The multivariate t-distribution

The multivariate t-distribution is a generalization of the univariate Student's t-distribution to multiple dimensions. It is commonly used for modeling heavy-tailed multivariate data and in robust statistics.

dist_multivariate_t(df = 1, mu = 0, sigma = diag(1))

Arguments

df: A numeric vector of degrees of freedom (must be positive).
mu: A list of numeric vectors for the distribution location parameter.
sigma: A list of matrices for the distribution scale matrix.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multivariate_t.html

In the following, let $\mathbf{X}$ be a multivariate t random vector with degrees of freedom df = $\nu$, location parameter mu = $\boldsymbol{\mu}$, and scale matrix sigma = $\boldsymbol{\Sigma}$.

Support: $\mathbf{x} \in \mathbb{R}^k$, where $k$ is the dimension of the distribution

Mean: $\boldsymbol{\mu}$ for $\nu > 1$, undefined otherwise

Covariance matrix:

$$ \text{Cov}(\mathbf{X}) = \frac{\nu}{\nu - 2} \boldsymbol{\Sigma} $$

for $\nu > 2$, undefined otherwise

Probability density function (p.d.f):

$$ f(\mathbf{x}) = \frac{\Gamma\left(\frac{\nu + k}{2}\right)} {\Gamma\left(\frac{\nu}{2}\right) \nu^{k/2} \pi^{k/2} |\boldsymbol{\Sigma}|^{1/2}} \left[1 + \frac{1}{\nu}(\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu})\right]^{-\frac{\nu + k}{2}} $$

where $k$ is the dimension of the distribution and $\Gamma(\cdot)$ is the gamma function.

Cumulative distribution function (c.d.f):

$$ F(\mathbf{t}) = \int_{-\infty}^{t_1} \cdots \int_{-\infty}^{t_k} f(\mathbf{x}) \, d\mathbf{x} $$

This integral does not have a closed form solution and is approximated numerically.

Quantile function:

The equicoordinate quantile function finds $q$ such that:

$$ P(X_1 \leq q, \ldots, X_k \leq q) = p $$

This does not have a closed form solution and is approximated numerically.

The marginal quantile function for each dimension $i$ is:

$$ Q_i(p) = \mu_i + \sqrt{\Sigma_{ii}} \cdot t_{\nu}^{-1}(p) $$

where $t_{\nu}^{-1}(p)$ is the quantile function of the univariate Student's t-distribution with $\nu$ degrees of freedom, and $\Sigma_{ii}$ is the $i$-th diagonal element of sigma.

Examples

dist <- dist_multivariate_t(
  df = 5,
  mu = list(c(1, 2)),
  sigma = list(matrix(c(4, 2, 2, 3), ncol = 2))
)
dimnames(dist) <- c("x", "y")
dist
#> <distribution[1]>
#> [1] MVT[2](5)

mean(dist)
#>      x y
#> [1,] 1 2
variance(dist)
#>             x y
#> [1,] 6.666667 5
support(dist)
#> <support_region[1]>
#> [1] R^2
generate(dist, 10)
#> [[1]]
#>                x          y
#>  [1,] -0.3308566  1.3370236
#>  [2,] -1.1270206  0.8409425
#>  [3,]  3.3641800  1.4620519
#>  [4,]  1.9347398  3.8888503
#>  [5,]  0.2964536  3.5556578
#>  [6,]  5.1773121  5.0947348
#>  [7,] -1.3326590  3.5913609
#>  [8,]  0.5244520  2.5669033
#>  [9,]  1.3997217 -0.6972666
#> [10,]  2.2500461  0.3456535
#> 

density(dist, cbind(2, 1))
#> [1] 0.02404305
density(dist, cbind(2, 1), log = TRUE)
#> [1] -3.727909

cdf(dist, 4)
#> [1] 0.7617795

quantile(dist, 0.7)
#>             x        y
#> [1,] 3.542593 3.542593
quantile(dist, 0.7, kind = "marginal")
#>             x        y
#> [1,] 2.118859 2.968961

Arguments

Details

See also

Examples