- Neyman-Pearson lemma
statistics, the Neyman-Pearson lemma states that when performing a hypothesis test between two point hypotheses "H"0: "θ"="θ"0 and "H"1: "θ"="θ"1, then the likelihood-ratio testwhich rejects "H"0 in favour of "H"1 when
is the most powerful test of size "α" for a threshold η. 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 ratiois 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 is related to the size of the ratio (i.e. whether a large statistic corresponds to a small ratio or to a large one).
If we define the rejection region of the null hypothesis, as , and any other test will have a different rejection region that we define as . Furthermore define the function of region, and parameter hence this is the probability of the data falling in region R, given parameter .
For both tests to have significance level , it must be true that, however it is useful to break these down into integrals over distinct regions.
Setting and equating the above two expression, yields that
Comparing the power of the two tests, which are 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 distributeddata is
We can compute the
likelihood ratioto 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 functionof . So we should reject if is sufficiently small. The rejection threshold depends on the size of the test.
Receiver operating characteristic
* cite journal
title=On the Problem of the Most Efficient Tests of Statistical Hypotheses
Jerzy Neyman, Egon Pearson
Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character
* [http://cnx.org/content/m11548/latest/ cnx.org: Neyman-Pearson criterion]
MIT OpenCourseWarelecture notes: [http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-443Fall2003/18B765F6-A398-48BF-A893-49A4965DED98/0/lec19.pdf most powerful tests] , [http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-443Fall2003/D6F12E47-A9A2-4FE0-AC3C-588B6A5EE5B6/0/lec20.pdf uniformly most powerful tests]
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