Neyman Structure

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What is a Neyman structure?

A Neyman structure describes a property of a hypothesis test that ensures its power function — the probability of rejecting the null hypothesis when it is false — is constant for all values of a sufficient statistic. A sufficient statistic is the best possible estimator for a population parameter.

The Neyman Structure was introduced in 1954 by Polish-American mathematician and statistician Jerzy Neyman [1] in connection with the problem of constructing similar tests in hypothesis testing. Neyman’s contributions to statistics — and other activities in scientific planning, organization and collaboration between different disciplines — were considered revolutionary at the time [2].

Neyman structure definition

The Neyman structure ensures that the critical function (a function that maps test statistic values to the rejection region) of a test does not depend on the value of the sufficient statistic, regardless of the value of the population parameter. The rejection region determines whether we reject the null hypothesis or not.

Formally, a Neyman structure is defined as follows [3]:

If T is a sufficient statistic for ψ in the classical sense, then a test Φ has Neyman structure relative to G and T if E_θ[Φ(X) | T = t] is constant as a function of t P_θ-a.s. for all θ ∈ G

Let’s break the definition down in steps:

• T is a sufficient statistic for ψ in the classical sense. This means that T contains all the information about ψ that is in the original sample X.
• A test Φ has Neyman structure relative to G and T. This means that the test Φ has the same power function for all values of T. In other words, the probability of rejecting the null hypothesis when it is true (a Type I error) does not depend on the value of T.
• Eθ[Φ(X) | T = t] is constant as a function of t Pθ-a.s. for all θ ∈ G. This means that the expected value of Φ(X) given T = t is constant for all values of t and for all values of θ in the set G.
  • The term “a.s.” means almost surely and indicates that the statement holds true for values of θ in the set G with probability 1. Without the use of the term a.s., the definition would only be valid for some values of θ, which would make it less useful.
  • Set G refers to the set of all values of θ for which the test Φ has Neyman structure. In other words, G is the set of all values of θ for which the expected value of Φ(X) given T = t is constant as a function of t.

Applications of the Neyman structure

 

The Neyman structure is particularly useful because it sometimes makes it easier to identify UMPU (uniformly most powerful unbiased) tests. An UMPU test is a test that has the highest power among all tests of the same significance level (α-level). If we can find statistic T such that the distribution of X conditional on T has one-dimensional parameter, then we can look for the UMPU test among all tests that have a α-level level conditional on T. Then this test will also be UMPU level α unconditionally [4]. In other words, if we can reduce a problem to testing a one-dimensional parameter, then we can find the UMPU test using a standard procedure.

For example, let’s say we want to test the hypothesis that the mean of a normal distribution equals zero. We can use the statistic T = X to test this hypothesis. The distribution of X conditional on T is a one-dimensional normal distribution with mean T. We can look at all tests for the UMPU test that have level α conditional on T. This can be done using a standard procedure, such as the Neyman-Pearson lemma.

In addition to being independent from sufficient statistics, the Neyman Structure can also be applied to many different types of tests including those conducted on sample sizes, populations, proportions and more. This allows us to create accurate tests on various topics without having to worry about relying on inaccurate statistics due to outside factors. Additionally, this structure helps make sure that our tests produce results that are consistent with what we expect them to be so that we can trust their accuracy when making decisions based off of them.

References

1. J. Neyman, “Current problems of mathematical statistics” , Proc. Internat. Congress Mathematicians (Amsterdam, 1954) , 1 , Noordhoff & North-Holland (1957) pp. 349–370
2. Hult, H. Lecture 9. 17. Hypothesis Testing. https://www.math.kth.se/matstat/gru/Statistical%20inference/Lecture5_2015.pdf
3. 1.3.6.7.2. Critical Values of the Student’s t Distribution
4. The Legacy of Jerzy Neyman. Supplements.