This winter, in my ACT8595 course, I asked my students (that was some homework) to prove that it was possible to derive the conditional distribution when we have a Student-*t* random vector (and to get the analytical expression of the later). But before, let us recall a standard result about the Gaussian vector. If is a Gaussian random vector, i.e.

then has a Gaussian distribution. More precisely, it is a distribution, with

and is the Schur complement of the block of the matrix ,

Observe that is also related to well known quantity: in the bivariate case, where and are univariate Gaussian variables,

which is the slope in the linear regression of on .

In the case of the Student-*t* distribution, the conditional distrubution will not be a Student-*t* distribution anymore, but it will still be an elliptical distribution, and some interpretations of various quantities can actually be obtained.

The density of the multivariate centred Student-*t* distribution, with unit variance, and parameters and is

If we consider the following blocks,

then we can get that marginal distributions have a centred Student-*t* distribution, with unit variance, and parameters and ,

Then, to derive the conditional density, we can use Bayes formula,

One can write (as in Section 9.1 in Tong, 1990, *The Multivariate Normal Distribution*, but other expressions can be found in Section 2.5 in Fang, Ng and Kotz, 1989, *Symmetric multivariate and related distributions*, or in Section 1.11 in Kotz and Nadarajah, 2004, *Multivariate t distributions and their applications*) this conditional density as

with

and

This conditional distribution is elliptical, ~~but it is not a Student-~~*t* distribution, except in the case where , or when the correlation matrix ~~ is the identity~~.

[

June 2016] actually, as shown in Ding (2016), this is a Student-tdistribution. “Kotz & Nadarajah (2004) and Nadarajah & Kotz (2005) failed to recognize that the conditional distribution of the MVT distribution is also a MVT distribution due to the complexity of the conditional density function[…]Conditionaldistributions of elliptically contoured distributions are also elliptically contoured distributions. But thisdoes not immediately guarantee that conditional distributions of the MVT distributions are also MVTdistributions without some further algebra.“

Now, if we look at the components of this density, we can observe that we have

which was mentioned previously, in the Gaussian case: the term on the right is the conditional mean,

and the bloc that appears at several places is the conditional variance,

Now, if we want to visualize that conditional density, let us plot it. The code below is based on Bayes formula

> library(mnormt) > r=.6 > R=matrix(c(1,r,r,1),2,2) > nu=4 > f2=function(x2) dt(x2,df=nu) > f =function(x) dmt(x,S=R,df=nu) > f1.2=function(x1,x2) f(c(x1,x2))/f2(x2)

In order to compare that conditional density with a Student-*t* one, let us define the density of a non-centred Student-*t* random variable,

> dstd=function(x,mu,s,nu) gamma((nu+1)/2)/ + (gamma(nu/2)*s*sqrt(pi*nu))* + (1+1/nu*(x-mu)^2/(s^2))^(-(nu+1)/2)

Here is the function we can use to plot those two densities,

> graphdensity=function(x2=-1.5){ + vectx1=seq(-3,3,length=251) + y=Vectorize(function(x) f1.2(x,x2))(vectx1) + plot(vectx1,y,type="l",col="red",ylim=c(0,.5), + xlab="",ylab="") + abline(v=r*x2,lty=2) + lines(vectx1,dstd(vectx1,x2*r,sqrt(1-r^2),nu),col="blue",lty=2)} > graphdensity(-1.5)

In the case where , the two lines are rather close (the difference migth come from computational issues)

> graphdensity(-1)

and just to conclude, a last one

> graphdensity(0)

Hi Arthur,

In a recent paper, we also had this result (Cambou, Hofert, Lemieux “Quasi-random numbers for copula models”, Statistics & Computing). I specifically mentioned that this doesn’t seem to be easily found, so I gave it. The referee then said that it’s ‘well known’ (but couldn’t tell us where to find it…) and required the proof to be removed. It’s indirectly mentioned in Fang/Kotz/Ng, but not really explicitly…

Cheers,

Marius

Hi Marius

I had the same feeling… it’s well known, but no one can tell you what it could be, or even worst, where to find it ! too bad…

Dear Prof. Charpentier,

In a recent paper (Simard, C. and Rémillard, B., 2015, Forecasting time series with multivariate copulas. Depend. Model., 3, 59–82.), a multivariate conditional Student distribution is used to model stationary time series. In their appendix B, they show that a Conditional Student distribution is also a Student distribution. Is there a difference (e.g. different assumptions) which explain that the results shown on this page are not the same?

Kind regards and thanks a lot for your amazing blog.

Guillaume Evin

Actually, it looks like that conditional distribution might be a Student-t distribution. There is a nice proof in Ding (2016), http://arxiv.org/pdf/1604.00561v1.pdf.