Why is Pearson’s ρ only an exhaustive measure of association if the joint distribution is multivariate normal?

This assertion was raised in the top response to this question. I think the ‘why’ question is sufficiently different that it warrants a new thread. Googling “exhaustive measure of association” did not produce any hits, and I’m not sure what that phrase means.

Answer

It might be best to understand “measure of association” in a multivariate distribution to consist of all properties that remain the same when the values are arbitrarily rescaled and recentered. Doing so can change the means and variances to any theoretically allowable values (variances must be positive; means can be anything).

The correlation coefficients (“Pearson’s ρ“) then completely determine a multivariate Normal distribution. One way to see this is to look at any formulaic definition, such as formulas for the density function or characteristic function. They involve only means, variances, and covariances–but covariances and correlations can be deduced from one another when you know the variances.

The multivariate Normal family is not the only family of distributions that enjoys this property. For example, any Multivariate t distribution (for degrees of freedom exceeding 2) has a well-defined correlation matrix and is completely determined by its first two moments, also.

Attribution
Source : Link , Question Author : user1205901 – Слава Україні , Answer Author : whuber

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