Inverse distributions: the logarithmic case
Sacchetti, Dario
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998), p. 785-795 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1998-01-01
Classification:  60E05,  60F05,  62E10 
@article{119053,
     author = {Dario Sacchetti},
     title = {Inverse distributions: the logarithmic case},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {39},
     year = {1998},
     pages = {785-795},
     zbl = {1060.62504},
     mrnumber = {1715467},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119053}
}

Sacchetti, Dario. Inverse distributions: the logarithmic case. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 785-795. http://gdmltest.u-ga.fr/item/119053/

Barndorff-Nielsen O. Information and exponential families in statistical inference, Wiley, New York, 1978. | MR 0489333

Dieudonné, J. Infinitesimal Calculus, Houghton Mifflin, Boston, 1971. | MR 0349286

Fichtenholz G.M. Functional Series, Gordon and Breach, Science Publishers, 1970. | MR 0463743 | Zbl 0213.35001

Guest G. Laplace Transform and an Introduction to Distributions, Ellis Horwood, 1991. | MR 1287158

Johnson N.L.; Kotz S. Discrete Distributions, Houghton Mifflin, Boston, 1969. | MR 0268996 | Zbl 1092.62010

Jorgensen B.; Martinez J.R.; Tsao M. Asymptotic behaviour of the variance function, Scand. J. Statist. 21 (1994), 223-243. (1994) | MR 1292637

Letac G. La reciprocité des familles exponentielles naturelles sur $\Bbb R$, C.R. Acad. Sci. Paris 303 Ser. I2 (1986), 61-64. (1986) | MR 0851270

Letac G. Lectures on natural exponential families and their variance functions, I.M.P.A., Rio de Janeiro, 1991. | MR 1182991 | Zbl 0983.62501

Letac G.; Mora M. Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990), 1-37. (1990) | MR 1041384 | Zbl 0714.62010

Mora M. Classification de fonctions variance cubiques des familles exponentielles sur $\Bbb R$, C.R. Acad. Sci. Paris Sér I Math. 302 (1986), 587-590. (1986) | MR 0844163

Morris C.N. Natural exponential families with quadratic variance functions, Ann. Statist. 10 (1982), 65-80. (1982) | MR 0642719 | Zbl 0498.62015

Sacchetti D. Inverse distribution: an example of non existence, Atti dell' Accademia delle Scienze Lettere ed Arti di Palermo, 1993.

Seshadri V. The Inverse Gaussian distribution, Oxford University Press, Oxford, 1993. | MR 1306281 | Zbl 0942.62011

Tweedie M.C.K. Inverse statistical variates, Nature 155 (1945), 453. (1945) | MR 0011907 | Zbl 0063.07892