Trivariate_LSDsim function

Simulates from the trivariate logarithmic series distribution

Trivariate_LSDsim(N, p1, p2, p3)

Arguments

• N: number of data points to be simulated
• p1: parameter $p1$ of the trivariate logarithmic series distribution
• p2: parameter $p2$ of the trivariate logarithmic series distribution
• p3: parameter $p3$ of the trivariate logarithmic series distribution

Returns

An $N \times 3$ matrix with $N$ simulated values from the trivariate logarithmic series distribution

Details

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

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 c("$\\tilde\n$", "$p_1=p_1/(1-p_2-p_3)$") 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 $x_1>0$.