Simulates from the bivariate logarithmic series distribution
Bivariate_LSDsim(N, p1, p2)number of data points to be simulated
parameter \(p_1\) of the bivariate logarithmic series distribution
parameter \(p_2\) of the bivariate logarithmic series distribution
An \(N \times 2\) matrix with \(N\) simulated values from the bivariate logarithmic series distribution
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 $$P(X_1=x_1,X_2=x_2)=\frac{\Gamma(x_1+x_2)}{x_1!x_2!} \frac{p_1^{x_1}p_2^{x_2}}{(-\log(1-p))},$$ 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 \(x_1>0\).