When doing thousands of simultaneous tests, such as in a gene association studies, the distribution of the resulting statistics does not always follow the theoretical distribution. This could be because the tests are not independent samples of an identical t, for instance. There are some methods of dealing with this, by fitting just the center of the observed values to estimate the distribution under the null hypothesis. Then the tails of the fitted distribution can be used to estimate Prob{reject null | null}. Here is a nice explanation of False Discovery Rates and from Brad Efron’s Large-scale simultaneous hypothesis testing: The choice of a null hypothesis:
We begin with a simple Bayes model. Suppose that the N z-values fall into two classes, “Uninteresting” or “Interesting”, corresponding to whether or not z_i is generated according to the null hypothesis, with prior probabilities p0 and p1 = 1 − p0 , for the classes; and that z_i has density either f_0(z) or f_1(z) depending on its class,
p0 = Prob{Uninteresting}, f_0(z) density if Uninteresting (Null)
p1 = Prob{Interesting}, f_1(z) density if Interesting (Non-Null) .
The smooth curve in Figure 1 estimates the mixture density f(z),
f(z) = p0 * f_0(z) + p1 * f_1(z) .
According to Bayes theorem the a posteriori probability of being in the Uninteresting class given z is
Prob{Uninteresting|z} = p0 * f_0(z)/f(z) .
Here we define the local false discovery rate to be
fdr(z) ≡ f_0(z)/f_(z) ,
ignoring the factor p0, so fdr(z) is an upper bound on Prob{Uninteresting|z}. In fact p0 can be roughly estimated, but we are assuming that p0 is near 1, say p0 ≥ 0.90, so fdr(z) is not a flagrant overestimator.
