Abstract:

The Bell distribution is a simple discrete distribution with one parameter. It has interesting properties, such as being a part of the one-parameter exponential family of distributions and being infinitely divisible. Moreover, the Bell distribution has a variance larger than the mean, indicating that it may be suitable for overdispersed data. With this performance, the Bell distribution is more useful than the Poisson distribution, which is the most popular model for count data. Inflated models have become quite popular in the recent applied statistical literature. In many scientific studies, we often experience situations in which the data consists of a large proportion of zeros and ones. To model count data with excess zeros and excess ones, this paper presents a zero-one inflated Bell distribution. Some properties of the zero-one inflated Bell distribution are also included, such as the probability mass function, probability generating function, moment about the origin, mean, and variance. In addition, in this paper, we investigate the parameter estimation of the zero-one inflated Bell distribution by using the maximum likelihood method. Finally, the insurance dataset is used to show how useful the zero-one inflated Bell distribution is in real life and to see how it compares to other well-known distributions in terms of fit.