# Neyman–Pearson lemma

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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) for alternatives in the set .

It is named for Jerzy Neyman and Egon Pearson.

In practice, the likelihood ratio is often used directly to construct tests — see Likelihood-ratio test. However it can also be used to suggest particular test-statistics that might be of interest or to suggest simplified tests — for this one considers algebraic manipulation of the ratio to see if there are key statistics 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).

## ProofEdit

Define the rejection region of the null hypothesis for the NP test as

Any other test will have a different rejection region that we define as . Furthermore define the function of region, and parameter

where this is the probability of the data falling in region R, given parameter .

For both tests to have significance level , it must be true that

However it is useful to break these down into integrals over distinct regions, given by

and

Setting and equating the above two expression, yields that

Comparing the powers of the two tests, which are and , one can see that

Now by the definition of ,

Hence the inequality holds.

## ExampleEdit

Let be a random sample from the distribution where the mean is known, and suppose that we wish to test for against . The likelihood for this set of normally distributed data is

We can compute the likelihood ratio to find the key statistic in this test and its effect on the test's outcome:

This ratio only depends on the data through . Therefore, by the Neyman-Pearson lemma, the most powerful test of this type of hypothesis for this data will depend only on . Also, by inspection, we can see that if , then is an increasing function of . So we should reject if is sufficiently large. The rejection threshold depends on the size of the test.

## See alsoEdit

## ReferencesEdit

- 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 linksEdit

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

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