Consider waiting for n events from a Poisson process with mean waiting time. Equivalently, consider estimating a variance s2 from m = 2n+1 (so n = (m-1)/2) observations from a Normal distibution. In both cases, the maximum likelihood estimate (MLE) follows a Gamma distribution with shape parameter n.
There are several sensible approached for testing a null value of the parameter against a two-sided alternative at a given level a. Rejection regions with probability a may be defined by identifying upper and lower regions each with probability a /2 (the "Equal Tail Probability" method); by identifying regions with equal or lower density with regards to the MLE (the "Equal Density" method), or by identifying regions with equal or lower values of the generalized likelihood ratio (GLR) statistic (the "GLR" method).
These three methods are compared in the following plots:
plot 1: n = 5 or m = 11
plot 2: n = 10 or m = 21
plot 3: n = 15 or m = 31
plot 4: n = 20 or m = 41
plot 5: n = 25 or m = 51