Due to the factorial in a poisson distribution, it becomes unpractical to estimate poisson models (for example, using maximum likelihood) when the observations are large. So, for example, if I am trying to estimate a model to explain the number of suicides in a given year (only annual data are available), and say, there are thousands of suicides every year, is it wrong to express suicides in hundreds, so that 2998 would be 29.98 ~= 30? In other words, is it wrong to change the unit of measurement to make the data manageable?
When you're dealing with a Poisson distribution with large values of \lambda (its parameter), it is common to use a normal approximation to the Poisson distribution.
As this site mentions, it's all right to use the normal approximation when \lambda gets over 20, and the approximation improves as \lambda gets even higher.
The Poisson distribution is defined only over the state space consisting of the non-negative integers, so rescaling and rounding is going to introduce odd things into your data.
Using the normal approx. for large Poisson statistics is VERY common.
In case of Poisson it is bad, since counts are counts -- their unit is an unity. On the other hand, if you'd use some advanced software like R, its Poisson handling functions will be aware of such large numbers and would use some numerical tricks to handle them.
Obviously I agree that normal approximation is another good approach.
Most statistical packages have a function to calculate the natural logarithm of the factorial directly (e.g. the lfactorial() function in R, the lnfactorial() function in Stata). This allows you to include the constant term in the log-likelihood if you want.
I'm afraid you can't do that. As @Baltimark states, with big lambda the distribution will be of more normal shape (symmetric), and with scaling it down it will no longer be poisson distrubution. Try the following code in R:
poi1 = rpois(100000, lambda = 5) # poisson poi2 = rpois(100000, lambda = 100)/20 # scaled-down poisson poi2_dens = density(poi2) hist(poi1, breaks = 0:30, freq = F, ylim = range(poi2_dens$y)) lines(poi2_dens, col = "red")
The result is below:
You can see that the downscaled poisson (red line) is completely different from the poisson distribution.
You can simply ignore the 'factorial' when using maximum likelihood. Here is the reasoning for your suicides example. Let:
λ : Be the expected number of suicides per year
ki: Be the number of suicides in year i.
Then you would maximize the log-likelihood as:
LL = ∑ ( ki log(λ) - λ - ki! )
Maximizing the above is equivalent to maximizing the following as ki! is a constant :
LL' = ∑ ( ki log(λ) - λ )
Could explain why the factorial is an issue? Am I missing something?