Difference between revisions of "Similarity region"

Revision as of 14:53, 7 June 2020

similar region

A generally used abbreviation of the term "critical region similar to a sample space" as used in mathematical statistics for a critical region with non-randomized similarity of a statistical test.

Let be a random variable taking values in a sample space , , and consider testing the compound hypothesis : against the alternative : . Suppose that in order to test against , a non-randomized similar test of level () has been constructed, with critical function , . As this test is non-randomized,

(1)

where is a certain set in , called the critical set for the test (according to this test, the hypothesis is rejected in favour of if the event is observed in an experiment). Also, the constructed test is a similar test, which means that

(2)

It follows from (1) and (2) that the critical region of a non-randomized similar test has the property:

Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called a "region similar to the sample space" , in the sense that the two probabilities and are independent of .

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2]	B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[3]	J. Neyman, E.S. Pearson, "On the problem of the most efficient tests of statistical hypotheses" Philos. Trans. Roy. Soc. London Ser. A , 231 (1933) pp. 289–337
[4]	E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation I" Sankhyā , 10 (1950) pp. 305–340
[5]	E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation II" Sankhyā , 15 (1955) pp. 219–236

How to Cite This Entry:
Similarity region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similarity_region&oldid=49424

This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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 ''similar region''
 A generally used abbreviation of the term  "critical region similar to a sample space"  as used in mathematical statistics for a [[Critical region|critical region]] with non-randomized similarity of a statistical test.
-Let  $  X $
+Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851701.png" /> be a random variable taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851703.png" />, and consider testing the compound hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851704.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851705.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851706.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851707.png" />. Suppose that in order to test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851708.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851709.png" />, a non-randomized [[Similar test|similar test]] of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517011.png" />) has been constructed, with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517013.png" />. As this test is non-randomized,
-be a random variable taking values in a sample space  $  ( \mathfrak X , \mathfrak B , {\mathsf P} _  \theta  ) $,
-$  \theta \in \Theta $,
-and consider testing the compound hypothesis  $  H _ {0} $:
-$  \theta \in \Theta _ {0} \subset  \Theta $
-against the alternative  $  H _ {1} $:
-$  \theta \in \Theta _ {1} = \Theta \setminus  \Theta _ {0} $.
-Suppose that in order to test  $  H _ {0} $
-against  $  H _ {1} $,
-a non-randomized [[Similar test|similar test]] of level  $  \alpha $(
-$  0 < \alpha < 1 $)
-has been constructed, with critical function  $  \phi ( x) $,
-$  x \in \mathfrak X $.
-As this test is non-randomized,
-$$ \tag{1 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
-\phi ( x)  = \
-\left \{
-where  $  K $
+where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517015.png" /> is a certain set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517016.png" />, called the critical set for the test (according to this test, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517017.png" /> is rejected in favour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517018.png" /> if the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517019.png" /> is observed in an experiment). Also, the constructed test is a similar test, which means that
-is a certain set in  $  \mathfrak X $,
-called the critical set for the test (according to this test, the hypothesis  $  H _ {0} $
-is rejected in favour of  $  H _ {1} $
-if the event  $  \{ X \in K \} $
-is observed in an experiment). Also, the constructed test is a similar test, which means that
-$$ \tag{2 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
-\int\limits _ { \mathfrak X } \phi ( x)  d {\mathsf P} _  \theta   =  \alpha
-\  \textrm{ for }  \textrm{ all }  \theta \in \Theta _ {0} .
-$$
-It follows from (1) and (2) that the critical region  $  K $
+It follows from (1) and (2) that the critical region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517021.png" /> of a non-randomized similar test has the property:
-of a non-randomized similar test has the property:
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517022.png" /></td> </tr></table>
-{\mathsf P} _  \theta  \{ X \in K \}  =  \alpha
-\  \textrm{ for }  \textrm{ all }  \theta \in \Theta _ {0} .
-$$
-Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called  $  K $
+Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517023.png" /> a  "region similar to the sample space"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517024.png" />, in the sense that the two probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517026.png" /> are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517027.png" />.
-a  "region similar to the sample space"   $  \mathfrak X $,
-in the sense that the two probabilities  $  {\mathsf P} _  \theta  \{ X \in K \} $
-and  $  {\mathsf P} _  \theta  \{ X \in \mathfrak X \} $
-are independent of  $  \theta \in \Theta _ {0} $.
 ====References====
 <table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Neyman,   E.S. Pearson,   "On the problem of the most efficient tests of statistical hypotheses"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''231'''  (1933)  pp. 289–337</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.L. Lehmann,   H. Scheffé,   "Completeness, similar regions, and unbiased estimation I"  ''Sankhyā'' , '''10'''  (1950)  pp. 305–340</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.L. Lehmann,   H. Scheffé,   "Completeness, similar regions, and unbiased estimation II"  ''Sankhyā'' , '''15'''  (1955)  pp. 219–236</TD></TR></table>

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Difference between revisions of "Similarity region"

Revision as of 14:53, 7 June 2020

References