Generating a quasi Poisson distribution, version 2

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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


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


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)

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.

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