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In statistics, the **Neyman-Pearson lemma** states that when performing a hypothesis test between two point hypotheses *H*_{0}: *θ*=*θ*_{0} and *H*_{1}: *θ*=*θ*_{1}, then the likelihood-ratio test which rejects *H*_{0} in favour of *H*_{1} when

is the **most powerful test** of size *α* for a threshold η. If the test is most powerful for all , it is said to be uniformly most powerful (UMP).

In practice, the likelihood ratio itself is not actually used in the test. Instead one computes the ratio to see how the key statistic in it is related to the size of the ratio (i.e. whether a large statistic corresponds to a small ratio or to a large one).

## Example[]

## See also[]

## References[]

- Jerzy Neyman, Egon Pearson (1933). On the Problem of the Most Efficient Tests of Statistical Hypotheses.
*Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character***231**: 289-337. - cnx.org: Neyman-Pearson criterion

## External links[]

- MIT OpenCourseWare lecture notes: most powerful tests, uniformly most powerful tests

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