Difference between revisions of "Asymptotically-unbiased test"
From Encyclopedia of Mathematics
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | 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 | The function | ||
| − | + | $$ | |
| + | \lim\limits _ {n \rightarrow \infty } {\mathsf P} ( R _ {n} \mid \theta ) | ||
| + | $$ | ||
| − | is called the asymptotic power function of the test | + | is called the asymptotic power function of the test $ R _ {n} $. |
Revision as of 18:48, 5 April 2020
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=18499
Asymptotically-unbiased test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-unbiased_test&oldid=18499
This article was adapted from an original article by O.V. Shalaevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article