Abstract
ABSTRACTThe 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.
Highlights
Let T be a real-valued test statistic, with probability measure P0 under the null hypothesis, denoted H0
We say that a random variable is sub-uniform if it is less variable than a uniform random variable, U, in the convex order
The convex order provides a formal platform for the treatment and interpretation of mid-p-values
Summary
Let T be a real-valued test statistic, with probability measure P0 under the null hypothesis, denoted H0. Note that we can work with an unrealised version of the randomised p-value, known as the fuzzy or abstract p-value (Geyer and Meeden, 2005), and either stop there — leaving interpretation to the decision-maker — or propagate uncertainty through to any post-hoc analysis, e.g. multiple-testing (Kulinskaya and Lewin, 2009; Habiger, 2015). It can allow breaches of the nominal level, the mid-p-value is often deemed to better represent the evidence against the null hypothesis than the ordinary or randomised p-values.
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