  The notion of p-value is a fundamental concept in statistical inference and
has been widely used for reporting outcomes of hypothesis tests. However,
p-value is often misinterpreted, misused or miscommunicated in practice. Part
of the issue is that existing definitions of p-value are often derived from
constructions under specific settings, and a general definition that directly
reflects the evidence of the null hypothesis is not yet available. In this
article, we first propose a general and rigorous definition of p-value that
fulfills two performance-based characteristics. The performance-based
definition subsumes all existing construction-based definitions of the p-value,
and justifies their interpretations. The paper further presents a specific
approach based on confidence distribution to formulate and calculate p-values.
This specific way of computing p values has two main advantages. First, it is
applicable for a wide range of hypothesis testing problems, including the
standard one- and two-sided tests, tests with interval-type null,
intersection-union tests, multivariate tests and so on. Second, it can
naturally lead to a coherent interpretation of p-value as evidence in support
of the null hypothesis, as well as a meaningful measure of degree of such
support. In particular, it places a meaning of a large p-value, e.g. p-value of
0.8 has more support than 0.5. Numerical examples are used to illustrate the
wide applicability and computational feasibility of our approach. We show that
our proposal is effective and can be applied broadly, without further
consideration of the form/size of the null space. As for existing testing
methods, the solutions have not been available or cannot be easily obtained.
