  The parametric Welch $t$-test and the non-parametric Wilcoxon-Mann-Whitney
test are the most commonly used two independent sample means tests. More recent
testing approaches include the non-parametric, empirical likelihood and
exponential empirical likelihood. However, the applicability of these
non-parametric likelihood testing procedures is limited partially because of
their tendency to inflate the type I error in small sized samples. In order to
circumvent the type I error problem, we propose simple calibrations using the
$t$ distribution and bootstrapping. The two non-parametric likelihood testing
procedures, with and without those calibrations, are then compared against the
Wilcoxon-Mann-Whitney test and the Welch $t$-test. The comparisons are
implemented via extensive Monte Carlo simulations on the grounds of type I
error and power in small/medium sized samples generated from various non-normal
populations. The simulation studies clearly demonstrate that a) the $t$
calibration improves the type I error of the empirical likelihood, b) bootstrap
calibration improves the type I error of both non-parametric likelihoods, c)
the Welch $t$-test with or without bootstrap calibration attains the type I
error and produces similar levels of power with the former testing procedures,
and d) the Wilcoxon-Mann-Whitney test produces inflated type I error while the
computation of an exact p-value is not feasible in the presence of ties with
discrete data. Further, an application to real gene expression data illustrates
the computational high cost and thus the impracticality of the non parametric
likelihoods. Overall, the Welch t-test, which is highly computationally
efficient and readily interpretable, is shown to be the best method when
testing equality of two population means.
