Likelihood Ratio Test

def. let $X_1,\dots ,X_n\sim f(;\theta )$. To compare hypotheses:

1. $H_0:\theta ={\theta }_0$

2. $H_1:\theta ={\theta }_1$ We define the Likelihood (Statistics) Ratio $\Lambda$ as:

$$\Lambda (X_1,...,X_n)=\frac{\mathcal{L}({\theta }_1;X_1,...,X_n)}{\mathcal{L}({\theta }_0;X_1,...,X_n)}$$

Then devise the Likelihood Ratio Test (LRT) for cutoff $c$:

$$\delta :\{\begin{array}{ll}{H}_0& \text{else }\\ H_1& \text{if }\Lambda >c\end{array}$$

The Significance of the LRT is in that it is the most powerful test:

thm. Neymann—Pearson Lemma. Given level $\alpha$ [=false positive rate = type I error rate], then the LRT is the Uniformly Most Powerful (UMP) test.

There is also another significant result for certain likelihood ratio. But first define:

def. Monotone Likelihood Ratio (MLR) is when the likelihood ratio $\Lambda$ is never decreasing for some variable $x$.

Then we can determine the UMP of such $\Lambda$:

thm. Karlin—Rubin Theorem. In a distribution with $\Lambda$ that is MLR for region $T(x)$ [statistic] for a sampling $X_1,\dots ,X_n$ in this distribution the UMP is:

$$\delta :\{\begin{array}{ll}{H}_0& \text{else }\\ H_1& \text{if }T>c\end{array}$$