# Density plot of the skew-t distribution

I am using the sgt package in R to recreate the plot from Hansen's paper ( available here http://www.ssc.wisc.edu/~bhansen/papers/ier_94.pdf on page 8) using random draws from the skew-t distribution.

I begin with $\eta=30$ using the following code:

x = rsgt(1000000, mu = 0, sigma = 1, lambda = 0.5, p = 2, q=30, mean.cent=TRUE, var.adj=TRUE)
t=density(x)
plot(t, xlim=c(-2, 2))

And I obtain a plot that is analogycal to the one given in the paper. However, using $\eta = 3$ or $\eta = 2.1$ (replace q with one of those values) results in much different plots, which look weird. Do you have any suggestions to how to solve this matter?

Edit: I include the plots I want to obtain and the ones I can obtain.

The one for $\eta=30$:

The one for $\eta=2.1$:

• It would be helpful if you could post the resulting plots here. – vonjd Feb 23 '16 at 8:31

The rsgt is a skewed generalized t distribution, whereas your picture is a skewed student-t distribution. Try using fGarch package.

Plot reproduced:

library(fGarch)
x<-seq(-2.5, +2.5, by=0.001)
plot(x,
fGarch::dsstd(x, mean = 0, sd = 1, nu = 30, xi = 1 + 0.5),
type = "l",
ylim=c(0, 2.4), lty = 1,
xlab="z",
ylab=expression(paste("g(z|",nu,",",lambda,")")),
main="CONDITIONAL DENSITY ESTIMATION")

lines(x,
fGarch::dsstd(x, mean = 0, sd = 1, nu = 3.0, xi = 1 + 0.5),
type = "l",
ylim=c(0, 2.4),
lty = 2)

lines(x,
fGarch::dsstd(x, mean = 0, sd = 1, nu = 2.1, xi = 1 + 0.5),
type = "l",
ylim=c(0, 2.4),
lty = 5)

legend(x="topleft", legend = c(expression(paste(eta,"=2.1")),
expression(paste(eta,"=3.0")),
expression(paste(eta,"=30"))),
lty=c(5,2,1))


• Many thanks for this solution! One more question: Why are you using xi = 1 + 0.5 instead of $0.5$? Because I was under the impression that this version of the skew-t Distribution by Fernandez and Steel is different from the one of Hansen. – Masher Feb 23 '16 at 21:02
• And why could I obtain the plot for $\eta = 30$ using the method proposed by me and not the other values? Is my random sampling wrong? I am also interested in random sampling from this distribution, that is why this is important to me. – Masher Feb 23 '16 at 21:16
• the $alpha=0.5$ is a right skew, but for sstd the xi=1 is no skew, so making it right-skew, you need to add the +0.5 – rbm Feb 24 '16 at 20:47
• Thank you! And could you tell me what is wrong with my method for other values of the parameters, as for $\eta = 30$ the method I described seems to work just fine. – Masher Feb 24 '16 at 21:00