View source for Asymptotically-unbiased test

<!--
a0138301.png
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/A013/A.0103830 Asymptotically\AAhunbiased test
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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} $.