Neyman–Pearson lemma
In statistics, the Neyman–Pearson lemma describes the existence and uniqueness of the likelihood ratio as a powerful test in certain contexts. It was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman–Pearson lemma is part of the Neyman–Pearson theory of statistical testing, which introduced concepts such as errors of the second kind, power function, and inductive behavior.Wald: Chapter II: The Neyman–Pearson Theory of Testing a Statistical Hypothesis: The previous Fisherian theory of significance testing postulated only one hypothesis. By introducing a competing hypothesis, the Neyman–Pearsonian flavor of statistical testing allows investigating the two types of errors. The trivial cases where one always rejects or fails to reject the null hypothesis are of little interest but it does prove that one must not relinquish control over one type of error while calibrating the other. Neyman and Pearson accordingly proceeded to restrict their attention to the class of all level tests while subsequently minimizing type II error, traditionally denoted by. Their seminal 1933 paper, which includes the Neyman–Pearson lemma showed not only the existence of tests with the most power that retain a prespecified level of type I error, but also provided a way to construct such tests. The Karlin-Rubin theorem extended the Neyman–Pearson lemma to settings involving composite hypotheses with monotone likelihood ratios.
Statement
Consider a test with hypotheses and, where the probability density function is for.For any hypothesis test with rejection set, and any, we say that it satisfies condition if
- * That is, the test has size .
- such that
- That is, we have a strict likelihood ratio test, except on a negligible subset.
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.
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 isWe 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 the economics of land value. 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 they can buy – i.e. the land parcel with the largest utility, whose price is at most their 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.Uses in electrical engineering
The Neyman–Pearson lemma is quite useful in electronics engineering, namely in the design and use of radar systems, digital communication systems, and in signal processing systems.In radar systems, the Neyman–Pearson lemma is used in first setting the rate of missed detections to a desired level, and then minimizing the rate of false alarms, or vice versa.
Neither false alarms nor missed detections can be set at arbitrarily low rates, including zero. All of the above goes also for many systems in signal processing.
Uses in particle physics
The Neyman–Pearson lemma is applied to the construction of analysis-specific likelihood-ratios, used to e.g. test for signatures of new physics against the nominal Standard Model prediction in proton–proton collision datasets collected at the LHC.Discovery of the lemma
The work that led to the lemma started around 1927. Neyman later wrote about this in a book chapter:Neyman described the actual discovery of the lemma as follows. Paragraph breaks have been inserted.