I am working with a multivariate t distribution, say of dimension p. Given a point P = (x1, ..., xp) in the sample space I need to calculate the probability of the region of the sample space enclosed by the hyper-surface of all points with the same probability density as at P.
So far I have been working with this problem in two dimensions: p=2. In that case the constant-density hypersurface is an ellipse, the coordinates of which are easily calculated. I could not see any analytic solution to the integration problem so instead I just performed numerical integration of the bivariate density function over the region (the interior of the ellipse) to get its probability.
Now I have to extend the problem to working with higher dimensions, maybe up to p=10. Performing numerical integration on a 10-dimensional region with a curved boundary sounds very computationally expensive, so I would like to explore whether there might be an analytic solution, or at least maybe some sort of shortcut that can cut down the computational load.
If anybody has any suggestions I would be very grateful.
The task giving rise to this challenge is the need to assess how extreme (unlikely) is a hypothetical scenario involving significant moves in several different financial market variables, given assumptions about their volatilities and dependency structure.
For completeness, the multivariate t distribution I'm working with has density function: $$f(\mathbf x; \mathbf \Sigma,\mathbf\mu,\nu)=C_{\mathbf\Sigma} \left(1+\frac{1}{\nu}(\mathbf x-\mathbf{\mu})^T \mathbf \Sigma^{-1} (\mathbf x-\mathbf{\mu}) \right)^{-\frac{\nu +p}2}$$ where $\mathbf\Sigma$ is the "scale matrix" (related to the covariance matrix), $\mathbf\mu$ is the mean vector, $\nu$ is the degrees of freedom of the distribution and $C_{\mathbf\Sigma}$ is a constant that depends on $\mathbf\Sigma$.
Thank you
Andrew
PS: The quantity $\sqrt{(\mathbf x-\mathbf{\mu})^T \mathbf \Sigma^{-1} (\mathbf x-\mathbf{\mu})}$ is called the "Mahalanobis Distance" of the point $\mathbf x$ from its mean $\mathbf\mu$. That doesn't help me solve the problem but I thought I should mention it here as it may help in discoverability of this question and any answers for others who are also confronting this problem.
I've found the answer, courtesy of formula 6.5 in the following paper: "On the Multivariate t Distribution" by Michael Roth, Linköping University Electronic Press, 2013, http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-91686
The paper shows that the squared Mahalanobis distance (see formula in OP) of a point in the sample space, divided by $p$, has an F distribution with parameters $(p,\nu)$. So to find the area enclosed by a given elliptical contour hypersurface, we just calculate the Mahalanobis distance $h$ for any of its points. Then the probability weight enclosed by the hypersurface is $F_F(h^2/p; p, \nu)$ where $F_F$ is the cumulative distribution function of the F distribution with the given parameters.
Conversely, to find the hypersurface enclosing probability weight $B$, we just calculate its Mahalanobis distance as $h=\sqrt{p \times F_F{}^{-1}(B)}$. We assume the parameters $\mathbf\mu, \mathbf\Sigma$ are known. So we can then determine the hypersurface as the set of all points for which the expression in the OP for the Mahalanobis distance equals $h$.
Say we have an eigendecomposition of $\mathbf\Sigma$ that enables us to write the squared Mahalanobis distance as $$\mathbf u^T \mathbf \Lambda\mathbf u$$ where $\mathbf u=\mathbf Q(\mathbf x - \mathbf \mu)$, $\mathbf \Lambda$ is the diagonal eigenvalue matrix, and $\mathbf Q$ is the matrix whose columns are eigenvectors in the original basis.
Then the dimension of the ellipsoid along the $k$-th principal axis (eg "major axis" or "minor axis" for the bivariate case) is equal to $\frac{h^2}{\mathbf e_k{}^T \mathbf \Lambda\mathbf e_k} = \frac{h^2}{\lambda_k}$ where $\mathbf e_k$ is a vector comprising all zeros except for a 1 in the $k$th component, and $\lambda_k$ is the $k$-th eigenvalue.