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

Shaul K Bar-Lev

doi:10.3390/math9131568

Abstract

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(αn∑i=1nXi)=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 .

Highlights

The notion of a distribution function, F, which is reproductive with respect to a parameter θ ∈ Θ, a nonempty set of R, was first introduced by Wilks [1] as follows: Let X1 and X2 be independent r.v.s with distributions F (.; θ1 ) and F (.; θ2 ), respectively
Where steepness of F means that the mean parameter space of F coincides with the interior of its common convex support. This definition of steepness will be further elaborated in the sequel. Their implementation of the notion of reproducibility of natural exponential families (NEFs) led to the characterization of the class of steep NEFs having power variance functions (VFs)
Bar-Lev and Enis [5] due to their steepness restriction. The latter assumption did not allow them to characterize the subclass of non-steep NEFs having VFs with negative power

Summary

Introduction

The notion of a distribution function, F, which is reproductive with respect to a parameter θ ∈ Θ, a nonempty set of R, was first introduced by Wilks [1] as follows: Let X1 and X2 be independent r.v.s with distributions F (.; θ1 ) and F (.; θ2 ), respectively. Bar-Lev and Enis [5] implemented their Definition 1 to the class of natural exponential families (NEFs) by imposing the following restrictions:. This definition of steepness will be further elaborated in the sequel Their implementation of the notion of reproducibility of NEFs led to the characterization of the class of steep NEFs having power variance functions (VFs). The analysis of the notion of reproducibility from a different angle than that formulated in Definition 1 and Equation (1) was given in Bar-Lev and Cassalis [6,7] They defined an NEF F to be reproducible in the broad sense, as follows. Bar-Lev and Enis [5] due to their steepness restriction The latter assumption did not allow them to characterize the subclass of non-steep NEFs having VFs with negative power.

Some General Noteworthy Comments on the Notion of Reproducibility

NEFs—Preliminaries and Characterization by the Reproducibility Property

Some Preliminaries on NEFs

Conclusions and Topics for Further Research