Comptes Rendus
no. 8
Probability Theory
The Bessel ratio distribution
[Distribution du rapport de deux probabilités de type Bessel]

Présenté par : Marc Yor

Saralees Nadarajah1 ; Samuel Kotz2
1 School of Mathematics, University of Manchester, Manchester M60 1QD, UK
2 Department of Engineering Management and Systems Engineering, George Washington University, Washington, DC 20052, USA
Comptes Rendus. Mathématique, Volume 343 (2006) no. 8, pp. 531-534.

Soient X et Y deux variables aléatoires ; on en déduit la valeur du rapport X/Y dans le cas où X et Y sont des variables aléatoires dont les densités de probabilités sont de type Bessel.

Let X and Y be two random variables; then the exact distribution of the ratio X/Y is derived when X and Y are independent Bessel function random variables.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.09.031
Saralees Nadarajah 1 ; Samuel Kotz 2

1 School of Mathematics, University of Manchester, Manchester M60 1QD, UK
2 Department of Engineering Management and Systems Engineering, George Washington University, Washington, DC 20052, USA 
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Saralees Nadarajah; Samuel Kotz. The Bessel ratio distribution. Comptes Rendus. Mathématique, Volume 343 (2006) no. 8, pp. 531-534. doi : 10.1016/j.crma.2006.09.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.031/

[1] B.C. Bhattacharyya The use of McKay's Bessel function curves for graduating frequency distributions, Sankhyā, Volume 6 (1942), pp. 175-182

[2] Y.P. Chaubey; A.B.M. Nur Enayet Talukder Exact moments of a ratio of two positive quadratic forms in normal variables, Communications in Statistics—Theory and Methods, Volume 12 (1983), pp. 675-679

[3] I.S. Gradshteyn; I.M. Ryzhik Table of Integrals, Series, and Products, Academic Press, San Diego, 2000

[4] S. Kotz; T.J. Kozubowski; K. Podgorski The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance, Birkhäuser, Boston, 2001

[5] J. von Neumann Distribution of the ratio of the mean square successive difference to the variance, Annals of Mathematical Statistics, Volume 12 (1941), pp. 367-395

[6] S.B. Provost; E.M. Rudiuk The exact density function of the ratio of two dependent linear combinations of chi-square variable, Annals of the Institute of Statistical Mathematics, Volume 46 (1994), pp. 557-571

[7] A.P. Prudnikov; Y.A. Brychkov; O.I. Marichev Integrals and Series, vols. 1–3, Gordon and Breach Science Publishers, Amsterdam, 1986

[8] T. Toyoda; K. Ohtani Testing equality between sets of coefficients after a preliminary test for equality of disturbance variances in two linear regressions, Journal of Econometrics, Volume 31 (1986), pp. 67-80

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