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Help with multidimensional integral (Dirichlet type)

  1. Jun 30, 2008 #1
    In a research problem (related to statistics), I've encountered a k-dimensional Dirichlet integral of the form:

    [tex]f_k(x) = \int_{x_c}^\infty dx_1 \ldots \int_{x_c}^\infty dx_k\, \delta\left(x-\left(x_1+\ldots+x_k\right)\right)\ x_1^{-s} \ldots x_k^{-s}[/tex]

    where [tex]s>0[/tex]. Essentially I am integrating the function [tex]x_1^{-s} \ldots x_k^{-s}[/tex] over the plane defined by [tex]x_1+\ldots+x_k=x[/tex], with the criterion [tex]x_k \geq x_c[/tex] providing the necessary cutoff so that the integral converges.

    While I have been able to write a program to numerically integrate this, I am wondering if there is a closed form solution. It seems that this integral is related in some way to a k-dimensional generalization of the incomplete beta function or an integral of a Dirichlet probability distribution. However, when trying to generalize some of the derivations I've found in the literature, I find I run into trouble because of the cutoff (the integral is usually done for [tex]s<0[/tex]). I've been able to derive

    [tex]f_2(x) = x^{(-2(s-1)-1)}\, \theta\left(x-2x_c\right)\, \beta\left(\frac{x_c}{x},1-\frac{x_c}{x},1-s,1-s\right)[/tex] (where [tex]\theta(x)[/tex] is the Heaviside step function and [tex]\beta(x)[/tex] is the generalized incomplete beta function)

    but have been unable to generalize the canonical derivations to higher k. It seems like the answer should look something like:

    [tex]f_k(x)=x^{(-k(s-1)-1)}\, \theta\left(x-kx_c\right)\, g_k\left(\frac{x_c}{x},s\right)[/tex]

    where [tex]g_k\left(\frac{x_c}{x},s\right)[/tex] is the desired generalization.

    Does anybody here have experience with these type of statistical integrals, or could point me to some literature? It seems like this problem must be solved, but I don't have much experience with these sorts of problems. I am also wondering if the approach taken in this paper, which evaluates the integral by a sum of the integrand evaluated at the vertices of the plane, can be applied to this problem (I suspect that it cannot?) Thanks.
     
  2. jcsd
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