# What is a generalisation of t-test to the case of multivariate distribution?

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?

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.