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The settings: we have a sequence of numbers generated from a normal distribution with $\mu=1;\sigma^2 = 1 $. And with every new observation collected, we'd like to test for: $H_0:\mu_0=0$ (two-sided).

From what I've learned in school, there is the Wald's group sequential for when comparing two simple hypothesis, and I would like to know what is available for complex hypothesis.

What I am looking for is:

  1. Some reference text (article/book) that describes the solution for this (or a similar) case.

  2. If there is some implementation in R for performing such a test.

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A bit late, but this might be helpful if somebody stumbles upon this question in the future:

Firstly, it is important to distinguish 'group sequential testing' and 'testing after every new observation', which is sometimes referred to as fully sequential testing. In group sequential testing, the null hypothesis is tested after group of observations are available and, in practice, this is what is used. Fully sequential tests are rarely used in my experience.

For fully sequential testing, the origin publication by Wald is certainly a good reference.

Concerning group sequential testing, the book Group Sequential Methods with Applications to Clinical Trials by Jennison and Turnbull is the standard text book. It is well-written and all relevant statistical models are discussed.

Group sequential and fully sequential tests are for example implemented in the R packages sequential and gsDesign.

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