Efficient estimation of COM-Poisson regression and a generalized additive model

The Conway-Maxwell-Poisson (CMP) or COM-Poisson regression is a popular model for count data due to its ability to capture both under dispersion and over dispersion. However, CMP regression is limited when dealing with complex nonlinear relationships. With today's wide availability of count data, especially due to the growing collection of data on human and social behavior, there is need for count data models that can capture complex nonlinear relationships. One useful approach is additive models; but, there has been no additive model implementation for the CMP distribution. To fill this void, we first propose a flexible estimation framework for CMP regression based on iterative reweighed least squares (IRLS) and then extend this model to allow for additive components using a penalized splines approach. Because the CMP distribution belongs to the exponential family, convergence of IRLS is guaranteed under some regularity conditions. Further, it is also known that IRLS provides smaller standard errors compared to gradient-based methods. We illustrate the usefulness of this approach through extensive simulation studies and using real data from a bike sharing system in Washington, DC.