A2005
Title: Exact distribution of some functions of the ordered multinomial counts: Computational aspects and applications
Authors: Sergio Venturini - Università Cattolica del Sacro Cuore (Italy) [presenting]
Marco Bonetti - Bocconi University (Italy)
Abstract: Testing whether a multinomial distribution is consistent with an equal-probability null hypothesis arises across a broad range of statistical applications. An alternative to classical test statistics is based on the order statistics of the multinomial counts: the maximum $N_<1>$, minimum $N_<m>$, and range $R=N_<1>-N_<m>$. Computing the exact distributions of these statistics is a non-trivial computational problem. An explicit recursive derivation of the exact CDFs of $N_<1>$ and $N_<m>$ is provided based on direct conditioning on individual multinomial counts and the closure property of the multinomial distribution. The derivation proceeds through nested binomial probabilities, making the probabilistic content of each step transparent. Formal equivalence between this construction and the matrix-based approach is established. It is also shown that the exact power of tests based on $N_<1>$ and $N_<m>$ can be computed analytically under structured one-sided alternatives, enabling closed-form sample size determination. The XOMultinom R package unifies these methods in a computationally efficient implementation and allows calculation of power functions and sample sizes. Applications involving sample size determination and the signalling of miscalibration in clinical risk prediction models illustrate the use of the package.
