# Neymanâ€“Pearson lemma

The Neyman -Pearson lemma ( Neyman - Pearson also of fundamental lemma ) is a set of mathematical statistics, which makes an optimality statement about the construction of a hypothesis test. Subject of the Neyman -Pearson lemma is the simplest scenario of a hypothesis test, also called the Neyman - Pearson test: Both the null hypothesis and the alternative hypothesis is simple, that is, they correspond to a single probability distribution whose associated probability densities will hereinafter be referred to and. Then, the statement of the Neyman -Pearson lemma, one of the strongest test obtained by a decision in which the null hypothesis is rejected if the Likelihoodquotient falls below a certain value.

The lemma is named after Jerzy Neyman and Egon Pearson.

## Situation

Wanted is a possible " good " hypothesis test to reach a decision between null and alternative hypothesis with high reliability. It is assumed that null and alternative hypothesis respectively correspond precisely to comply with one of the results of observation probability distribution. Under these conditions, for each setting of a rejection region, the probability of a false test decision can be calculated precisely: in detail, it is the two probabilities of an error of the first type and an error of the second kind Therefore, in a predetermined by the significance level limit for an error first type the theoretically conceivable test decisions are particularly easy compared in qualitative terms with each other.

## Formal Description of the situation

Observed realizations of a real random vector with dimension above the measuring room. Unknown is the exact distribution of. To be tested the hypothesis " " against the alternative " " for two probability measures on the given measurement space. The extent and have densities or with respect to the Lebesgue measure, ie they describe continuous distributions over the.

A decision procedure is characterized now by defining a rejection region, which will make it the basic hypothesis if and only rejects if the observed realization of in is. This test must not exceed a predetermined level,

That is, the probability of falsely rejecting the basic hypothesis of the so-called type 1 error must not be greater than. Among all the tests that comply with this standard, called those the strongest test that maximizes the so-called test strength, ie a minimal type 2 error,

The probability of an erroneous Nichtverwerfen the basic hypothesis has.

## Formulation

### The Neyman -Pearson lemma

Under the above situation, you look for a realization of the extended likelihood ratio

The case is included for completeness defined, since it enters with no positive probability.

Now is a test of the hypothesis, " " against the alternative " " at a given level if and only optimal ( strongest test) if a exists, so his rejection region demands

Met. The almost certain properties of 2 and 3 in this case relate to the probability measure, ie they must almost certainly occur and respect.

Meets the requirements of a rejection region 1 - 3rd, is called a Neyman - Pearson also this area. In discrete models, such a rejection region exists only at certain levels to a pre surrounded level must fully exploit if necessary resorting to randomized tests.

### Special cases

By the above lemma is not considered to have at least the following special cases:

- The rejection region is the most powerful test for the test level, ie the test does not have a type 1 error. The corresponding test parameter.
- The rejection region is the most powerful test of level, because he has the strength test, ie the test does not have a type 2 error. The corresponding test parameter.