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Regarding the assumption that $X$ and $Y$ need to be normally jointly distributed, when applying Pearson's correlation test...
what is a good test (function and package) in R to accomplish this task?

In this case, will low "p-values" represent that $X$ and $Y$ have Gaussian bivariate distribution, or the opposite?

CV has already a similar question, but it relies specifically on Matlab solutions.

Although this is a simple question, I think it is a good repository for CV.

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    $\begingroup$ FYI: Mardia's test is available in the psych package for R. See here. $\endgroup$
    – COOLSerdash
    Commented Jun 22, 2013 at 15:38
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    $\begingroup$ Doornik-Hansen test is quite well based. e.g. cran.r-project.org/web/packages/normwhn.test/normwhn.test.pdf (beware the typo Doornick) $\endgroup$
    – Nick Cox
    Commented Jun 22, 2013 at 15:49
  • $\begingroup$ I am game for that. $\endgroup$
    – Nick Cox
    Commented Jun 22, 2013 at 17:01
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    $\begingroup$ The assumption isn't really a necessary one. If they are normally distributed then the r is the MLE and it's maximally efficient. But non-normal distributions also can be examined using Pearson's R. Furthermore, normality tests are generally frowned upon. It's better to plot and examine the data for approximate normality (which is the requirement, not an exact test against absolute normality). $\endgroup$
    – John
    Commented Oct 24, 2013 at 1:36

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The Doornik-Hansen test is quite well based.

See e.g. http://www.cran.r-project.org/web/packages/normwhn.test/normwhn.test.pdf

Beware the typo "Doornick" in web searches.

Note that several citations are of a discussion paper from 1994 and miss the definitive later paper

Doornik, J.A. and Hansen, H. 2008. An omnibus test for univariate and multivariate normality. Oxford Bulletin of Economics and Statistics 70: 927-939.

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