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

Let $T$ be the Student one- or two-sample $t$-, $F$-, or Welch statistic. Now release the underlying assumptions of normality, independence and identical distribution and consider a more general case where one only assumes that the vector of data has a continuous joint density. We determine asymptotic expressions for $\mathbf{P}(T>u)$ as $u\to \infty$ for this case. The approximations are particularly accurate for small sample sizes and may be used, 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 extreme significance levels are used. We give numerous examples and complement our results by an investigation of the convergence speed - both theoretically, by deriving exact bounds for absolute and relative errors, and by means of a simulation study.