| Summary: | This paper is concerned with the formulation and computation of average problems on the multinomial and negative multinomial models. It can be deduced that the multinomial and negative multinomial models admit complementary geometric structures. Firstly, we investigate these geometric structures by providing various useful pre-derived expressions of some fundamental geometric quantities, such as Fisher-Riemannian metrics, <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-connections and <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-curvatures. Then, we proceed to consider some average methods based on these geometric structures. Specifically, we study the formulation and computation of the midpoint of two points and the Karcher mean of multiple points. In conclusion, we find some parallel results for the average problems on these two complementary models.
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