Hotelling test

-test

A test intended for testing a hypothesis according to which the true value of the unknown vector of mathematical expectation of a non-degenerate -dimensional normal law whose covariance matrix is also unknown, is the vector . Hotelling's test is based on the following result. Let be independent -dimensional random vectors, , subject to the non-degenerate normal law , and let

where

and

are maximum-likelihood estimators for the unknown parameters and . Then the statistic

has the non-central Fisher -distribution with and degrees of freedom and non-centrality parameter

the statistic has the Hotelling -distribution. Consequently, to test the hypothesis : against the alternative : one can compute the values of the statistic based on realizations of the independent random vectors from the non-degenerate -dimensional normal law , which under the hypothesis has the central -distribution with and degrees of freedom. Using Hotelling's test with significance level , must be rejected if , where is the -quantile of the -distribution. The connection between Hotelling's test and the generalized likelihood-ratio test should be mentioned. Let

be the likelihood function computed from the sample . The generalized likelihood-ratio test for testing the simple hypothesis : against the compound alternative : is constructed from the statistic

The statistic and the statistics and are related by:

For testing the hypothesis : , Hotelling's test is uniformly most powerful among all tests that are invariant under similarity transformations (see Most-powerful test; Invariant test).

References

[1]	T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1984)
[2]	C.R. Rao, "Linear statistical inference and its applications" , Wiley (1973)