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NeymanPearson lemmaFrom Wikipedia, the free encyclopedia

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In statistics, the Neyman-Pearson lemma, named after Jerzy Neyman and Egon Pearson, states that when performing a

hypothesis test between two point hypothesesH0: = 0 andH1: = 1, then the likelihood-ratio test which rejectsH0in favour ofH1 when

where

is the most powerful test of size for a threshold . 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).

Contents

1 Proof

2 Example

3 See also

4 References

5 External links

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

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Now by the definition of ,

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 an increasing 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

Statistical power

References

Jerzy Neyman, Egon Pearson (1933). "On the Problem of the Most Efficient Tests of Statistical Hypotheses".

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or

Physical Character231: 289337. doi:10.1098/rsta.1933.0009 (http://dx.doi.org/10.1098%2Frsta.1933.0009) .

JSTOR 91247 (http://www.jstor.org/stable/91247) .

cnx.org: Neyman-Pearson criterion (http://cnx.org/content/m11548/latest/)

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 (http://cscs.umich.edu/~crshalizi/weblog/630.html)

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