4.2 Neyman-Pearson Lemma

Assumption throughout this section: both hypotheses are simple.

When both the null and alternative hypotheses are simple, we can talk about the most powerful tests (or the best critical region).

Denote \(X = (X_1, \dots, X_n)\) and recall that the space that this lives in is a state space \(S\).

Definition 4.4 Let \(C\) be a subset of the state space. Then we say that \(C\) is the best critical region of size \(\alpha\) for testing the simple hypothesis \(H_0: \theta = \theta_0\) against the alternative simple hypothesis \(H_1: \theta = theta_1\) if

  1. \(P_{\theta_0}( (X_1, \dots, X_n) \in C ) = \alpha\)

  2. And for every subset \(A\) of the state space \[ \mathbb{P}_{\theta_0}( X \in A) = \alpha \implies \mathbb{P}_{\theta_1}(X \in C) \geq \mathbb{P}_{\theta_1} ( X \in A)\]

Recall the likelihood function \[\mathcal{L}(\theta;x) = \prod_{i=1}^n f(x_i;\theta)\] where \(x = (x_1, \dots, x_n)\).

Theorem 4.1 (Neyman-Pearson Theorem) Let \(X_1, \dots, X_n\) be a sample from a family of distributions \(f(x;\theta)\), where \(\theta \in \{\theta_0, \theta_1\}\). Let \(k\) be a positive number and \(C\) be a subset of the state space such that

  1. \(\displaystyle{\frac{\mathcal{L}(\theta_0;x)}{\mathcal{L}(\theta_1;x)}} \leq k\) for each point \(x\in C\).
  2. \(\displaystyle\frac{\mathcal{L}(\theta_0;x)}{\mathcal{L}(\theta_1;x)} \geq k\) for each point \(x\in C^c\).
  3. \(\alpha = P_{\theta_0}(X \in C)\).

Then \(C\) is a bests critical region of size \(\alpha\) for testing the hypothesis \(H_0: \theta= \theta_0\) against the alternative hypothesis \(H_1: \theta = \theta_1\).