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
Contents
where
is the most powerful test at significance level α for a threshold η. If the test is most powerful for all
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
where
Any other test will have a different rejection region that we define as
For the test with critical region
It will be useful to break these down into integrals over distinct regions:
Setting
Comparing the powers of the two tests,
We show that the left hand inequality holds: Now by the definition of
Example
Let
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
Application in economics
A variant of the Neyman–Pearson lemma has found an application in the seemingly-unrelated domain of economy with land. One of the fundamental problems in consumer theory is calculating the demand function of the consumer given the prices. In particular, given a heterogeneous land-estate, a price measure over the land, and a subjective utility measure over the land, the consumer's problem is to calculate the best land parcel that he can buy – i.e, the land parcel with the largest utility, whose price is at most his budget. It turns out that this problem is very similar to the problem of finding the most powerful statistical test, and so the Neyman–Pearson lemma can be used.