## Abstract

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(X_{1})=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.

Original language | English |
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Article number | 1568 |

Journal | Mathematics |

Volume | 9 |

Issue number | 13 |

DOIs | |

State | Published - 1 Jul 2021 |

Externally published | Yes |

### 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.

## Keywords

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

## ASJC Scopus subject areas

- Mathematics (all)