On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Shaul K. Bar-Lev
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Shaul K. Bar-Lev: Faculty of Industrial Engineering and Technology Management, Holon Institute of Technology, Holon 5810201, Israel

Mathematics, 2021, vol. 9, issue 13, 1-11

Abstract: Let F = F ? : ? ? ? ? R be a family of probability distributions indexed by a parameter ? and let X 1 , ? , X n be i.i.d. r.v.’s with L ( X 1 ) = F ? ? F . Then, F is said to be reproducible if for all ? ? ? and n ? N , there exists a sequence ( ? n ) n ? 1 and a mapping g n : ? ? ? , ? ? g n ( ? ) such that L ( ? n ? i = 1 n X i ) = F g n ( ? ) ? 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 ? n and g n ( ? ) . 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 .

Keywords: natural exponential families; reproducibility; infinite divisibility; variance function; functional equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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