Preferred point α-manifold and Amari's α-connections

We consider a new mathematical object called preferred point α-manifold, which is a natural extension of Lauritzen (1987) and Critchley et al. (1993). A sufficient and necessary condition is given to guarantee that preferred point α-geometry subsumes a statistical manifold. We obtained some properties related to Amari's expected geometry, in particular the duality in this new geometry. We also defined a new preferred point α-estimate to investigate the duality from the asymptotic statistical viewpoint. The mixture family is discussed as an illustration.