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

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Let F={Fθ: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)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αnn i=1Xi)=Fgn(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 gn(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.

Original languageEnglish
Article number1568
Issue number13
StatePublished - 1 Jul 2021
Externally publishedYes

Bibliographical note

Funding Information:
This research was funded by the Spanish Ministry of Education, Culture and Sports, grant number CAS18/00474 (José Castillejo program).

Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.


  • Functional equation
  • Infinite divisibility
  • Natural exponential families
  • Reproducibility
  • Variance function

ASJC Scopus subject areas

  • Mathematics (all)


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