Lecture 11. B) Testing Procedure
Testing Procedure
Suppose [math]X_{1}..X_{n}[/math] is a random sample with a pmf/pdf [math]f\left(\left.\cdot\right|\theta\right)[/math] where [math]\theta\in\Theta[/math] is unknown. Consider the test
[math]H_{0}:\theta\in\Theta_{0}\,vs.\,H_{1}:\theta\in\Theta_{1}[/math]
A testing procedure is a rule for choosing between [math]H_{0}[/math] and [math]H_{1}.[/math]
For example, in the normal example above, if the data come out relatively low, then we may opt for hypothesis [math]\mu=0[/math]; whereas we may opt for the alternative hypothesis that [math]\mu\gt 0[/math] if the data are relatively high.
There is no obvious way how we should define the decision rule. However, for any rule, we can define a data region that corresponds to supporting one of the alternatives.
Let [math]C\subseteq\mathbb{R}^{n}[/math]. The rule "reject [math]H_{0}[/math] iff [math]\left(X_{1}..X_{n}\right)\in C[/math]" (i.e., if the data fall in the region) is a testing procedure with critical region [math]C[/math].
Example: Normal
Suppose [math]X_{i}\overset{iid}{\sim}N\left(\mu,1\right)[/math], where [math]\mu\geq0[/math] is unknown, and [math]H_{0}:\mu=0\,vs.\,H_{1}:\mu\gt 0.[/math]
A possible decision rule is to reject [math]H_{0}[/math] if [math]\overline{X}\gt 2[/math]. Of course, we could have selected a different right-hand side, like 3, or even a function of [math]n[/math] to account for the fact that higher samples produce more precise means.
For our current example, the critical region is given by [math]C=\left\{ \left(X_{1}..X_{n}\right)^{'}:\frac{\sum_{i=1}^{n}X_{i}}{n}\gt 2\right\}[/math].
- We call [math]\frac{\sum_{i=1}^{n}X_{i}}{n}[/math] the test-statistic: It’s a statistic that will be used to decide the test result.
- We call [math]2[/math] the test threshold or the critical value.
- One can practically always write tests as [math]T\left(X\right)\gt c[/math].
Hypothesis testing involves choosing a test statistic (left-hand side) and a critical value (right-hand side). Depending on the data, the condition is either satisfied or not (so, a test produces a binary outcome).
We will first discuss critical value selection (for generic tests or specific ones). I.e., we will focus on the right-hand side. In the next lecture, we discuss test selection.
But before we get in too deep, something completely different.
