Inverse distributions: the logarithmic case

Dario Sacchetti

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 4, page 785-795
  • ISSN: 0010-2628

Abstract

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In this paper it is proved that the distribution of the logarithmic series is not invertible while it is found to be invertible if corrected by a suitable affinity. The inverse distribution of the corrected logarithmic series is then derived. Moreover the asymptotic behaviour of the variance function of the logarithmic distribution is determined. It is also proved that the variance function of the inverse distribution of the corrected logarithmic distribution has a cubic asymptotic behaviour.

How to cite

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Sacchetti, Dario. "Inverse distributions: the logarithmic case." Commentationes Mathematicae Universitatis Carolinae 39.4 (1998): 785-795. <http://eudml.org/doc/248279>.

@article{Sacchetti1998,
abstract = {In this paper it is proved that the distribution of the logarithmic series is not invertible while it is found to be invertible if corrected by a suitable affinity. The inverse distribution of the corrected logarithmic series is then derived. Moreover the asymptotic behaviour of the variance function of the logarithmic distribution is determined. It is also proved that the variance function of the inverse distribution of the corrected logarithmic distribution has a cubic asymptotic behaviour.},
author = {Sacchetti, Dario},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {natural exponential family; Laplace transform; variance function; logarithmic distribution; inverse distribution; variance function},
language = {eng},
number = {4},
pages = {785-795},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Inverse distributions: the logarithmic case},
url = {http://eudml.org/doc/248279},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Sacchetti, Dario
TI - Inverse distributions: the logarithmic case
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 4
SP - 785
EP - 795
AB - In this paper it is proved that the distribution of the logarithmic series is not invertible while it is found to be invertible if corrected by a suitable affinity. The inverse distribution of the corrected logarithmic series is then derived. Moreover the asymptotic behaviour of the variance function of the logarithmic distribution is determined. It is also proved that the variance function of the inverse distribution of the corrected logarithmic distribution has a cubic asymptotic behaviour.
LA - eng
KW - natural exponential family; Laplace transform; variance function; logarithmic distribution; inverse distribution; variance function
UR - http://eudml.org/doc/248279
ER -

References

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  10. Mora M., Classification de fonctions variance cubiques des familles exponentielles sur ℝ , C.R. Acad. Sci. Paris Sér I Math. 302 (1986), 587-590. (1986) MR0844163
  11. Morris C.N., Natural exponential families with quadratic variance functions, Ann. Statist. 10 (1982), 65-80. (1982) Zbl0498.62015MR0642719
  12. Sacchetti D., Inverse distribution: an example of non existence, Atti dell' Accademia delle Scienze Lettere ed Arti di Palermo, 1993. 
  13. Seshadri V., The Inverse Gaussian distribution, Oxford University Press, Oxford, 1993. Zbl0942.62011MR1306281
  14. Tweedie M.C.K., Inverse statistical variates, Nature 155 (1945), 453. (1945) Zbl0063.07892MR0011907

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