Seeking for shortest paths between negative binomial distributions. Application to the statistical analysis of counts data.

The statistical analysis of counts of living organisms brings information about the collective behavior of species. This task is often implemented in a nonparametric setting, but parametric distributions such as the negative binomial (NB) distributions studied here, are also very useful for modeling populations abundance. Considering the Riemannian manifold NB(DR) of NB distributions equipped with the Rao metrics DR, one can compute geodesic distances between species, which can be considered as absolute. But computing such a distance requires solving a second-order nonlinear differential equation, whose solution cannot be always found in an acceptable length of time with enough precision.Manté and Kidé (2016) proposed numerical remedies to this problem, which are completed here by Poisson Approximation combined with Differential Geometry techniques. The performances of the proposed method are investigated, and it is illustrated by displaying distributions of counts of marine species through multidimensional scaling (MDS) of the table of computed Rao's distances between species