# Goodness of fit to Poisson Distribution

What are some of the well known statistical tests to measure goodness of fit of observed random variables to a poisson distribution? I know the Kolmogorov-Smirnov test is one such, are there any others out there?

1) There are two issues with the Kolmogorov-Smirnov* -

a) it assumes the distribution is completely specified, with no estimated parameters. If you estimate parameters a KS becomes a form of Lilliefors test (in this case for Poisson-ness), and you need different critical values

b) it assumes the distribution is continuous

both impact the calculation of p-values, and both make it less likely to reject.

*(and the Cramer-von Mises and the Anderson Darling, and any other test that assumes a continuous, completely specified null)

Unless you don't mind a potentially highly-conservative test (of unknown size), you have to adjust the calculation of the significance for both of these; simulation would be called for.

2) on the other hand, a vanilla chi-square goodness of fit is a terrible idea when testing something that's ordered, as a Poisson is. By ignoring ordering, it's really not very sensitive to the more interesting alternatives - it throws away power against directly interesting alternatives like overdispersion, instead spending its power against things like 'an excess of even numbers over odd numbers'. As a result its power against interesting alternatives is generally even lower than the vanilla KS but without the compensation of the much lower type I error rate.

I think this is even worse.

3) on the gripping hand, you can partition the chi-squared into components that do respect the ordering via the use of orthogonal polynomials, and drop off the less interesting highest-order components. In this particular case you'd use polynomials orthogonal to the Poisson p.f.

This is an approach taken in Rayner and Best's little 1989 book on Smooth Tests of Goodness of Fit (they have a newer one on smooth tests in R that might make your life easier)

Alternatively, see papers like this one:

http://www.jstor.org/discover/10.2307/1403470

4) However, depending on why you're doing it, it may be better to reconsider the whole enterprise...

The discussion in questions like these carry over to most goodness of fit tests ... and indeed often to most tests of assumptions in general:

Is normality testing 'essentially useless'?

What tests do I use to confirm that residuals are normally distributed?

• Its nor clear to me what's not clear about the statement but the chi-squared test is spending a large fraction of its power budget on high-frequency deviations from expected - rapidly changing positive and negative deviations - leaving relatively less of it for the more interesting low-frequency deviations like you'd see if there was overdispersion - more probability near 0 and at higher values, less in the middle than with a Poisson with the same mean. Aug 5, 2020 at 17:44

The KS-Test and other tests such as Anderson Darling are used for continuous distributions. For discrete distributions, you can use the Chi-Square goodness of fit test, which is based on comparing the #observed events vs. the number of expected based on the expected number for your distribution. If the parameter is known for the Poisson distribution you would obviously use that, more likely you will estimate the parameter using MLE, which reduces the degrees of freedom in your Chi-sq test. An example is here; you would just adapt it to your specific distribution: http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm