Neyman–Pearson lemma

In statistics, the Neyman–Pearson lemma, named after Jerzy Neyman and Egon Pearson, states that when performing a hypothesis test between two simple hypotheses H₀: θ = θ₀ and H₁: θ = θ₁, the likelihood-ratio test which rejects H₀ in favour of H₁ when

where

is the most powerful test at significance level α for a threshold η. If the test is most powerful for all ${\displaystyle \theta _{1}\in \Theta _{1}}$, it is said to be uniformly most powerful (UMP) for alternatives in the set ${\displaystyle \Theta _{1}\,}$.

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