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

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 $E(X) \to 0$.