Parametric hypothesis testing for non-normal data Are there any methods to make parametric hypothesis testing assuming that data is sampled from a known but non-normal continuous distribution?
I'm glad to see a solution to any particular distribution. I'll be happy if there is some kind of cookbook or bunch of scientific papers for different distributions.
P.S. It's more for theoretical curiosity's sake rather than practical purposes.
 A: Given a particular parametric assumption, and a suitable test statistic, if you can compute the distribution of the test statistic under the null, you can perform a hypothesis test.
(Simulation could be used to compute p-values where such a calculation is not tractable/convenient.)
The main issue is then "Given a parametric distributional assumption, how do we find a good test statistic?". This boils down to finding test statistics that have good power under that parametric assumption.
This task - and the tests that result - is one focus of statistical theory, and many books used in statistics degree programs discuss it in detail. 
Some useful Wikipedia links:
Likelihood-ratio test 
Score test
Wald test
A: With enough data (whatever "enough" means), a t-test will suffice.  Briefly, the central limit theorem says that the sampling distribution of the sample mean is normal with mean $\mu$ and variance $\sigma^2$. Because we have to estimate $\sigma$, that means we can use the t test to test for a difference in means.
The story is different when we don't have "enough" data. As Dave mentions, generalized linear models allow us to test for differences in means between to groups with additional restrictions on the likelihood (that is, the distribution of the data.  Namely, the data must belong to the exponential family e.g. Normal, Poisson, Binomial, Gamma, with possible extensions like the beta binomial, negative binomial, and so on).  In this case, we can fit a GLM using a binary indicator or group assignment (e.g. 1 for test, 0 for control).  The significance of the difference between groups is determined by the Wald test for the associated coefficient, with the estimate of the difference depending on what the link function is.  We can even use GLM to estimate the mean if a single group.
