Anscombe transform & Anderson-Darling test failure I created a synthetic constant image (all the pixel values are the same) on which I generated a Poisson distribution using a Poisson noise generator, with mean values set to input pixel values. 
Then, I applied the Anscombe transform and checked if the output distribution fit the normal distribution by using the Anderson-Darling test with p=0.05. 
Surprisingly, all the tests failed (I generated several Poisson distributions), do you have any explanation for this? Should we expect that the tests succeed (as the Anscombe transform is supposed to yield an APPROXIMATE normal distribution)?
 A: The Anderson-Darling test is a test of exact normality, your transformation is supposed to give approximate normality, so the null hypothesis for AD is false and the correct decision is to reject the null.
Also note that the AD (and other) tests are rule out tests, if they reject the null then that means your data is not consistent with the null hypothesis of exact normality.  If the test does not reject the null hypothesis that does not mean that the data is normal, it may be (probably is) that you do not have enough power to find the difference.
The real issue is how good is your approximation or is the data normal enough?  This is the much more interesting question, but the standard normality tests do not answer this question.  Plots can help you get a better feel for this, but the final decision is going to be subjective.
A: With large sample sizes, the Anderson-Darling test is prone to reject very large samples from otherwise acceptable distributions.
If you are dealing with thousands of data points, the  Ryan-Joiner or Shapiro-Wilk tests should be used.
Note: Changing the test should be something that, in the future, is done based on sample size information and not as a result of choosing hypothesis tests until you are successful.  
Another word of caution: it is possible to change tests to something more robust for a larger sample size even though the data looks decent.  Using the data from this discussion on the topic of hypothesis tests, the large "normal" data still fails.  (I tested the data with a stabilized normal probability plot, and it becomes apparent that the seemingly normal data actually fails on multiple points, and not even in the tails.)
A: Finally, I found out that if we have a large number of samples (cf. Wikipedia), AD test can fail. Hence, I tested this with  100 samples and the test succeeded.
