Density of a Gamma distributed variable X and a (zero-truncated) Poisson variable Y if their joint distribution is defined via a copula
Usage
density_joint(x, y, mu, delta, lambda, theta, family,zt)
Arguments
x
vector at which the density is evaluated
y
vector at which the density is evaluated
mu
expectation of the Gamma distribution
delta
dispersion parameter of the Gamma distribution
lambda
parameter of the zero-truncated Poisson distribution
theta
copula parameter
family
an integer defining the bivariate copula family: 1 = Gauss, 3 = Clayton, 4=Gumbel, 5=Frank
zt
logical. If zt=TRUE, we use a zero-truncated Poisson variable. Otherwise, we use a Poisson variable. Default is TRUE.
Details
For a Gamma distributed variable X and a (zero truncated) Possion variable Y, their joint density function is given by
$$f_{XY}(x,y)=f_X(x) \left(D_u(F_Y(y),F_X(x)|\theta) - D_u(F_Y(y-1),F_X(x)|\theta) \right)\,.$$Here $D_u$ is the h-function of a copula famila family with copula parameter theta.
References
N. Kraemer, E. Brechmann, D. Silvestrini, C. Czado (2013): Total loss estimation using copula-based regression models. Insurance: Mathematics and Economics 53 (3), 829 - 839.