Likelihood Ratio Test

Learning Outcomes

• Likelihood Ratio Test

Likelihood Ratio Test

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 \(\Lambda\le k\)

Examples

One-Sample t-test

\(H_0:\ \mu=\mu_0\)

\(H_a:\ \mu\ne\mu_0\)

\[ X_1,\ldots,X_n\sim N(\mu,\sigma^2) \]