Uniformly Most Powerful Tests and the Karlin-Rubin Theorem

Example 2 of the Neyman-Pearson lemma note noticed something that looked almost accidental: the most powerful test of $H_0:\mu=\mu_0$ against $H_1:\mu=\mu_1$ turned out to be the same test — reject for $\bar X>x_0$ — no matter which value $\mu_1>\mu_0$ was chosen as the specific alternative. This note shows that was no accident at all. A test that is simultaneously most powerful against every simple alternative in a composite $H_1$ is called uniformly most powerful (UMP), and the Karlin-Rubin theorem identifies exactly the structural property — the monotone likelihood ratio — that guarantees a UMP test exists for one-sided hypotheses, and hands over its exact form for free. This is where the Neyman-Pearson machinery graduates from a special-case result about two fully specified distributions to a genuinely practical theorem covering an entire family of composite testing problems: exponential families, which cover nearly every named distribution used in quantitative finance, always qualify.

Learning objectives. By the end of this note you should be able to: