Kullback–Leibler Asymmetry and Information Homogeneity in Natural Exponential Families

Natural exponential families admit two complementary structural descriptions: through their variance functions and through the information geometry generated by relative entropy and Fisher information. This paper establishes a direct connection between these descriptions. For a regular one-dimensional natural exponential family, we show that a fixed global direction of Kullback–Leibler asymmetry is equivalent to monotonicity of the variance function and, equivalently, to a fixed sign of the standardized skewness. In fact, a local ordering condition around each parameter point already forces the same ordering globally. We then reparametrize the family by Fisher–Rao arc length, which coincides with the classical variance-stabilizing coordinate, and study local spatial homogeneity of information divergence. Local homogeneity of the directed Kullback–Leibler divergence, of its antisymmetric part, of the magnitude of that asymmetry, or local radiality of Jeffreys divergence are shown to be equivalent. Each condition forces constant standardized skewness and consequently characterizes, up to a nonsingular affine transformation, precisely the normal location and fixed-shape gamma natural exponential families. Thus, weak local information conditions determine the global structure of the family. We further show that the global conclusions already follow from a sequential local ordering in the first theorem, and that position-independence of either the leading cubic KL-asymmetry coefficient or the quartic Jeffreys-excess coefficient alone yields the same normal/gamma rigidity.