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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 <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,

<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>

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

<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>

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:

<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>

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" />.

====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>