(This article was first published on

**Freakonometrics - Tag - R-english**, and kindly contributed to R-bloggers)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

whereand in R, a negative binomial distribution can be parametrized using two parameters, out of the following ones

- the size,
- the probability,
- the mean,

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)

return(r)

}

> 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

> 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

To

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