What is Neyman–Pearson Lemma: Why the Likelihood Ratio Test Is Most Powerful

Neyman–Pearson Lemma: Most Powerful Tests

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The Neyman–Pearson Lemma is one of the most important results in mathematical statistics tutoring. It provides the foundation for the likelihood ratio test and explains how to construct the most powerful test for simple hypotheses. This page breaks down the statement, intuition, and exam‑relevant logic behind the lemma.

Reject \(H_0\) when \(\frac{f_1(X)}{f_0(X)} > k\)

where \(f_0\) and \(f_1\) are the densities under the null and alternative.

Why is the likelihood ratio test most powerful? Because it maximizes the probability of rejecting the null when the alternative is true, subject to a fixed Type I error rate. The lemma shows that no other test can achieve higher power without exceeding the allowed significance level. This optimality makes the likelihood ratio test the backbone of modern hypothesis testing.

1. Specify the hypotheses. \(H_0: \theta = \theta_0\) vs. \(H_1: \theta = \theta_1\).
2. Write the likelihoods. \(L_0 = f(X \mid \theta_0)\), \(L_1 = f(X \mid \theta_1)\).
3. Form the likelihood ratio. \(\Lambda(X) = \frac{L_1}{L_0}\).
4. Choose a rejection region. Reject \(H_0\) when \(\Lambda(X) > k\).
5. Determine k from the size constraint. Choose \(k\) so that \(P_{H_0}(\Lambda(X) > k) = \alpha\).
6. Conclude optimality. No other test of size α has higher power.

Suppose \(X \sim N(\theta, 1)\). Test \(H_0: \theta = 0\) vs. \(H_1: \theta = 1\).

Likelihood ratio:

\(\Lambda(x) = \exp(x – 1/2)\)

Reject \(H_0\) when \(\Lambda(x) > k\), which is equivalent to:

\(x > c\)

for some cutoff \(c\) chosen to achieve size α. This is exactly the most powerful test.

• Thinking the lemma applies to composite hypotheses (it does not).
• Confusing “most powerful” with “uniformly most powerful.”
• Forgetting that the lemma requires simple vs. simple hypotheses.
• Misinterpreting the likelihood ratio as a probability.

The Neyman–Pearson Lemma is the foundation of likelihood ratio tests, uniformly most powerful tests, and classical hypothesis testing. It appears in nearly every graduate statistics exam and is essential for understanding optimal decision rules.

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