χ 2 tests in the Riemannian manifold of negative binomial distributions: a geometrical approach. Application to ecological data

Claude Manté (1)

1 MIO - Institut méditerranéen d'océanologie

The statistical analysis of counts of living organisms brings information about the collective behavior of species (schooling, habitat preference, etc), possibly depending on their socio-biological characteristics (aggregation, growth rate, reproductive power, survival rate, etc). The negative binomial (NB) distribution is widely used to model such data, but the parametric approach suers from an important nuisance: the visual distance between parameters cannot be considered as a relevant distance between distributions, because these parameters are not commensurable in general (dierent ecological meaning, dierent ranges, ...). On the contrary, considering the Riemannian manifold N B(D R) of NB distributions equipped with the Rao metrics D R , one can compute intrinsic distances between species which can be considered as absolute. Suppose now D R (A, B) is small; does this means that species A and B have similar characteristics? We rst tackle this point by focusing on geometrical aspects of the χ 2 goodness-of-t test for distributions in N B(D R), and question its feasibility in connection with the position of the distributions. We afterward focus on a kin problem: performing two-sample χ 2 tests and building condence regions in the same geometrical setting. Our purpose is illustrated by processing eld experiment data studied by Bliss and Fisher in the fties. Notations Let's introduce rst some notation. Consider a Riemannian manifold M, and a parametric curve α : [a, b] → M; its rst derivative will be denotedα. We will also consider for any θ ∈ M the local norm V g (θ) associated with the metrics g on the tangent space T θ M : ∀ V ∈ T θ M, V g (θ) := V t .g(θ).V. (1) A parametric probability distribution L i will be identied with its coordinates with respect to some chosen parametrization; for instance, we will write L i ≡ (φ i , µ i) for some negative binomial distribution. Finally, logical propositions will be combined by using the classical connectors ∨ (or) and ∧ (and).

Mots clés: Chi square test; Fisher-Rao distance; Riemannian manifolds; Multidimensional scaling

Domaines: Statistiques [math.ST]

Claude Manté. χ 2 tests in the Riemannian manifold of negative binomial distributions: a geometrical approach. Application to ecological data. 2020. ⟨hal-02931054⟩