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Here and there, I mentioned two codes to generated quasiPoisson random variables. And in both of them, the negative binomial approximation seems to be wrong. Recall that the negative binomial distribution is

where
and in R, a negative binomial distribution can be parametrized using two parameters, out of the following ones
• the size
• the probability
• the mean
Recall also that the variance for a negative binomial distribution should be
Here, we consider a distribution such that  and . In the previous posts, I used

i.e.
rqpois = function(n, lambda, phi) {
mu = lambda
k = mu/(phi * mu - 1)
r1 = rnbinom(n, mu = mu, size = k)
r2 = rnbinom(n, size=phi*mu/(phi-1),prob=1/phi)
k = mu/phi/(1-1/phi)
r3 = rnbinom(n, mu = mu, size = k)
r4 = rnbinom(n, size=mu/phi/(1-1/phi),prob=1/phi)
r = cbind(r1,r2,r3,r4)
return(r)
}
but as we can see below, none of those two functions work,
> N=rqpois(1000000,2,4)
> mean(N[,1])
[1] 2.001992
> mean(N[,2])
[1] 8.000033
> var(N[,1])/ mean(N[,1])
[1] 7.97444
> var(N[,2])/ mean(N[,2])
[1] 4.002022
with the first one, the expected value is correct, while it is the overdispersion parameter for the second. Now, if we do the maths correctly, it comes

which should now work,
> mean(N[,3])
[1] 2.001667
> mean(N[,4])
[1] 2.002776
> var(N[,3])/ mean(N[,3])
[1] 3.999318
> var(N[,4])/ mean(N[,4])
[1] 4.009647
So, finally it is better when we do the maths well.