On the notion of reproducibility and its full implementation to natural exponential families

Let F={F_θ:θ∈Θ⊂R} be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X₁)=F_θ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (α_n)_n≥1 and a mapping g_n:Θ→Θ,θ⟼g_n(θ) such that L(α_n∑ⁿ _i=1X_i)=F_gn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and g_n(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties.