Non-centered Skew Normal

Modeling
1

Is the implementation of a non-centered skew normal distribution really as simple as this?

data{
  int N;
  vector[N] x;  // observed values
  real tau;  // homoskedastic error

}



parameters {

  vector[N] y_raw; // normalized variable
  real mu, alpha;
  real<lower=0> sigma;


}


transformed parameters {
  vector[N] y;

 // transform
  y = mu + sigma * y_raw;


}

model {

  sigma ~ std_normal();
  mu ~ std_normal();
  alpha ~ normal(0,3);
 
 y_raw  ~ skew_normal(0,1,alpha);

  x ~ normal(y, tau);

}

I have done some simulations and fits and it seems to work properly with no divergences... just feel simpler than it should be. I suppose the priors on alpha could affect the parameterization... but not sure.
2

Your parameterization estimate is different from skew_normal of the centered.
There is a discussion and an posting in the old Google Stan Forum. It follows the
definition in wikipedia Skew normal distribution - Wikipedia
Ben Goodrich wrote:

For the general Wikipedia parameterization of the skew-normal, I think it would be

`// some priors on alpha, xi, and omegax ~ normal(xi, omega);lp__ <- lp__ + log(normal_cdf(alpha * x, xi, omega));`

but if xi = 0 and omega = 1, then

`// some prior on alphax ~ normal(0,1);`

`lp__ <- lp__ + log(Phi(alpha * x));`

> This is additional motivation for CDFs on the log scale
> in addition to numerical stability of truncation.

Indeed.
1 Like
3

@andre.pfeuffer thanks! do you have the link for the full post? I couldn’t find it and I would like to read the full post.

4

Skew normal. Link to old Stan forum

1 Like
5

Thanks. This seemed to do the trick perfectly.