The Arithmetic–Geometric Gap as a Heterogeneity Diagnostic: Structure, Generalizations, and Statistical Theory

For a weight vector w on the simplex and a positive bounded signal c, the gap Δ = A(c;w) −G(c;w) between the weighted arithmetic and geometric means is a single scalar measuring theheterogeneity of c. This paper gives a unified account of the object in two parts. Part I establishesthe deterministic structure from one organizing principle: the logarithmic gap MLD =ln(A/G) is a Bregman information, equal at once to the Atkinson and Theil indices, to aKullback–Leibler divergence from an abundance-tilted distribution, and to the minimal averageBregman divergence to the arithmetic mean. From this follow sharp two-sided variance bounds,an additive decomposition that is the chain rule for relative entropy, majorization monotonicity,an embedding in the power-mean/diversity family, divergence-comparison corollaries, a lift topositive-definite matrices with a sharp Frobenius stability bound, and an honest categoricalplacement. Part II develops the statistical theory that the algebra cannot supply. Becausethe gap is a Bregman information, the influence function of its plug-in estimator is exactly thecentered pointwise Bregman divergence; this yields a central limit theorem with variance equalto the variance of that divergence, a nonasymptotic Efron–Stein bound, and—via the nonnegativityof the Bregman remainder—a one-sided Bernstein bound whose Gaussian core uses thetrue variance while the diagnostic’s bottom-sensitivity enters only logarithmically, locating the“fragility” in the tail scale rather than the variance. The same influence function decomposesblock-by-block for grouped sampling and lifts to sampled positive-definite matrices through thematrix Bregman divergence. Finally, the gap is shown to be a homogeneity test statistic witha weighted-χ2 null, local power equal to the Pearson test, and a quantified small-count failuremode with a bootstrap remedy. We claim no new inequalities; the contribution is the unification,several exact identities, and a statistical theory organized throughout by the Bregman structure.Every claim is verified by simulation.