Lecture 12. F) Test Equivalence

<< Example: LRT

Equivalence Between LRT and LM Tests >>

Test equivalence

We now show that both the LM and Wald tests approximate the LRT test.

• The LM test is obtained by expanding the LRT via Taylor expansion around [math]\theta_{0}[/math].
• The Wald test is obtained by expanding the LRT via Taylor expansion around [math]\widehat{\theta}_{ML}[/math].

For simplicity, below we will write [math]l\left(\theta\right)[/math] rather than [math]l\left(\left.\theta\right|x_{1}..x_{n}\right).[/math]

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<< Example: LRT

Equivalence Between LRT and LM Tests >>