Joint density of X and Y

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Description

Density of a Gamma distributed variable X and a (zero-truncated) Poisson variable Y if their joint distribution is defined via a copula

Usage

1
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) ≤ft(D_u(F_Y(y),F_X(x)|θ) - D_u(F_Y(y-1),F_X(x)|θ) \right)\,.

Here D_u is the h-function of a copula famila family with copula parameter theta.

Note

We allow two options: If mu and lambda are vectors of the same length as x and y, we evaluate the density for the corresponding parameter values. Otherwise, lambda and nu have to be numbers, and the parameters are the same for all entries of x and y.

Author(s)

Nicole Kraemer

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.

See Also

density_conditional, D_u

Examples

1
out<-density_joint(2,3,mu=1,delta=1,lambda=4,theta=0.5,family=1)