Neyman–Pearson lemma

In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933.[1] It shows that the likelihood-ratio test is the most powerful test, among all possible statistical tests.

PropositionEdit

Suppose one is performing a hypothesis test between two simple hypotheses   and   using the likelihood-ratio test with likelihood-ratio threshold  , which rejects   in favour of   at a significance level of

 

where

 

and   is the likelihood function. Then, the Neyman–Pearson lemma states that the likelihood ratio,  , is the most powerful test at significance level  .

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).

ProofEdit

Define the rejection region of the null hypothesis for the Neyman–Pearson (NP) test as

 

where   is chosen so that  

Any alternative test will have a different rejection region that we denote by  .

The probability of the data falling within either region   or   given parameter   is

 

For the test with critical region   to have significance level  , it must be true that  , hence

 

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

 

where   is the complement of region R. Setting  , these two expressions and the above inequality yield that

 

The powers of the two tests are   and  , and we would like to prove that:

 

However, as shown above this is equivalent to:

 

in what follows we show that the above 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 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 economicsEdit

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 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.[2]

Uses in electrical engineeringEdit

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 (low) 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 physicsEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ Neyman, J.; Pearson, E. S. (1933-02-16). "IX. On the problem of the most efficient tests of statistical hypotheses". Phil. Trans. R. Soc. Lond. A. 231 (694–706): 289–337. doi:10.1098/rsta.1933.0009. ISSN 0264-3952.
  2. ^ Berliant, M. (1984). "A characterization of the demand for land". Journal of Economic Theory. 33 (2): 289–300. doi:10.1016/0022-0531(84)90091-7.
  • E. L. Lehmann, Joseph P. Romano, Testing statistical hypotheses, Springer, 2008, p. 60

External linksEdit