View source for Invariant test

A statistical test based on an [[Invariant statistic|invariant statistic]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523401.png" /> be a sampling space and suppose that the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523402.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523403.png" /> is tested against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523404.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523405.png" />, where the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523406.png" /> is invariant under the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523407.png" /> of one-to-one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523408.png" />-measurable transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i0523409.png" /> onto itself, that is,

<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/i/i052/i052340/i05234010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234011.png" /> is the element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234012.png" /> of one-to-one transformations of the probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234013.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234014.png" />, defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234016.png" /> according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234018.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234019.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234020.png" />, in testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234021.png" /> it is natural to use a test based on an invariant statistic with respect to this same group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234022.png" />. Such a test is called an invariant test, and the class of all invariant tests is the same as the class of tests based on a maximal invariant. In the theory of invariant tests, the Hunt–Stein theorem plays an important role: If the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234023.png" /> is invariant under the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234024.png" />, then there exists a maximin test in the class of invariant tests for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052340/i05234025.png" />. An invariant test is a special case of an invariant statistical procedure (see [[Invariance of a statistical procedure|Invariance of a statistical procedure]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Schmetterer,   "Introduction to mathematical statistics" , Springer  (1974)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.J. Hall,   R.A. Wijsman,   J.K. Chosh,   "The relationships between sufficiency and invariance with applications in sequential analysis"  ''Ann. Math. Stat.'' , '''36'''  (1965)  pp. 575</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.P. Klimov,   "Invariant inferences in statistics" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Zacks,   "The theory of statistical inference" , Wiley  (1971)</TD></TR></table>