Generating a quasi Poisson distribution, version 2

arthur charpentier

11 years ago

[This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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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.

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