Simulates from the trivariate logarithmic series distribution
Trivariate_LSDsim(N, p1, p2, p3)number of data points to be simulated
parameter \(p1\) of the trivariate logarithmic series distribution
parameter \(p2\) of the trivariate logarithmic series distribution
parameter \(p3\) of the trivariate logarithmic series distribution
An \(N \times 3\) matrix with \(N\) simulated values from the trivariate logarithmic series distribution
The probability mass function of a random vector \(X=(X_1,X_2,X_3)' \) following the trivariate logarithmic series distribution with parameters \(0<p_1, p_2, p_3<1\) with \(p:=p_1+p_2+p_3<1\) is given by $$P(X_1=x_1,X_2=x_2,X_3=x_3)=\frac{\Gamma(x_1+x_2+x_3)}{x_1!x_2!x_3!} \frac{p_1^{x_1}p_2^{x_2}p_3^{x_3}}{(-\log(1-p))},$$ for \(x_1,x_2,x_3=0,1,2,\dots\) such that \(x_1+x_2+x_3>0\).
The simulation proceeds in two steps: First, \(X_1\) is simulated from the modified logarithmic distribution with parameters \(\tilde p_1=p_1/(1-p_2-p_3)\) and \(\delta_1=\log(1-p_2-p_3)/\log(1-p)\). Then we simulate \((X_2,X_3)'\) conditional on \(X_1\). We note that \((X_2,X_3)'|X_1=x_1\) follows the bivariate logarithmic series distribution with parameters \((p_2,p_3)\) when \(x_1=0\), and the bivariate negative binomial distribution with parameters \((x_1,p_2,p_3)\) when \(x_1>0\).