Estimation of the Minimum Probability of a Multinomial Distribution

The estimation of the minimum probability of a multinomial distribution is important for a variety of application areas. However, standard estimators such as the maximum likelihood estimator and the Laplace smoothing estimator fail to function reasonably in many situations as, for small sample sizes, these estimators are fully deterministic and completely ignore the data. Inspired by a smooth approximation of the minimum used in optimization theory, we introduce a new estimator, which takes advantage of the entire data set. We consider both the cases with a known and an unknown number of categories. We categorize the asymptotic distributions of the proposed estimator and conduct a small-scale simulation study to better understand its finite sample performance.