On the T-test
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The T-test is probably the most popular statistical test; it is routinely recommended by the textbooks. The applicability of the test relies upon the validity of normal or Student's approximation to the distribution of Student's statistic t_n. However, the latter assumption is not valid as often as assumed. We show that normal or Student's approximation to Ł(t_n) does not hold uniformly even in the class P_n of samples from zero-mean unit-variance bounded distributions. We present lower bounds to the corresponding error. The fact that a non-parametric test is not applicable uniformly to samples from the class P_n seems to be established for the first time. It means the T-test can be misleading, and should not be recommended in its present form. We suggest a generalisation of the test that allows for variability of possible limiting/approximating distributions to Ł(t_n).
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