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I have a set of N measurements of a p-dimensional vector, and another set of N measurements of a different p-dimensional vector. Each measurement is contaminated with random gaussian noise.

How can I determine statistically whether the two measured vectors have different directions?

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  • $\begingroup$ I'm wondering whether I can use the Multivariate Paired Hotelling's T-Square test on the vectors after scaling to norm=1. That's a generalization of the t-test to the multivariate case, so the null hypothesis is that the population means of the two P-dimensional random vectors are equal. If the null hypothesis is rejected, that would indicate that the directions are different. $\endgroup$ Jan 4 '16 at 14:03
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I don't think there is an existing test for any given number of dimensions. However, you could (brute force) bootstrap the 95% confidence interval of directions of one set of measurements, and then see if the average vector of the other set of measurements is withing that interval or not. This will probably only work if N is several times P - not sure how many times.

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  • $\begingroup$ That's a good suggestion -- bootstrap is always the solution when we don't have a direct test. Now, why you must have N>P? $\endgroup$ Jan 4 '16 at 13:51

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