Catalyst Project: Modeling Count Data with the Conway-Maxwell-Poisson Distribution

催化剂项目支持历史悠久的黑人学院和大学致力于建立教职员工的研究能力，以加强科学、技术、工程和数学本科教育和研究。预计该奖项将进一步提高教职员工的研究能力，改善学院的研究和教学，并让本科生参与研究经验。北卡罗来纳中央州立大学的这个项目将研究概率分布的数学建模，并将为本科生提供一个机会，通过研究经验增强他们的统计知识。这位研究人员与乔治城大学的教职员工建立了强有力的合作关系。计数数据出现在各种情况下，并且经常用泊松分布建模。然而，在许多情况下，数据不满足泊松分布的均值和方差相等的性质。本项目将研究双参数Conway-Maxwell-Poisson(CMP)分布，该分布在方差大于平均值的情况下考虑过度分散，在方差小于平均值的情况下说明欠分散。CMP分布将泊松分布、伯努利分布和几何分布作为特例。这个项目的第一个目标是进行一项模拟研究，将CMP分布的特性与混合泊松概率分布的特性进行比较，例如泊松-对数正态分布、泊松-林德利分布和泊松-逆高斯分布，这些分布传统上被用来对过度分散的数据进行建模。第二个目标是推导和发展双变量CMP值分布的推断方法。为了展示CMP分布及其扩展的灵活性，将考虑对真实世界数据的应用。这项研究将扩大可用于建立计数数据模型的概率分布的类别。

Catalyst Projects provide support for Historically Black Colleges and Universities to work towards establishing research capacity of faculty to strengthen science, technology, engineering and mathematics undergraduate education and research. It is expected that the award will further the faculty member's research capability, improve research and teaching at the institution, and involve undergraduate students in research experiences. This project at North Carolina Central State University will investigate mathematical modeling with probability distributions and will provide an opportunity for undergraduate students to enhance their statistical knowledge through research experiences. The researcher has established a strong collaboration with faculty at Georgetown University.Count data arise in a variety of situations and are often modeled with a Poisson distribution. In many cases, however, the data do not satisfy the Poisson distribution's property of equal mean and variance. This project will examine the two-parameter Conway-Maxwell-Poisson (CMP) distribution that accounts for over-dispersion where the variance is greater than the mean and under-dispersion where the variance is less than the mean. The CMP distribution captures the Poisson, Bernoulli, and geometric distributions as special cases. The first goal of this project is to conduct a simulation study which compares properties of the CMP distribution to those of mixed Poisson probability distributions, such as the Poisson-lognormal, Poisson-Lindley and Poisson-inverse Gaussian, that have traditionally been used to model over-dispersed data. The second goal is to derive and develop inferential methods for a bivariate CMP distribution. To demonstrate the flexibility of the CMP distribution and its extensions, applications to real-world data will be considered. This research will expand the class of probability distributions that are available to model count data.