Monotonicity Properties of the Multinomial Distribution

Abstract

Let $X = (X_1, \cdots, X_k)$ have the multinomial distribution, given by \begin{equation*} \tag{0.1} \operatorname{Pr}\{X = x\} = n!\prod^k_{i=1} (p^{xi}_i/(x_i!))\end{equation*} where $x = (x_1, \cdots, x_k), \sum^k_{i=1} x_i = n$ and $\sum^k_{i=1}p_i = 1$, and let \begin{equation*} \tag{0.2} C(p_1, \cdots, p_m) = \operatorname{Pr}\{X_i \geqq s_i; i = 1, \cdots, m\}\end{equation*} where $\sum^m_{i=1}s_i\leqq n$ and $m \leqq \min (k - 1, n)$. We show that $C(p_1, \cdots, p_m)$ is nondecreasing in $p_i$ for $i = 1, \cdots, m$ and that for $s_i = s_j$, \begin{equation*} \tag{0.3} C(p_1, \cdots, p_m) \leqq C_{ij}(p_1, \cdots, p_m)\quad\text{and}\end{equation*} \begin{equation*} \tag{0.4} C(p_1, \cdots, p_m) \geqq C_{ijt}(p_1, \cdots, p_m)\end{equation*} where $C_{ij}(p_1, \cdots, p_m)$ is obtained from $C(p_1, \cdots, p_m)$ by substituting $p = \frac{1}{2}(p_i + p_j)$ for $p_i$ and $p_j$ and $C_{ijt}(p_1, \cdots, p_m)$ is obtained from $C(p_1, \cdots, p_m)$ by substituting $t$ for $p_i$ and $p_i + p_j - t$ for $p_j$ where $0 \leqq t \leqq \min (p_i, p_j)$. These and similar results are shown. An application of these results to a multiple decision problem is indicated.