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Let $T$ be the Student one- or two-sample $t$-, $F$-, or Welch statistic. Nowrelease the underlying assumptions of normality, independence and identicaldistribution and consider a more general case where one only assumes that thevector of data has a continuous joint density. We determine asymptoticexpressions for $\mathbf{P}(T>u)$ as $u\to \infty$ for this case. Theapproximations are particularly accurate for small sample sizes and may beused, for example, in the analysis of High-Throughput Screening experiments,where the number of replicates can be as low as two to five and often extremesignificance levels are used. We give numerous examples and complement ourresults by an investigation of the convergence speed - both theoretically, byderiving exact bounds for absolute and relative errors, and by means of asimulation study.

Tail approximations for the Student $t$-, $F$-, and Welch statistics for non-normal and not necessarily i.i.d. random variables