1

In the p-variate normal distribution ($\mathbf{N_p(\mu,\Sigma)}$), the solid ellipsoid of x values satisfying

$(\mathbf{x-\mu)'\Sigma^{-1}(\mathbf{x}-\mu)}\leq \chi^2_p(\alpha)$

has probability $1-\alpha$ , where $\chi^2_p(\alpha)$ is the upper $(100\alpha)$th percentile of a chi-square distribution with p degrees of freedom.

Is there a similar result for the Multivariate Laplace (symmetric) distribution? That is, which ellipsoid has probability $1-\alpha$ in the Multivariate Laplace case?

Helder Alves Arruda
  • 11
  • 1
    Welcome to Cross Validated! In the case of a multivariate normal distribution, there is a standard definition. I do not know of a standard definition of a multivariate Laplace distribution. Is there one? How the marginal distributions relates to each other will be critical to creating the joint distribution. For instance, comments like these could be written about Laplace margins instead of Gaussian. – Dave Oct 08 '22 at 17:58
  • Thanks for the answer! I found a similar result for the multivariate t distribution. I saw drawings of the contours of a bivariate laplace, but it was not explained how the contours were made. If there is a way to get an estimate of such ellipsoids using R, it would be very helpful! – Helder Alves Arruda Oct 08 '22 at 18:11
  • How would you make contour lines of any other multivariable function? – Dave Oct 08 '22 at 21:32
  • I'm interested in elliptical distributions, whose contours are ellipsoids. For example, for p-variate t the contours are given by: (x - \mu)' \Sigma^-1 (x - \mu) < p F(\alpha, p, v). Where F(\alpha, p, v) is the upper (100α)th percentile of the F distribution and v is the degrees of freedom. – Helder Alves Arruda Oct 08 '22 at 23:33
  • What happens when you plot the contour lines of a multivariate Laplace, such as independent margins? – Dave Oct 09 '22 at 00:28
  • I don't know how to plot contour lines, but I think if they are independent and identically distributed we will have circles. – Helder Alves Arruda Oct 09 '22 at 02:41
  • 1
    There is an analogous result: because this is an elliptic distribution, the contours are also ellipsoids. Instead of a chi-squared distribution governing the radial variable, though, the distribution has a density proportional to a monomial times a Bessel K function. Unless you are using a powerful symbolic platform like Mathematica, you will likely need to find the ellipsoid through numeric integration and a root finder. – whuber Oct 09 '22 at 13:43
  • Thanks for the answer! Do you know of any references that mention this topic? – Helder Alves Arruda Oct 09 '22 at 19:58

0 Answers0