) depend on the desired type I and type II errors, and . They may be chosen as follows: and
In other words, and must be decided beforehand in order to set the thresholds appropriately. The numerical value will depend on the application. The reason for using approximation signs is that, in the discrete case, the signal may cross the threshold between samples. Thus, depending on the penalty of making an error and the sampling frequency, one might set the thresholds more aggressively. Of course, the exact bounds may be used in the continuous case.
Example
A textbook example is parameter estimation of a probability distribution function. Let us consider the exponential distribution:
:
The hypotheses are simply and , with . Then the log-likelihood function (LLF) for one sample is
:
The cumulative sum of the LLFs for all x is
:
Accordingly, the stopping rule is: