Bivariate negative binomial distribution for 2d count data Bivariate negative binomial distribution
The probability mass function (PMF) of a bivariate negative binomial distribution ($\mathsf{BNBin}$) is given by [1]:
$$P(X=x, Y=y) = \frac{(a + x + y - 1)!}{(a-1)! x! y!} p_0^a p_1^x p_2^y $$
where $a, p_0, p_1, p_2 > 0$ and $p_0 + p_1 + p_2 = 1$. It can be shown that the marginal $P(X)$ of this distribution follows a negative binomial ($\mathsf{NBin}$) with parameters $r=a$ and $p=\frac{p_1}{1 - p_2}$ (and similarly $r=a$ and $p=\frac{p_2}{1 - p_1}$ for $P(Y)$).

Small motivating example
Consider the following 2d count dataset:




$X$
0
0
1
1
1
1
2
2
3
3
4
4
4
4
4
5
5
7
8
9




$Y$
7
6
5
5
4
3
3
3
3
3
3
2
2
2
2
1
0
0
0
0




The marginal distributions are both $\mathsf{NBin}$:

However, the joint distribution $P(X, Y)$ cannot effectively be modeled as a  $\mathsf{BNBin}$. This is because the covariance of $\mathsf{BNBin}$, $\text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = \frac{ap_1p_2}{p_0}$, is always positive, but the variables $X$ and $Y$ are negatively correlated (the sample covariance is approximately -0.77).

Questions

*

*Under what conditions is the $\mathsf{BNBin}$ distribution suitable for 2d count data?

*Is there any other 2d distribution for count data that generalises $\mathsf{BNBin}$? If not, how could $\mathsf{BNBin}$ be extended to deal with these failure cases? (e.g. negative correlation between both variables)

Any comment or suggestion will be greatly appreciated.

References:
[1] Dunn 1967, Characterization of the Bivariate Negative Binomial Distribution (pdf)
 A: Have you tried modelling your data with copulas?
In this case you can choose your margins independently from your dependence structure, meaning that you can choose your margins as negative binomial, but you can choose the dependence between them in such a way that you have (for example) a negative association between the variables.
To answer your questions:

*

*Your joint distribution (which is one possible bivariate distribution with negative binomial margins) is appropriate if there is reason to believe that your data are in line with the implied constraints on the parameters. Even if the margins are negative binomial, why should their parameters be related by $p=p_1/(1-p_2)$ for $X$ and $p=p_2/(1-p_1)$ for $Y$? Why should the correlation only be positive?

*There is an infinite number of possible distributions with negative binomial margins. Some will have a positive correlation (like the one you cite), others will not. Using copulas you can avoid linking the marginal parameters with the parameters that describe the dependence between the variables.

