View source for Hotelling test

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480802.png" />-test''

A test intended for testing a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480803.png" /> according to which the true value of the unknown vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480804.png" /> of mathematical expectation of a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480805.png" />-dimensional normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480806.png" /> whose covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480807.png" /> is also unknown, is the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480808.png" />. Hotelling's test is based on the following result. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h0480809.png" /> be independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808010.png" />-dimensional random vectors, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808011.png" />, subject to the non-degenerate normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808012.png" />, and let

<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/h/h048/h048080/h04808013.png" /></td> </tr></table>

where

<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/h/h048/h048080/h04808014.png" /></td> </tr></table>

and

<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/h/h048/h048080/h04808015.png" /></td> </tr></table>

are maximum-likelihood estimators for the unknown parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808017.png" />. Then the statistic

<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/h/h048/h048080/h04808018.png" /></td> </tr></table>

has the non-central [[Fisher-F-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808019.png" />-distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808021.png" /> degrees of freedom and non-centrality parameter

<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/h/h048/h048080/h04808022.png" /></td> </tr></table>

the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808023.png" /> has the [[Hotelling-T^2-distribution|Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808024.png" />-distribution]]. Consequently, to test the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808025.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808026.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808027.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808028.png" /> one can compute the values of the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808029.png" /> based on realizations of the independent random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808030.png" /> from the non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808031.png" />-dimensional normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808032.png" />, which under the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808033.png" /> has the central <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808034.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808036.png" /> degrees of freedom. Using Hotelling's test with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808038.png" /> must be rejected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808040.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808041.png" />-quantile of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808042.png" />-distribution. The connection between Hotelling's test and the generalized likelihood-ratio test should be mentioned. Let

<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/h/h048/h048080/h04808043.png" /></td> </tr></table>

<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/h/h048/h048080/h04808044.png" /></td> </tr></table>

be the likelihood function computed from the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808045.png" />. The generalized likelihood-ratio test for testing the simple hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808046.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808047.png" /> against the compound alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808048.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808049.png" /> is constructed from the statistic

<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/h/h048/h048080/h04808050.png" /></td> </tr></table>

The statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808051.png" /> and the statistics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808053.png" /> are related by:

<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/h/h048/h048080/h04808054.png" /></td> </tr></table>

For testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808055.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048080/h04808056.png" />, Hotelling's test is uniformly most powerful among all tests that are invariant under similarity transformations (see [[Most-powerful test|Most-powerful test]]; [[Invariant test|Invariant test]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.W. Anderson,   "An introduction to multivariate statistical analysis" , Wiley  (1984)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.R. Rao,   "Linear statistical inference and its applications" , Wiley  (1973)</TD></TR></table>