# Fitting a copula with Poisson marginals to data in R

First off, I know this is a question which requires an thorough answer, so I am coming here with a very humble attitude. I have limited knowledge about both copulas and R, so I will try to explain what I know and what my thoughts are:

I have a data set of n observations of $(X,Y)$, two correlated Poisson random variables, which comes in "set" - that is - I observe $(X_1,Y_1), (X_2,Y_2),...,(X_n,Y_n)$. I have read about using discrete marginals with copulas in:

http://www.actuaries.org/LIBRARY/ASTIN/vol37no2/475.pdf

and I realize it can be troublesome using Poisson marginals. I have to admit that I did not understand too much of the above article, but nevertheless I hope I am able to describe the joint distribution of $(X,Y)$ using some copula.

I understand that different copulas model different type of dependence, such as tail dependence and such. But this is not what I am looking for. For my data, it seems that X follows Y, and vice verca - so it seems like we have strong dependence on the "straight line".

I found the following about fitting copulas to data:

    library(copula)

## Toy example for gumbel copula with log-normal distribution
## (Taken from the documentation of copula::fitMvdc)

## Specify the copula
gumbel.cop <- gumbelCopula(3, dim=2)
myMvd <- mvdc(gumbel.cop, c("lnorm","lnorm"), list(list(meanlog = 1.17),
list(meanlog = 0.76)))
## Generate some random sample to test
x <- rmvdc(myMvd, 1000)

## Fit the random sample
fit <- fitMvdc(x, myMvd, c(1,1,2))
fit


Now, first of all this uses a gumbel copula. Could there be some other copula better fitting what I am looking for? I am curious to how 'the experts' choose which copula to work with. For example, if we are looking at financial data with strong tail dependence, it seems obvious, but for other 'types' of dependence, I am not sure if I understand the copula choice.

From the above code in R, could I just replace "lnorm" with "name of poisson distribution", and what would this name be? I cannot seem to find a list of names of the marginals.

Also, the above code generated data. I have my own data.

I know I am asking a lot here, but this is all a bit overwhelming to me...