# Skewed Multivariate Student-t distribution

**URL:** <https://discourse.mc-stan.org/t/skewed-multivariate-student-t-distribution/6760>\
**Category:** Modeling\
**Tags:** techniques\
**Created:** [November 28, 2018, 7:39am UTC](https://discourse.mc-stan.org/t/skewed-multivariate-student-t-distribution/6760 "2018-11-28T07:39:36Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![andre.pfeuffer](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@andre.pfeuffer](https://discourse.mc-stan.org/u/andre.pfeuffer)\
**Post date:** [November 28, 2018, 7:39am UTC](https://discourse.mc-stan.org/t/skewed-multivariate-student-t-distribution/6760/1 "2018-11-28T07:39:36Z")

</div>

GNU R package miscF has a function `rmvst` to draw samples from a skewed multivariate Student-t distribution. I translated it to Stan:

* * *

```stan
data {
  real<lower=1> nu;
  int p;
  vector[p] D;
  cov_matrix[p] Sigma;
}
transformed data {
  real sqrt_nu = sqrt(nu);
  matrix[p,p] L_Sigma = cholesky_decompose(Sigma);
}
parameters {
  vector<lower=0>[p] z;
  real<lower=0> w;
  vector[p] x;
}
transformed parameters {
  vector[p] Y = z .* D + sqrt(w * nu) * L_Sigma * x;
}
model {
  w ~ inv_chi_square(nu);
  z ~ normal(0, 1);
  x ~ normal(0, 1);
}

```

The main parts of R code are:

```
p <- nrow(D)
Y <- matrix(0, n, p)  

for(i in 1:n){
    u <- runif(p, 1/2, 1)
    z <- qnorm(u, 0, 1)
    w <- rgamma(1, nu/2, nu/2)
    Y[i,] <- mvrnorm(1, Mu+D%*%z, Sigma/w)
}
Y

```

I replaced `qnorm(runif(10000,0.5,1),0,1)` with `abs(rnorm(10000))`. Changed  
`gamma` to `inv_chi_squared`.

Can somebody confirm, I did it right? I looks “too easy” for my 2 cents. (I didn’t add the mu)

[multivariate\_skew\_t2a.R](https://canada1.discourse-cdn.com/flex030/uploads/mc_stan/original/2X/0/0cf0b0c9bb72062e698551f157e623779b4cb5d9.R) (833 Bytes)

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<div class="post-metadata">

**Author:** ![bbbales2](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bbbales2/32/77_2.png) [@bbbales2](https://discourse.mc-stan.org/u/bbbales2)\
**Post date:** [November 29, 2018, 2:46pm UTC](https://discourse.mc-stan.org/t/skewed-multivariate-student-t-distribution/6760/2 "2018-11-29T14:46:10Z")

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Just going by the little blurb in Wikipedia ([https://en.wikipedia.org/wiki/Multivariate\_t-distribution#Definition](https://en.wikipedia.org/wiki/Multivariate_t-distribution#Definition)), what you’ve written should imply:

z \sim N(0, \text{diag}(D))  
Y \sim \text{multivariate\_t}(\nu, z, \Sigma)

So it looks right enough to me :P?

I’d check the covariance of your samples against the analytical form on Wikipedia for various sigma and nu first. Just use \nu \> 2 or whatever so the covariance is finite. Then you can check your student t against the implementation in that package to make sure they’re in line.

edit: Made Latex prettier