We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for P(a < Y < b) for Y ~ Np(0,V) and for repeated integrals of the density of Y. When V-1y > 0 in R3 the expansion for P(Y < y) reduces to one given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3-11]. in terms of the moments of Np(0,V-1). This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials. These are given explicitly.
Mots clés : asymptotic expansion, Leibniz' rule, repeated integrals of products, multivariate Hermite polynomials, multivariate normal
@article{PS_2011__15__340_0,
author = {Withers, Christopher S. and Nadarajah, Saralees},
title = {Expansions for {Repeated} {Integrals} of {Products} with {Applications} to the {Multivariate} {Normal}},
journal = {ESAIM: Probability and Statistics},
pages = {340--357},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2010005},
mrnumber = {2870519},
zbl = {1266.60024},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2010005/}
}
TY - JOUR AU - Withers, Christopher S. AU - Nadarajah, Saralees TI - Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal JO - ESAIM: Probability and Statistics PY - 2011 SP - 340 EP - 357 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2010005/ DO - 10.1051/ps/2010005 LA - en ID - PS_2011__15__340_0 ER -
%0 Journal Article %A Withers, Christopher S. %A Nadarajah, Saralees %T Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal %J ESAIM: Probability and Statistics %D 2011 %P 340-357 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2010005/ %R 10.1051/ps/2010005 %G en %F PS_2011__15__340_0
Withers, Christopher S.; Nadarajah, Saralees. Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 340-357. doi : 10.1051/ps/2010005. http://www.numdam.org/articles/10.1051/ps/2010005/
[1] and , The bivariate Hermite polynomials up to order six. Scand. J. Stat. 6 (1978) 127-128. | MR | Zbl
[2] , Introduction of “Table of Hh functions”, of Airey (1931), xxvi-xxxvii, Mathematical Tables, 2nd edition 1946, 3th edition 1951. British Association for the Advancement of Science, London (1931), Vol. 1,
[3] and , A geometric derivation of the shape density. Adv. Appl. Prob. 23 (1991) 496-514. | MR | Zbl
[4] , Moments and cumulants of the multivariate normal distribution. Stoch. Anal. Appl. 6 (1988) 273-278. | MR | Zbl
[5] and , Advanced Multivariate Statistics with Matrices. Springer, New York (2005). | MR | Zbl
[6] , and , Continuous Multivariate Distributions. 2nd edition, Wiley, New York (2000) Vol. 1. | MR | Zbl
[7] and , Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge (2004). | MR | Zbl
[8] , Fisher's repeated normal integral function and shape distributions. J. Appl. Stat. 25 (1998) 231-235. | MR | Zbl
[9] , Handbook of Statistical Tables. Addison Wesley, Reading, Massachusetts (1962). | MR | Zbl
[10] and , The mean resultant length of the spherically projected normal distribution. Stat. Prob. Lett. 78 (2008) 557-563. | MR | Zbl
[11] , An asymptotic expansion for the multivariate normal distribution and Mills ratio. J. Res. Nat. Bureau Stand. B 68 (1964) 3-11. | MR | Zbl
[12] , Mills ratio for multivariate normal distributions. Journal of Research of the National Bureau of Standards B 66 (1962) 93-96. | Zbl
[13] , Lower bounds for the multivariate normal Mills ratio. Ann. Prob. 7 (1979) 547-551. | MR | Zbl
[14] , The Multivariate Normal Distribution. Springer Verlag, New York (1990). | MR | Zbl
[15] , Infuential observations in multivariate linear models. Scand. J. Stat. 22 (1995) 207-222. | MR | Zbl
[16] , A chain rule for differentiation with applications to multivariate Hermite polynomials. Bull. Aust. Math. Soc. 30 (1984) 247-250. | MR | Zbl
[17] , The moments of the multivariate normal. Bull. Aust. Math. Soc. 32 (1985) 103-108. | MR | Zbl
Cité par Sources :
