# 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 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

where

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 .

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 related to the size of the ratio (i.e. whether a large statistic corresponds to a small ratio or to a large one).

## Proof

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 probability of the data falling in region R, given parameter as

For both tests to have size , it must be true that

It will be useful to break these down into integrals over distinct regions:

and

Setting and equating the above two expression yields that

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

Hence the inequality holds.

## Example

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 a decreasing function of . So we should reject if is sufficiently large. The rejection threshold depends on the size of the test. In this example, the test statistic can be shown to be a scaled Chi-square distributed random variable and an exact critical value can be obtained.

## See also

## References

## External links

- Cosma Shalizi, a professor of statistics at Carnegie Mellon University, gives an intuitive derivation of the Neyman–Pearson Lemma using ideas from economics