NumCosmoMathStatsDistKernelST

Multivariate Student’s t kernel for NcmStatsDist.

The kernel of NcmStatsDistKernel with $\nu$ degrees of freedom, \begin{equation} \bar{K}(\chi^2) = \left(1 + \frac{\chi^2}{\nu}\right)^{-(\nu + d)/2}, \qquad u(\Sigma) = \frac{\Gamma(\nu/2)\,(\nu\pi)^{d/2}}{\Gamma\left((\nu + d)/2\right)}\sqrt{\det\Sigma}, \end{equation} that is, the density of the multivariate t distribution with location $\mu$ and scale matrix $h^2\Sigma$. For $\nu > 2$ its covariance is $\kappa h^2\Sigma$ with $\kappa = \nu/(\nu - 2)$; for $\nu \leq 2$ it has none. A sample is $x = \mu + h\sqrt{\nu/W}\,U^T z$, where $z$ holds $d$ independent standard normal variables, $W$ is a chi-squared variable with $\nu$ degrees of freedom and $\Sigma = U^T U$; see On Sampling from the Multivariate t Distribution, Marius Hofert. As $\nu \to \infty$ the kernel tends to NcmStatsDistKernelGauss; for finite $\nu$ it decays as a power of $\chi^2$.

The rule-of-thumb bandwidth is \begin{equation} h = \left[\frac{16 (\nu - 2)^2 (1 + d + \nu)(3 + d + \nu)} {(2 + d)(d + \nu)(2 + d + \nu)(d + 2\nu)(2 + d + 2\nu)\,n}\right]^{1/(d + 4)}, \end{equation} which minimizes the asymptotic mean integrated squared error for $n$ points drawn from a t density with $\nu$ degrees of freedom and scale matrix $\Sigma$. It is evaluated at $\nu = 3$ when $\nu < 3$, and tends to the Gaussian rule as $\nu \to \infty$.