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Is this true?

A UMP test exists for every testing problem

No — it is false.

The counterexample

$H_{0} : μ = 0$ vs $H_{1} : μ \neq = 0$ for $N (μ, 1)$

The MP test for $μ > 0$ rejects for large X̄, for $μ < 0$ for small X̄; no single test is best against both.

The kind of mistake this is

What the inference means

p-values, size, power and error types say specific things. This was not one of them.

Drill statements like this

statistics testing

Others that fail the same way

From Hypothesis Testing › Neyman–Pearson lemma and UMP tests

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