  The mid-p-value is a proposed improvement on the ordinary p-value for the
case where the test statistic is partially or completely discrete. In this
case, the ordinary p-value is conservative, meaning that its null distribution
is larger than a uniform distribution on the unit interval, in the usual
stochastic order. The mid-p-value is not conservative. However, its null
distribution is dominated by the uniform distribution in a different stochastic
order, called the convex order. The property leads us to discover some new
finite-sample and asymptotic bounds on functions of mid-p-values, which can be
used to combine results from different hypothesis tests conservatively, yet
more powerfully, using mid-p-values rather than p-values. Our methodology is
demonstrated on real data from a cyber-security application.
