Explanation for the thresholds in the sequential probability ratio test

Question

Recently I have learned about sequential analysis especially sequential probability ratio tests (after I have struggled a lot with cumulation of alpha - errrors). See also this question Sequential hypothesis testing in basic science.

My question is: What is the derviation or explanation of the tresholds of the stopping-rule (Again see Sequential probability ratio tests - Theory)? How do these thresholds prevent the errors from accumulation? I am talking about the scheme: $a < S_i < b$ where $S_i$ is the likelihood-ratio and $a:=\frac{\beta}{1-\alpha}$ and $b:=\frac{1-\beta}{\alpha}$. Why are a and b set this way ?

I'd prefer an intuitive explanation, but a mathematical one using not too many rarely known concepts is fine, too.

Answer 1

A first step in understanding this type of testing plan is to consider a Double-Sampling Plan for attributes. This type of plan is designed to determine whether a lot of product should be accepted or rejected based on sampling items, where each item in the lot can be categorized as either good or defective. The plan is defined by four numbers, $ n_{1} $, $ c_{1} $, $ n_{2} $, and $ c_{2} $. The plan is run as follows

• A sample of size $ n_{1} $ is taken
• If there are no more than $ c_{1} $ defects in the sample, then the lot is accepted.
• If there are more than $ c_{2} $ defects, then the lot is rejected.
• If the lot is neither accepted nor rejected then a sample of size $ n_{2} $ is taken.
• If the sum of the number of defectives in both samples is less than $ c_{2} $, then the lot is accepted.
• If the number the number of defects in both samples is greater than $ c_{2} $, then the lot is rejected.

In such a plan, both a Type I and Type II error are chosen BEFORE the plan is run, and that is how the numbers above are set.

There are plans called multiple sampling plans that instead of having two samples being taken, have some pre-determined number of samples being taken. Finally, when the multiplicity of the multiple sampling plan goes to infinity you get the sequential probability ratio test plan.