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# Hypothesis Testing

The goal of hypothesis testing is to select a subset of the parameter space $\Theta$.

Set $\Theta$ is first partitioned into disjoint subsets, $\Theta_{0}$ and $\Theta_{1}$, where $\Theta_{1}=\Theta\backslash\Theta_{0}$.

Then, we decide on a rule for choosing between $\Theta_{0}$ and $\Theta_{1}$.

Some Terminology

• A hypothesis is a statement about $\theta$.
• Null Hypothesis: $H_{0}:\theta\in\Theta_{0}$.
• Alternative Hypothesis: $H_{1}:\theta\in\Theta_{1}$.
• Maintained Hypothesis: $H:$$\theta\in\Theta$.

The goal of hypothesis testing is to decide for the null or the alternative hypothesis. Throughout the procedure, the maintained hypothesis is assumed.

A typical formulation of a hypothesis test is: $H_{0}:\theta\in\Theta_{0}\,vs.\,H_{1}:\theta\in\Theta_{1}$

## Example: Normal

Suppose $X_{i}\overset{iid}{\sim}N\left(\mu,1\right)$, where $\mu\geq0$ (maintained hypothesis) is unknown. The aim is to test whether $\mu=0$.

Notice that we can write the problem down in two equivalent formulations:

$H_{0}:\mu=0\,vs.\,H_{1}:\mu\gt 0$

or

$H_{0}:\mu\gt 0\,vs.\,H_{1}:\mu=0$

It is usually easier to consider the hypothesis test with the simple null hypothesis. The null (alternative) hypothesis is simple if $\Theta_{0}$($\Theta_{1}$) is a singleton. Otherwise, it is composite.