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When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero.

I am interested in a generalisation of this test to the case where instead a sample of numbers we have a sample of vectors. In other words, we sample from a multivariate distribution. For example, we have 1000 three dimensional vectors (which can be represented as a 1000 by 3 matrix).

My null hypothesis is that mean of this distribution is zero. The test have to take into account that the components of the vectors are not independent (correlated). Do such a test exist?

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A statistical test based on Hotelling T-squared statistic may be suitable. It assumes the (multivariate normal) vectors are independent from sample to sample, but has taken into account that the components within a vector sample may be correlated by the provision of a covariance matrix.

Unlike the univariate $t$-test, this test statistic is non-negative, and is more closely related to the $F$-distribution. Thus for a test with a null hypothesis that $\boldsymbol{\mu} = 0$, one will reject the null when the test statistics is large.

A note of caution that if you usually rely on CLT for your sample mean to converge to normal in the univariate case, the convergence rate for the multivariate case may be slower due to the number of dimensions in play, and the test may then provide unexpected results due to differing assumptions.

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