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What are good tests to check if the data is generated by multivariate jointly gaussian distribution.

I know $\chi^2$ test is commonly used as a goodness of fit metric but I did not understand how to use it for multidimensional random vector case. Most of the examples I saw were for univariate cases.

An example would help a lot.

Note: the covariance matrix isn't diagonal. No assumption of independent variables.

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You can use the Mardia's Test.

The test has R package : https://www.rdocumentation.org/packages/MVN/versions/4.0/topics/mardiaTest (https://www.youtube.com/watch?v=pBJNXhl4cxQ&feature=youtu.be)

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