Revisiting Estimation of Number of Trials in Binomial Distribution

Estimating the parameter $n$ when $p$ is known or simultaneous estimation of $n$ and $p$ of the binomial distribution based on $k\ge 1$ independent observations has been considered by many authors over the last several decades. A range of estimators have been proposed, and questions regarding asymptotic and small sample properties received adequate treatment. In this paper, we provide an extensive review and a comprehensive performance comparison of the estimators from the literature. We propose a conceptually simple estimator of $n$ that uses the marginal likelihood when $p$ is integrated out by simultaneous optimisation w.r.t. $n$ and the hyperparameters. We compare the proposed estimator with various existing estimators and find its performance competitive and, in some scenarios, superior.