An Additional Characterization of the Triple Consisting of Normal, Gamma, and Inverse Gaussian Distributions

Two characterizations of the triple consisting of the normal, gamma, and inverse Gaussian families are already known: one through the saddlepoint method, and another through the coincidence of the bilateral UMPU and generalized likelihood ratio tests. Bar-Lev and Reiser considered a steep two-parameter exponential family with canonical parameters (θ1,θ2), canonical statistics u1(X) and u2(X), and mean parameters (η1,η2), where ηi=Eθ[ui(X)], i=1,2. They showed that the relation θ2=−θ1ϕ′(η2) provides a sufficient condition for obtaining the normal, gamma, and inverse Gaussian families. In the present paper, we impose an additional mild local smoothness and positivity condition on the transformed kernel on the u2-scale and prove the converse. Thus, within this locally regular framework, the above relation is not only sufficient but also necessary for obtaining the same triple. Accordingly, this gives a third characterization of the triple consisting of the normal, gamma, and inverse Gaussian families. This points to an intrinsic structural phenomenon: three different and apparently unrelated characterizations single out precisely the same classical triple.