Week 13

We will discuss essential concepts to hypothesis testing.
Published

November 15, 2022

Learning Outcomes

Monday

  • Type I and Type II Errors
  • Power of Tests
  • Neyman-Pearson Lemma

Wednesday

  • Likelihood Ratio Test

Important Concepts

Type I and II Errors

With hypothesis testing, we are testing to either reject or fail to reject the \(H_0\). We usually choose a significance value of \(\alpha\) to determine how we will make our decision. In this case, \(\alpha\) represents the probability that we reject \(H_0\) given that \(H_0\) is true (\(P(Reject\ H_0| H_0\ true)\)). If we reject \(H_0\), given that it is true, we are committing Type I error. Generally speaking, we control Type I error by selecting a small \(\alpha\)

Type II error is the opposite case, where we fail to reject \(H_0\), given it is false. Similar to type I error, a hypothesis test will have a probability (\(P(Fail\ to\ reject\ H_0|H_0\ False)\)).

Power of Tests

Power

The power of a test is the probability of rejecting \(H_0\) when the true parameter is \(\theta\).

\[ \mathrm{power}(\theta) = \mathrm P(\mathrm{Reject}\ H_0\ \mathrm{when}\ \theta\ \mathrm{is true}) \]

Power and Type II Error

\[ \beta = 1 - \mathrm{power}(\theta_a) \]

Neyman-Pearson Lemma

Suppose you test the simple null hypothesis (\(H_0: \theta=\theta_0\)) vs a simple alternative hypothesis (\(H_a: \theta=\theta_1\)), based on a random sample with parameter \(\theta\). Let \(L(\theta)\) denote the likelihood function of the sample with parameter \(\theta\). Then for a given \(\alpha\), the test that maximizes the power at \(\theta_1\) has a rejection region determined by \[ \frac{L(\theta_0)}{L(\theta_1)}<k \]The value of \(k\) is chosen so that the test has the desired value for \(\alpha\). such a test is most powerful \(\alpha\)-level test for \(H_0\) vs \(H_a\)

Likelihood Ratio Test

The likelihood ratio test is used when you cannot find a uniformly most powerful test for a given set of hypothesis. This will yield a very good test that is decently powerful.

Hypothesis

\(H_0:\ \theta\in\Theta_0\)

\(H_a:\ \theta\in\Theta_a\)

  • \(\Theta = \Theta_0\cup\Theta_a\)

  • \(\Theta\) is the parameters space

Test Statistic

\[ \Lambda = \frac{L(\hat\theta_0)}{L(\hat\theta)}=\frac{f(x_1,\ldots,x_n;\hat\theta_0)}{f(x_1,\ldots,x_n;\hat\theta)} \]

  • \(\hat\theta_0=\underset{\theta\in\Theta_0}{\arg\max}\ L(\theta)\)

  • \(\hat\theta=\underset{\theta\in\Theta}{\arg\max}\ L(\theta)\): MLE

Decision

Reject \(H_0\) if \(\Lambda\le k\)

Resources

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Lecture Slides Videos
Monday Slides Video
Wednesday Slides Video