Asymptotically-unbiased test

A concept indicating that a statistical 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:

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

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} $.