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December, 1986 Reproducibility and Natural Exponential Families with Power Variance Functions
Shaul K. Bar-Lev, Peter Enis
Ann. Statist. 14(4): 1507-1522 (December, 1986). DOI: 10.1214/aos/1176350173


Let $X_1, \cdots, X_n$ be i.i.d. r.v.'s having common distribution belonging to a family $\mathscr{F} = \{F_\theta: \theta \in \Theta \subset R\}$ indexed by a parameter $\theta$. $\mathscr{F}$ is said to be reproducible if there exists a sequence $\{\alpha(n)\}$ such that $\mathscr{L}(\alpha(n)\sum^n_{i=1} X_i) \in \mathscr{F}$ for all $\theta \in \Theta$ and $n = 1, 2, \cdots$. This property is investigated in connection with linear exponential families of order 1 and its intimate relationship to such families having a power variance function is demonstrated. Moreover, the role of such families is examined, in a unified approach, with respect to properties relative to infinite divisibility, steepness, convolution, stability, self-decomposability, unimodality, and cumulants.


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Shaul K. Bar-Lev. Peter Enis. "Reproducibility and Natural Exponential Families with Power Variance Functions." Ann. Statist. 14 (4) 1507 - 1522, December, 1986.


Published: December, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0657.62016
MathSciNet: MR868315
Digital Object Identifier: 10.1214/aos/1176350173

Primary: 60E05
Secondary: 62E10

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 4 • December, 1986
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