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Assume I want to generate a sample of n-dimensional vector $z$ with a given correlation matrix $Corr(z)=R$. Some of the margins of its distribution are not normal (if you're interested here's the motivation behind my question), i.e. $z$ doesn't follow the multivariate normal distribution.
Let's assume that I have some imperfect method and I want to check if the sample correlation matrix is close to the theoretical one, i.e. I'd like to test the null hypothesis $H_0:R_{sample}=R$ vs $H_1:R_{sample}\neq R$
How can I do it?

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  • $\begingroup$ See this answer - the method of Ruscio and Kaczetow (2008) can help you creating the data matrix stats.stackexchange.com/a/43230/6082 $\endgroup$ – Felix S Jan 14 '13 at 8:51
  • $\begingroup$ I don't think so. But the code can be found here. $\endgroup$ – Felix S Jan 15 '13 at 6:13
  • $\begingroup$ Thanks for the code. However, I still would like to test the hypothesis if the sample generated with this code (or with any other) has the given correlation matrix R (i.e. H_0:R_sample=R, H_1: not H_0). What do you think how can this be achieved? $\endgroup$ – Max Li Jan 16 '13 at 15:46
  • $\begingroup$ I am a civil engineer specialized in probabilistis geotechnics. I not know if it is of any interest, but I have developed an algorith for the random drawing of not-normal multivariate statistical distribution functions. I keep it now secret, but when I think that the time is there I shall publish it. Thank you for your interest, Henk Bakker, Netherlands. $\endgroup$ – user130118 Sep 5 '16 at 19:53
  • $\begingroup$ It sounds like you might be talking about using a Gaussian copula model? I am not very knowledgeable in this area, but e.g. a Google Scholar search turns up this reference, which may be useful. Copulas are very common in finance, so you may also check on the Quant site. $\endgroup$ – GeoMatt22 Sep 5 '16 at 20:38
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Just some ideas. From your linked background post, your situation has normal and gamma marginals. You could try to fit some copula, maybe a gaussian copula. If you then also fit the margins, you can simulate from this combined model. You could try some different copulas and comparing them via AIC. But you should be aware that the correlation matrix estimate you get from the gaussian copula, is not necessarily a good estimator of the correlation matrix you have estimated from the untransformed data.

For the question about testing, testing correlation matrices seems to be a complicated problem. Here are some relevant papers:

Tests for correlation matrices,
Analysis of Correlation Matrices Using Covariance Structure Models,
Visualizing Tests for Equality of Covariance Matrices.

Distribution theory can be complicated. Maybe combine with bootstrapping?

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