Likelihood Ratio Tests (LRT)

• Reduced to MP or UMP tests when they exists.
• Guarantee good properties when $n$ is large in regular cases.

Definition

Data: $X_1,...,X_n$: iid $f(x|\theta)$
$\Omega$ = parameter space Likelihood: $L(\theta)=\prod^n_{i=1} f(x_i|\theta)$
Test $H_0: \theta \in \Omega_0$ against $H_0:\theta\in\Omega-\Omega_0$

LRTs reject when

$$ \lambda=\frac{\max_{\theta\in\Omega_0}L(\theta)}{\max_{\theta\in\Omega}L(\theta)} \leq k $$

If $\lambda=1$, no reason to reject $H_0$, and if $\lambda < 1$, we should consider if we reject $H_0$. $(0\leq\lambda\leq1)$

• $\lambda=1$ → most likely value is in $\Omega_0$
• $\lambda<1$ → most likely value is in $\Omega -\Omega_0$ → consider to reject $H_0$

In general,

$$ \lambda=\frac{\max_{\theta\in\Omega_0}L(\theta)}{\max_{\theta\in\Omega}L(\theta)} = \frac{L(\theta_0)}{L(\hat\theta^\text{MLE})}=g(\hat\theta) $$

$g(\hat\theta)$ is a function of $\hat\theta^\text{MLE}$

Note:

• In practice, find a specific RR of level $\alpha$, using the fact that $\sum X_i \sim \text{Poisson}(n\theta_0)$ under $H_0:\theta=\theta_0$
• In principle, $k$ gives us $c_1$ and $c_2$. In practice, divide the desired level $\alpha$ roughly evenly between two tails.
• In a discrete model, cannot get any desired $\alpha$ exactly.

Asymptotic distribution

LRT does not always produce a test statistic with a known distribution. However, we have a result on the asymptotic distribution of LRT.