Archive for Bessel functions

multiplying a Gaussian matrix and a Gaussian vector

Posted in Books with tags , , , , , on March 2, 2017 by xi'an

This arXived note by Pierre-Alexandre Mattei was actually inspired by one of my blog entries, itself written from a resolution of a question on X validated. The original result about the Laplace distribution actually dates at least to 1932 and a paper by Wishart and Bartlett!I am not sure the construct has clear statistical implications, but it is nonetheless a good calculus exercise.

The note produces an extension to the multivariate case. Where the Laplace distribution is harder to define, in that multiple constructions are possible. The current paper opts for a definition based on the characteristic function. Which leads to a rather unsavoury density with Bessel functions. It however satisfies the constructive definition of being a multivariate Normal multiplied by a χ variate plus a constant vector multiplied by the same squared χ variate. It can also be derived as the distribution of

Wy+||y||²μ

when W is a (p,q) matrix with iid Gaussian columns and y is a Gaussian vector with independent components. And μ is a vector of the proper dimension. When μ=0 the marginals remain Laplace.

Bessel integral

Posted in R, Statistics, University life with tags , , , on September 28, 2011 by xi'an

Pierre Pudlo and I worked this morning on a distribution related to philogenic philogenetic trees and got stuck on the following Bessel integral

\int_a^\infty e^{-bt}\,I_n(t)\,\text{d}t\qquad a,b>0

where In is the modified Bessel function of the first kind. We could not find better than formula 6.611(4) in Gradshteyn and Ryzhik. which is for a=0… Anyone in for a closed form formula, even involving special functions?