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, then the likelihood-ratio test which rejects H0 in favour of H1 when
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).
Define the rejection region of the null hypothesis for the NP test as
where is chosen so that .
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 the test with critical region to have level , it must be true that , hence
It will be useful to break these down into integrals over distinct regions:
Setting , these two expressions and the above inequality yield that
Comparing the powers of the two tests, and , one can see that
Now by the definition of ,
Hence the inequality holds.
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.
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 stastistical test, and so the Neyman–Pearson lemma can be used.
- Neyman, Jerzy; Pearson, Egon S. (1933). "On the Problem of the Most Efficient Tests of Statistical Hypotheses". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 231 (694–706): 289–337. Bibcode:1933RSPTA.231..289N. doi:10.1098/rsta.1933.0009. JSTOR 91247.
- cnx.org: Neyman–Pearson criterion
- Cosma Shalizi, a professor of statistics at Carnegie Mellon University, gives an intuitive derivation of the Neyman–Pearson Lemma using ideas from economics