The Multi-set Allocation Occupancy function and inequality (MAO function and MAO inequality): the foundation of Generalized hypergeometric distribution theory

TL;DR

This paper introduces a unified formula for all higher moments of the MAO distribution, based on the MAO function and norm, and establishes a new inequality and asymptotic properties.

Contribution

It develops a novel, elegant formula for higher moments of the MAO distribution and introduces the MAO inequality under proximity conditions.

Findings

Unified moment formulas simplify complex relations

Established a new MAO inequality under specific conditions

Proved asymptotic properties of the MAO distribution

Abstract

In our previous work, we studied the Generalized Hypergeometric Distribution (GHGD), which we refer to as the Multi-set Allocation Occupancy (MAO) distribution. We derived formulas for its expectation and variance for any number of subsets $T$ and overlap count $t$ ($1 \le t \le T$), and established an asymptotic property. However, these formulas were complex, and higher moments were not derived. Through further study, we have established a novel function that describes all higher moments of the MAO distribution with a unified, elegant formula. The core definitions are the MAO function $g(A_1, A_2, \dots, A_r) = \prod_{i=1}^{T} (m_i)_{k_i} \cdot (n-m_i)_{r-k_i}$ and the MAO norm $\|(p_1, \dots, p_r)\|_T = \frac{\sum_{A_1, \dots, A_r \subseteq [T] \; : \; |A_j|=p_j} g(A_1, \dots, A_r)}{((n)_r)^{T-1}}$, where $p_i$ is the size of subset $A_i$, $m_i < n$, and $(x)_r$ is the falling factorial. Using these definitions, the intricate moment relations simplify into a unified form: the $\nu$-th raw moment of $p(x_{=t})$ and $p(x_{\ge t})$ can be calculated as $E(x_{=t}^\nu) = \sum_{1 \le i \le \nu} s_{\nu,i} \|t^i\|$ and $E(x_{\ge t}^\nu) = \sum_{1 \le i \le \nu} s_{\nu,i} \|[t, T]^i\|$, where $s_{\nu,i}$ are Stirling numbers of the second kind and $[t,T] = \{t, t+1, \dots, T\}$. Furthermore, based on the MAO norm, we formulate a novel MAO inequality under the proximity condition $\max(p_i) - \min(p_i) \le 1$: $\prod_{1\le i \le r} \|(p_i)\|_T \ge \|(p_1, \dots, p_r)\|_T$. A direct corollary is the asymptotic property of the MAO distribution: $E(X) > \text{Var}(X)$ and $E(X) - \text{Var}(X) = o(E(X))$ as $E(X) \to 0$.