I have what I believe are an interesting QQ plot of the residuals of some sports data fitted using Poisson regression. My model is actually a live model to predict the number of remaining events in the game (basketball, soccer etc) from a set of explanatory variables. It looks also very much like the data is Poisson distributed, with equal mean and variance.
Residuals are as we know $r_i=Y_i-\mu_i$, where $Y_i$ are the observations, and $\mu_i$ the fitted mean of $Y_i$ ($Y_n$ may have different explanatory variables than, for example, $Y_{n-1}$).
Here is the first residual QQ plot of fitting my model to soccer match data (whole 90 minutes):
Interpreting QQ plots if I want to check for normality of some data is not too hard. It is easy to see that the residuals have too thick tails. But assuming the data is without a doubt Poisson distributed, something must therefore be wrong with $\mu_i$. Looking at the lower tail, I cannot reach any other conclusion than that the values of $\mu_i$ generally are too big - such that $r_i$ becomes too small. But also, looking at the upper tail, I cannot reach any other conclusion than that $\mu_i$ is too small - such that $r_i$ becomes too big. Is this a contradiction, or merely just a reflection that $\mu_i$ is not fitted good enough for extreme cases?
