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A Classification of Reproducible Natural Exponential Families in the Broad Sense


Let μ be a positive Radon measure on \(\mathbb{R}\) having a Laplace transform L μ and F=F(μ) be a natural exponential family (NEF) generated by μ. A positive Radon measure μ λ with λ>0 and its associated NEF F λ =F(μ λ ) are called the λth convolution powers of μ and F, respectively, if \(L_{\mu _\lambda } = L_\mu ^\lambda\). Let f α,β (F) be the image of an NEF F under the affine transformation f α,β :xαx+β. If for a given NEF F, there exists a triple (α,β,λ) in \(\mathbb{R}^3 \) such that f α,β (F)=F λ , we call F a reproducible NEF in the broad sense. In other words, an NEF F is reproducible in the broad sense if a convolution power of F equals an affine transformation of F. Clearly, for λ=1, F is reproducible in the broad sense if there exists an affinity under which F is invariant. In this paper we obtain a complete classification of the class of reproducible NEF's in the broad sense. We show that this class is composed of infinitely divisible NEF's and that it contains the class of NEF's having exponential and power variance functions as well as NEF's constituting discrete versions of the latter NEF's. We also provide a characterization of the reproducible NEF's in the broad sense in terms of their associated exponential dispersion models.

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  1. 1.

    Bar-Lev, S. K., and Enis, P. (1986). Reproducibility and natural exponential families with power variance-functions. Ann. Statist. 14, 1507–1522.

    Google Scholar 

  2. 2.

    Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory, Wiley, New York.

    Google Scholar 

  3. 3.

    Casalis, M. (1991). Familles exponentielles naturelles sur ℝd invariantes par un groupe. I. Stat. Rev. 59, 241–262.

    Google Scholar 

  4. 4.

    Devroye, L. (1986). Non Uniform Random Variate Generation, Springer-Verlag, New York.

    Google Scholar 

  5. 5.

    Jørgensen, B. (1986). Some properties of exponential dispersion models. Scand. J. Statist. 13, 187–198.

    Google Scholar 

  6. 6.

    Jørgensen, B. (1987). Exponential dispersion models (with discussion). J. Roy. Statist. Soc. B 49, 127–162.

    Google Scholar 

  7. 7.

    Jørgensen, B. (1989). Lectures on the Theory of Exponential Dispersion Models and Analysis of Deviance, Monografias de Matematica, Vol. 23, Instituto de Mathematica Pura Aplicada, Rio de Janeiro.

    Google Scholar 

  8. 8.

    Jørgensen, B. (1997). The Theory of Dispersion Models, Monographs on Statistics and Applied Probability, Vol. 76, Chapmann and Hall.

  9. 9.

    Letac, G., and Mora, M. (1990). Natural exponential families with cubic variance functions. Ann. Statist. 18, 1–37.

    Google Scholar 

  10. 10.

    Letac, G. (1992). Lectures on Natural Exponential Families and Their Variance Functions, Monografias de Matematica, Vol. 50, Instituto de Mathematica Pura Aplicada, Rio de Janeiro.

    Google Scholar 

  11. 11.

    Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65–80.

    Google Scholar 

  12. 12.

    Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In Ghosh, J. K., and Roy, J. (eds.), Statistics: Applications and New Directions, Proceedings of the Indian Statistical Institute Golden Jubilee International Conference, Indian Statistical Institute, Calcutta, pp. 579–604.

    Google Scholar 

  13. 13.

    Wilks, S. S. (1963). Mathematical Statistics, Wiley, New York.

    Google Scholar 

Download references


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Bar-Lev, S.K., Casalis, M. A Classification of Reproducible Natural Exponential Families in the Broad Sense. Journal of Theoretical Probability 16, 175–196 (2003).

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  • Exponential dispersion models
  • infinitely divisible distributions
  • natural exponential families
  • reproducibility
  • variance functions