  Concentration inequalities have become increasingly popular in machine
learning, probability, and statistical research. Using concentration
inequalities, one can construct confidence intervals (CIs) for many quantities
of interest. Unfortunately, many of these CIs require the knowledge of
population variances, which are generally unknown, making these CIs impractical
for numerical application. However, recent results regarding the simultaneous
bounding of the probabilities of quantities of interest and their variances
have permitted the construction of empirical CIs, where variances are replaced
by their sample estimators. Among these new results are two-sided empirical CIs
for U-statistics, which are useful for the construction of CIs for a rich class
of parameters. In this article, we derive a number of new one-sided empirical
CIs for U-statistics and their variances. We show that our one-sided CIs can be
used to construct tighter two-sided CIs for U-statistics, than those currently
reported. We also demonstrate how our CIs can be used to construct new
empirical CIs for the mean, which provide tighter bounds than currently known
CIs for the same number of observations, under various settings.
