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I have n observations $X_1, ..., X_n$ each of which has d dimensions $X_i=(X_{i1},...,X_{id})$, the dimensions are independent from another, and the values are continuous.

I want to know if my observations follow a certain distribution (uniform or gaussian). What is a general way to do this?

Can I just create a d-dimensional histogram of my data and then use the usual chi-squared test $\sum_i\frac{(E_i - O_i)^2}{E_i}$? And if so, does the multi-dimensionality change my degrees of freedom?

mexx
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    Yes, you can proceed in this fashion. But there are many subtle considerations concerning how to estimate the distribution parameters and bin the data; and they come to the fore with multivariate data. See a general account at https://stats.stackexchange.com/a/17148/919. Please pay particular attention to the two points following the clause "Things went wrong because ..." – whuber Feb 27 '20 at 13:56

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