. Author manuscript; available in PMC: 2019 Sep 25.

Published in final edited form as: Biom J. 2018 Mar 25;60(3):616–638. doi: 10.1002/bimj.201600262

Table 1.

Multinomial distribution with probabilities (p₁, ⋯, p_m) with m = 100 and n = 20

Pattern	Description	(p₁, ⋯, p_m)^[1]	Other Parameters
1	Constant P₁ = p₂ ⋯ = p_m	P_i = 1/m, i = 1, ⋯, m
2	Monotonically increasing P₁ < p₂ < ⋯ < p_m	P₁^# = 0.006, p_i = p₁ + (i − 1)d, i = 2, ⋯, m	$d = 2 \frac{1 / m - p_{1}}{m - 1}$
3	Constant followed by monotonically increasing P₁ = ⋯ = p_k < p_k+1 < ⋯ < p_m	k^# = 50, $p_{i} = \frac{P_{k}}{k}, i = 1, \dots, k$ p_i = p_k + (i − k)d, i = k + 1, ⋯, m	$P_{k}^{#} = \sum_{i = 1}^{k} p_{i} = 0.2$ , $d = \frac{2}{m - k} (\frac{1 - P_{k} + \frac{P_{k}}{k}}{m - k + 1} - \frac{P_{k}}{k})$
4	Monotonically increasing followed by constant P₁ < ⋯ < p_k = p_k+1 = ⋯ = p_m	k^# = 50 $P_{k}^{#} = \sum_{i = k + 1}^{m} p_{i} = 0.65$ p_i = p_k − (k − i)d, i = 1, ⋯, k − 1	$P_{k}^{#} = \sum_{i = k + 1}^{m} p_{i} = 0.65$ $d = \frac{2}{k - 1} (\frac{P_{k}}{m - k} - \frac{1 - P_{k}}{k})$
5	Constant, monotonically increasing followed by constant $p_{1} = p_{2} \dots = p_{k_{1}} < p_{k_{1} + 1} < \dots < p_{k_{2}} = \dots = p_{m}$	k₁^# = 20, k₂^# = 70, $p_{i} = \frac{P_{k_{1}}}{k_{1}}, i = 1, \dots, k_{1}$ $p_{i} = p_{k_{1}} + (i - k_{1}) d, i = k_{1} + 1, \dots, k_{2}$	$P_{k_{1}}^{#} = \sum_{i = 1}^{k_{1}} p_{i} = 0.05$ , $d = \frac{2 (1 - m P_{k_{1}} / k_{1})}{(k_{2} - k_{1}) [2 * (m - k_{2}) + (k_{2} - k_{1} + 1)]}$
6	Monotonically increasing, constant followed by monotonically increasing $p_{1} < p_{2} \dots < p_{k_{1}} = p_{k_{1} + 1} = \dots = p_{k_{2}} < \dots < p_{m}$	k₁^# = 25, k₂^# = 75 $p_{i} = \frac{P_{k_{2}}}{k_{2} - k_{1}}, i = k_{1}, \dots, k_{2}$ $p_{i} = p_{k_{1}} - (k_{1} - i) d_{1}, i = 1, \dots, k_{1} - 1$ $p_{i} = p_{k_{2}} + (i - k_{2}) d_{2}, i = k_{2} + 1, \dots, m$ $Note: \frac{k_{1}}{2} \frac{P_{k_{2}}}{k_{2} - k_{1}} < P_{k_{1}} < k_{1} \frac{P_{k_{2}}}{k_{2} - k_{1}}$	$P_{k_{1}}^{#} = \sum_{i = 1}^{k_{1}} p_{i} = 0.15, P_{k_{2}}^{#} = \sum_{i = k_{1} + 1}^{k_{2}} p_{i} = 0.5$ $d_{1} = \frac{2}{k_{1} - 1} (\frac{P_{k_{2}}}{k_{2} - k_{1}} - \frac{P_{k_{1}}}{k_{1}})$ , $d_{2} = \frac{2}{m - k_{2}} (\frac{1 - P_{k_{1}} - P_{k_{2}} + \frac{P_{k_{1}}}{k_{1}}}{m - k_{2} + 1} - \frac{P_{k_{2}}}{k_{2} - k_{1}})$

^[1]

(p₁, ⋯, p_m) are probabilities and $\sum_{i = 1}^{m} p_{i} = 1$ .

^#:

required parameters.