Lecture 11. A) Hypothesis Testing

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Hypothesis Testing

The goal of hypothesis testing is to select a subset of the parameter space [math]\Theta[/math].

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

Then, we decide on a rule for choosing between [math]\Theta_{0}[/math] and [math]\Theta_{1}[/math].

Some Terminology

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

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: [math]H_{0}:\theta\in\Theta_{0}\,vs.\,H_{1}:\theta\in\Theta_{1}[/math]

Example: Normal

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

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

[math]H_{0}:\mu=0\,vs.\,H_{1}:\mu\gt 0[/math]

or

[math]H_{0}:\mu\gt 0\,vs.\,H_{1}:\mu=0[/math]

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