Why does no uniformly most powerful test exist for two-sided alternatives?

Because the most powerful test against an alternative on one side differs from the one against the other side, so no single test can be most powerful against both at once; restricting to unbiased tests resolves the conflict.

What does the monotone likelihood ratio property buy you?

It guarantees that a simple one-sided test based on a single statistic is uniformly most powerful, so optimality for the whole one-sided alternative follows without checking each alternative separately.

Uniformly Most Powerful Tests

A uniformly most powerful test is most powerful against every alternative at once; such tests exist for one-sided problems with monotone likelihood ratio and are sought within restricted classes otherwise.

Definition

A test is uniformly most powerful at a given size if, among all tests of that size, it has the greatest power simultaneously against every distribution in the alternative hypothesis.

Scope

This topic covers composite hypotheses, the monotone likelihood ratio property and the families that possess it, the existence of uniformly most powerful tests for one-sided alternatives, the nonexistence of such tests for two-sided alternatives, and the restriction to unbiased or invariant tests that restores optimality, including uniformly most powerful unbiased tests in exponential families.

Core questions

What is the monotone likelihood ratio property, and which families have it?
Why do uniformly most powerful tests exist for one-sided but not two-sided alternatives?
How does restricting to unbiased tests recover an optimal two-sided test?
How does invariance reduce a problem so that a uniformly most powerful test exists?

Key theories

Monotone likelihood ratio and one-sided tests: If the likelihood ratio is monotone in a statistic, the test rejecting for large values of that statistic is uniformly most powerful for the corresponding one-sided alternative, extending the Neyman-Pearson lemma to a composite alternative.
Uniformly most powerful unbiased tests: For two-sided alternatives no uniformly most powerful test exists, but within the class of unbiased tests an optimal test does, and in exponential families it takes an explicit two-tailed form.

Clinical relevance

Standard one-sided z and t tests used in trials and quality control are uniformly most powerful for their problems, so the theory explains why these familiar procedures are not merely conventional but optimal among size-controlled tests.

History

Building on the Neyman-Pearson lemma of 1933, Lehmann systematized uniformly most powerful, unbiased, and invariant tests in his 1959 monograph Testing Statistical Hypotheses, later revised with Romano, which remains the standard reference.

Key figures

Erich L. Lehmann
Jerzy Neyman
Egon Pearson
Joseph P. Romano

Seminal works

lehmannRomano2005

Frequently asked questions

Why does no uniformly most powerful test exist for two-sided alternatives?: Because the most powerful test against an alternative on one side differs from the one against the other side, so no single test can be most powerful against both at once; restricting to unbiased tests resolves the conflict.
What does the monotone likelihood ratio property buy you?: It guarantees that a simple one-sided test based on a single statistic is uniformly most powerful, so optimality for the whole one-sided alternative follows without checking each alternative separately.