Meta-Analysis of Mid--Values: Some New Results based on the Convex Order.

The mid--value is a proposed improvement on the ordinary -value for the case where the test statistic is partially or completely discrete. In this case, the ordinary -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--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--values, which can be used to combine results from different hypothesis tests conservatively, yet more powerfully, using mid--values rather than -values. Our methodology is demonstrated on real data from a cyber-security application.