# What is the point of a likelihood ratio test?

Suppose you have:

1. $$n$$ data points, i.i.d. $$X_i \forall i \in 1,2,3,...n$$
2. $$H_0: X_i \sim \mathcal{N}(0,1)$$
3. $$H_1: X_i \sim \mathcal{N}(0,4)$$

You know the distribution of $$X_i$$ under both $$H_o$$ and $$H_1$$

The problem is to choose $$H_0$$ or $$H_1$$ that results in a given False Positive Rate (Type 1 error) under $$H_0$$

The likelihood ratio test as described here goes as follows:

1. Define the likelihood ratio $$R = \frac{\mathcal{L}(H_1 \mid X)}{\mathcal{L}(H_0 \mid X)}$$ = $$\frac{P(X;H_1)}{P(X; H_0)}$$
2. Set $$R$$ equal to some threshold $$\xi$$
3. Decide a level of Type 1 error $$\alpha$$ under $$H_0$$ that you want
4. Find the threshold $$\xi$$ such that $$P(R > \xi ; H_0) = \alpha$$
5. Reject $$H_0$$ if $$R > \xi$$

This seems like a lot of that that didn't need to be. We could directly have set $$P(X;H_0) = \alpha$$ to be the decision boundary. That is to say if $$P(X;H_0)$$ is lesser than $$\alpha$$ then reject the null hypothesis. Why the extra steps of coming via $$\xi$$? What purpose does the likelihood ratio test serve?

I don't understand your first suggestion: What does $$P(X;H_0) = \alpha$$ mean? (A probability measure can only be applied to 'events'; like the event $$\{R>\zeta\}$$ for the likelihood ratio test.)
• $P(X;H_0) = \alpha$ is indeed illegible. I have made an update to the post. Basically I wanted to say that if the probability of observing the observed data under the assumption of the null hypothesis is lesser than $\alpha$ then reject the null. Why do you go through the entire $\xi$ business? Jul 22, 2022 at 9:55
• I am afraid I do not see how the example is relevant. If I go in with an $\alpha$ in mind then I have set $\beta$ as well since $\alpha$ uniquely identifies a cutoff point (denoted as $q_{\alpha/2}$ in case of a standard normal). So, what I am saying is that the moment I decide what the type 1 error rate has to be the cutoff and $\beta$ are decided as well. So what additional purpose is served by the likelihood ratio test Jul 27, 2022 at 11:23