Relationship between H\"older Divergence and Functional Density Power Divergence: Intersection and Generalization

In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom -- the H\"older divergence and the functional density power divergence (FDPD) -- based on their intersection and generalization. These divergence families include the density power divergence and the -divergence as special cases. First, we prove that the intersection of the H\"older divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001). Subsequently, motivated by the fact that H\"older's inequality is used in the proofs of nonnegativity for both the H\"older divergence and the FDPD, we define a generalized divergence family, referred to as the -H\"older divergence. The nonnegativity of the -H\"older divergence is established through a combination of the inequalities used to prove the nonnegativity of the H\"older divergence and the FDPD. Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the -H\"older divergence. Finally, we prove that imposing the mathematical structure of the H\"older score on a composite scoring rule results in the -H\"older divergence.