I am busy fitting a Poisson regression with the ML method, whose integers are very large. Indeed the range of response variable is 10-3000! The log-likelihood gives -infinite values because R cannot compute large factorials (e.g. 500!). I do not want to use normal approximation. How can I fix this? Must I use Stirling's approximation? Any suggestion is really appreciated.
Have you perhaps tried fitting a negative binomial regression model (https://stats.idre.ucla.edu/stata/dae/negative-binomial-regression/)? This might help if the range of your variable causes overdispersion.
You may rescale the outcomes by dividing them to a positive large integer and then interpolate the factorial of real numbers with the gamma function.
In the expression of the likelihood of Poisson probabilities the factorials of the observed numbers may (shall?) be neglected because they are mere multiplicative constants converted in additive constants when getting to the log-likelihood, and hence discarded by differentiation. All the test computations or parameter estimations I can figure out are insensitive to these factorials.
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