The Robustness of Hotelling'sT2

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Abstract
Monte Carlo methods are used to study the effect of inequality of covariance matrices on the distribution of Hotelling's T 2 statistic. Samples are drawn from multivariate normal distributions and from them the distribution of T 2 is approximated. One minus the significance level is calculated for various departures from equality, for tests with a supposed significance level of .05 and .01. The power of the tests is also given. The number of variates considered is 1, 2, 3, 5, 7, and 10. The effect of sample size on level of significance and power is studied; both equal and unequal sample sizes are used. Let ∑1 and ∑2 be the two covariance matrices. Graphs are presented for one minus the significance level versus d, for the case where all eigenvalues of the matrix ∑2∑1 −1 are equal to d. One minus the significance level is also plotted against the sample size for the case of equal sample size and equal eigenvalues of ∑2∑1 −1. Power curves are also included for several ratios of sample size. In general it may be said that as the number of variates increases, or as the sample size decreases, the actual level of significance increases. Equal sample sizes help in keeping the level of significance close to the supposed level, but do not help in maintaining the power of the test.

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