I have three sets of data from some experiments. I fitted each set to different distributions, and each one fits a different distribution. For example, Gamma, Weibull, and Lognormal. If I want to compare these sets what methods would you suggest? I am not sure, but I think I read somewhere that I can use Student t test and ANOVA only if I have the normal distribution. Therefore in my case what would be the approach that I should take? Similar questions were asked in this link and this link but they were not clear for me. Thank you
ANOVA is quite robust to violations of normality if the variances are roughly equal and the sample sizes are similar as well. Both are not given here which is a problem.
You could use Welch t-tests instead of ANOVA and all those problems are solved. A Welch t-test does not need equal variances or sample size. You would do a t-test for group 1 against 2, 1 against 3 and 2 against 3. For each test, you allocate 1/3 of your alpha level (Bonferroni correction).
However, per central limit theorem, you need 30 samples per group to do t-tests on non-normal data. This remains a problem unless you can have this amount of data.