P-Value

statistics

Definition

P-Value

A p-value $p(\mathbf{X})$ is a test statistic satisfying $0\le p(\mathbf{X})\le 1$, which is a function of the random sample $\mathbf{X}$. It is used to give small values of $p(\mathbf{X})$ reason to reject $H_0$.

The p-value is a measure of how “surprising” the observed sample is under $H_0$.

Example

The p-value can be defined as:

$$p(\mathbf{X})=\{\begin{array}{ll}{F}_{H_0}[W(\mathbf{X})]& \text{for left-tailed tests}\\ 1-F_{H_0}[W(\mathbf{X})]& \text{for right-tailed tests}\\ 2\cdot \min \{F_{H_0}[W(\mathbf{X})],1-F_{H_0}[W(\mathbf{X})]\}& \text{for two-sided tests}\end{array}$$

where $F_{H_0}$ is the cdf of the conservative distribution ${\mathbb{P}}_{H_0}$ specified under the null hypothesis $H_0$, corresponding to the parameter value ${\mu }_0$.