推导联合占用类别的精确表示式，经穷举枚举和蒙特卡洛模拟验证，为多集合占用统计提供严格有限总体基础。
中文摘要 AI 辅助
设A₁,…,A_T是从大小为n的有限总体中独立均匀选取的子集，其规定基数为m₁,…,m_T。对每个总体元素，定义其占用水平为包含该元素的选中子集的数量。令x_t和x≥t分别表示占用水平恰好为t和至少为t的元素数量。2025年的研究引入了高阶占用矩的通用多集合分配占用（MAO）表示。本文建立了该表示的联合概率解释，并为任意联合占用类别和矩阶数提供了严格的统一推导。具体而言，对任意B₁,…,B_ℓ⊆{0,1,…,T}，我们证明了精确表示式F_ℓ(B₁,…,B_ℓ)=G_T(B₁,…,B_ℓ)/(n)_{ℓ}^{T-1}，其中G_T(B₁,…,B_ℓ)是对应的广义MAO横截和。该恒等式在单一精确有限总体框架内统一了任意占用类别的联合阶乘矩。特别地，令B≥t={t,t+1,…,T}，则精确和阈值占用计数的阶乘矩可通过特例化得到：E[(x_t)ℓ]=F_ℓ({t},…,{t})，E[(x≥t)ℓ]=F_ℓ(B≥t,…,B≥t)。混合阶乘矩、原始矩、方差和协方差均可通过标准变换从同一表示中推导得出。这些公式通过可行参数范围内的穷举枚举和蒙特卡洛模拟得到验证。所得理论为精确和阈值多集合占用统计提供了严格且统一的有限总体基础。
英文摘要
Let $A_1,\ldots,A_T$ be independent uniformly selected subsets of a finite population of size $n$, with prescribed cardinalities $m_1,\ldots,m_T$. For each population element, define its occupancy level as the number of selected subsets containing it. Let $x_t$ and $x_{\geq t}$ denote the numbers of elements with occupancy exactly $t$ and at least $t$, respectively. The 2025 work introduced a general multi-set allocation occupancy (MAO) representation for higher-order occupancy moments. The present paper establishes its joint-probabilistic interpretation and provides a rigorous unified derivation for arbitrary joint occupancy categories and moment orders. Specifically, for arbitrary $B_1,\ldots,B_\ell\subseteq\{0,1,\ldots,T\}$, we prove the exact representation $F_\ell(B_1,\ldots,B_\ell)=G_T(B_1,\ldots,B_\ell)/(n)_\ell^{T-1}$, where $G_T(B_1,\ldots,B_\ell)$ is the corresponding generalized MAO transversal sum. This identity unifies the joint factorial moments of arbitrary occupancy categories within a single exact finite-population framework. In particular, writing $B_{\geq t}=\{t,t+1,\ldots,T\}$, the factorial moments of exact- and threshold-occupancy counts are obtained as the specializations $\mathbb{E}[(x_t)_\ell]=F_\ell(\{t\},\ldots,\{t\})$ and $\mathbb{E}[(x_{\geq t})_\ell]=F_\ell(B_{\geq t},\ldots,B_{\geq t})$. Mixed factorial moments, raw moments, variances, and covariances follow from the same representation through standard transformations. The formulas are verified by exhaustive enumeration over feasible parameter ranges and by Monte Carlo simulation. The resulting theory provides a rigorous and unified finite-population foundation for exact and threshold multi-set occupancy statistics.
