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Asymptotically-unbiased test

From Encyclopedia of Mathematics
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A concept indicating that the test is unbiased in the limit. For example, in the case of $ n $ independent samples from a one-dimensional distribution depending on a parameter $ \theta \in \Omega $, let $ H $ be the null hypothesis: $ \theta \in \Omega _ {H} $, and let $ K $ be the alternative:

$$ \theta \in \Omega _ {K} ,\ \Omega _ {H} \cup \Omega _ {K} = \Omega ,\ \ \Omega _ {H} \cup \Omega _ {K} = \emptyset . $$

The critical set $ R _ {n} $ in the $ n $- dimensional Euclidean space, $ n=1, 2 \dots $ is an asymptotically-unbiased test of the hypothesis $ H $ with level $ \alpha $ if

$$ \lim\limits _ {n \rightarrow \infty } {\mathsf P} ( R _ {n} \mid \theta ) \leq \alpha , \ \theta \in \Omega _ {H} , $$

$$ \alpha \leq \lim\limits _ {n \rightarrow \infty } {\mathsf P} ( R _ {n} \mid \theta ),\ \theta \in \Omega _ {K} . $$

The function

$$ \lim\limits _ {n \rightarrow \infty } {\mathsf P} ( R _ {n} \mid \theta ) $$

is called the asymptotic power function of the test $ R _ {n} $.

How to Cite This Entry:
Asymptotically-unbiased test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-unbiased_test&oldid=45237
This article was adapted from an original article by O.V. Shalaevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article