Asymptotically-unbiased test

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 $R _ {n}$.