Šidák correction for t-test

One of the application of Student's t-test is to test the location of one sequence of independent and identically distributed random variables. If we want to test the locations of multiple sequences of such variables, Šidák correction should be applied in order to calibrate the level of the Student's t-test. Moreover, if we want to test the locations of nearly infinitely many sequences of variables, then Šidák correction should be used, but with caution. More specifically, the validity of Šidák correction depends on how fast the number of sequences goes to infinity.

Suppose we are interested in m different null hypotheses, ${\displaystyle H_{1},...,H_{m}}$, and would like to check if all of them are true. Now the hypothesis test scheme becomes

Let ${\displaystyle \alpha }$ be the level of this test, that is, the probability that we falsely reject ${\displaystyle H_{null}}$ when it is true. Now we aim to design a test with certain level ${\displaystyle \alpha }$. Suppose when testing each null hypothesis ${\displaystyle H_{i}}$, the statistic we use is ${\displaystyle t_{i}}$. If these ${\displaystyle t_{i}}$'s are independent, then a test for ${\displaystyle H_{null}}$ can be developed by the following procedures, known as Šidák correction.

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