Inverse trinomial distribution: further properties and parameter estimation

Abstract This paper studies the inverse trinomial distribution, a one-dimensional random walk distribution with an absorbing barrier that extends the negative binomial model. The main theoretical contribution is to show that the inverse trinomial distribution admits a mixed Poisson representation, placing it within a flexible and well-studied class of count distributions and thereby widening its scope for practical modelling. Several probabilistic properties that are important for applications are established, including infinite divisibility, log-convexity, log-concavity, and discrete self-decomposability; in particular, log-convexity implies a non-increasing hazard function. On the methodological side, we propose a simple L 1 -based estimation procedure derived from the probability generating function. This estimator is computationally straightforward and is shown to perform well in finite samples. Its practical usefulness is illustrated using two real datasets, one on public infrastructure–related accidents and another on insurance claims. The resulting fits are compared across different distributions with those obtained via maximum likelihood estimation and an L 2 -based approach.