Well I don't know that much about the NR method used to find the cdf of the factor.The method I have seen is the following: basically you are interested to know the cdf (F) of Z=S_1+S_2where S_1, S_2 are two student rv.An easy way to find this is to simulate ("n" times) S_1 and S_2 and then look at the empirical distribution of Z, F_n. *This can be done by a nonparametric estimation of the cdf of Z. *Or you can use :F_n(z)=1/n sum_{i=1}^n 1(Z_i<z)If you do a large simulation then your result for the empirical distribution Z_n should be close to the "real" distribution of Z (as F_n converges uniformly to F, from the Glivenko-Cantelli theorem).

Hi Meteor,What exactly are S_1 and S_2 ?I mean, I am not able to understand why we are talking of adding two student t random variables here?Tarun

just to make a clarification (possibly trivial) .. I am trying to implement the "double t copula" and not the "student t copula".

As I mentioned in my previous post S_1 and S_2 are two student random variable so Z is not student (since student distribution is not stable).This is exactly the point of the double t copula: you have two factors which have student distribution (whereas in the student copula you have three factors, 2 with gaussian distribution and one with a chi square).

Hi meteor,Thanks a lot! I got the point!From the implementation perspective, will it be enough to generate an empirical distribution just once and "hard-code" it in the software?Secondly, once I have the empirical CDF, to get the inverseCDF do I have to use some kind of an approximation there?Thirdly, what should be the size of "n" for me to assume the Glivenko-Cantelli theorem in my implementation?Thanks again.Tarun

1)Yes just simulate the (scaled) sum of of the student once and use the empirical distribution for all your inversions. 2)Correct, you need some kind of approximation. You just need to estimate the empirical cdf F_n(z) at a lot of points (ie lots of z) and then aproximate as follows, suppose z\in(z_i,z_j) (j>i) then set F(z)=F_n(z_i); if your intervals (z_i,z_j) are small then the aproximation should be close.3)For G-C theorem n goes to infinity. But I guess with n=10^9 you shouldn't be that wrong. You can even choose n to be larger since you are doing that simulation once (and then using its result after).

tarunmakhija your question about the convergence between the emprical and real cdf spotted my interest.By the law of large number we have that the empirical measure (mu_n) is convering to the real one (mu) and this convergence is uniform, if the cdf are smooth by G-C lemma.Furtheremore it is possible to compute the rate at which the the probability that emprical measure (mu_n) is converging to a wrong measure (nu) or more precisely to a ball surounding that measure. This probability is given by the Sanov's theorem and is equal to :exp(-n H(mu|nu))where H(mu|nu) is the relative entropy and is greater or equal to zeroSo the convergence between the emprical and real measure happens exponetially fast.I would be interested if people have some comments/opinion on that.