Bivariate_LSDsim function

Simulates from the bivariate logarithmic series distribution

Bivariate_LSDsim(N, p1, p2)

Arguments

• N: number of data points to be simulated
• p1: parameter $p_1$ of the bivariate logarithmic series distribution
• p2: parameter $p_2$ of the bivariate logarithmic series distribution

Returns

An $N \times 2$ matrix with $N$ simulated values from the bivariate logarithmic series distribution

Details

The probability mass function of a random vector $X=(X_1,X_2)'$

following the bivariate logarithmic series distribution with parameters $0<p_1, p_2<1$ with $p:=p_1+p_2<1$ is given by

for $x_1,x_2=0,1,2,\dots$ such that $x_1+x_2>0$. The simulation proceeds in two steps: First, $X_1$

is simulated from the modified logarithmic distribution with parameters $\tilde p_1=p_1/(1-p_2)$ and $\delta_1=\log(1-p_2)/\log(1-p)$. Then we simulate $X_2$ conditional on $X_1$. We note that $X_2|X_1=x_1$ follows the logarithmic series distribution with parameter $p_2$ when $x_1=0$, and the negative binomial distribution with parameters $(x_1,p_2)$ when $x_1>0$.