Neyman–Pearson lemma and UMP tests

Neyman–Pearson lemma

For $H_{0}: \theta = \theta_{0}$ vs $H_{1}: \theta = \theta_{1} ($both simple), the most powerful size$-\alpha$ test rejects when

$L(\theta_{1})/L(\theta_{0}) > k$, with k chosen so the size is $\alpha$.

From MP to UMP

If the family has monotone likelihood ratio in T (exponential families do), then the one-sided test "reject if T > c" is UMP for $H_{0}: \theta \le \theta_{0}$ vs $H_{1}: \theta > \theta_{0}$.

Two-sided alternatives generally admit no UMP test — the best one-sided tests point in opposite directions. One then uses UMP unbiased tests or the likelihood ratio test.

Direction matters

If larger $\theta$ makes T smaller, the rejection region flips. For a Gamma$(4, \lambda)$ sample, $T = \sum{}X_{i}$ has $2\lambda{}T$ ~ $\chi^{2}_{4n}$, and larger $\lambda$ shrinks T, so the UMP test for $\lambda > 1$ rejects for small T.

Size and power

• Size = P(reject | $H_{0})$; power = P(reject | $H_{1}) = 1 - \beta$.
• Power increases with n, with |$\theta_{1} - \theta_{0}$|, and with $\alpha$.
• A test with power < size is worse than random (biased test).

Likelihood ratio test

$\Lambda =$ $sup_{\Theta_{0}}$ $L / sup_\Theta L$; reject for small $\Lambda$. Under regularity, $**-2 \ln \Lambda \to \chi^{2}_r**$ where r = the number of restricted parameters (Wilks). This is the general-purpose test when no UMP exists.

Key takeaways

• NP: simple vs simple, likelihood ratio, threshold from the size.
• MLR ⇒ one-sided UMP; two-sided usually has none.
• Check whether the statistic increases or decreases with the parameter before writing the rejection region.