Goodness of fit test for bivariate circular distributions I'm trying to implement a goodness-of-fit test for a bivariate distribution on the torus. In the univariate case, i use the distribution function to transform the data so it presents a uniform distribution, and from there i can use Rayleigh's test to obtain a significance level in correspondence with a $\chi^2_2$ distribution. My efforts were initially to apply a similar solution for the $(\theta_1,\theta_2)$ case, but it seems not a trivial problem. Literature did not provide a definite answer so far, so my question is. 
How do you perform a goodness-of-fit test for bivariate angular distributions on the torus?
Is there a general methodology or should i specify the family of the distribution?
Thanks!
 A: The most general approach for the univariate case is based on bootstrapping:


*

*Find the cumulative distribution function for the model $f(\theta \vert \boldsymbol\psi)$ under scrutiny, say $F(\theta \vert \boldsymbol\psi)$, where $\theta$ is an angle or direction, and $\boldsymbol\psi$ are some parameters of the model, which are usually estimated. For many circular models, computing $F(\theta \vert \boldsymbol\psi)$ involves numerical integration. 

*For your sample $\boldsymbol\theta = \theta_1, \dots, \theta_n$, compute the values $2 \pi F(\theta_1), \dots, 2 \pi F(\theta_n)$.

*Choose a test statistic for circular uniformity, for example Kuiper's, Watson's, Rayleigh, or Rao Spacing test. Calculate the value for this test statistic for the set of values $2 \pi F(\theta_1), \dots, 2 \pi F(\theta_n)$. 

*Generate a bootstrap sample of test statistics for uniform circular data sets with the same sample size as the original data.

*Obtain the proportion of values in the bootstrap sample which exceed the test statistic in for the set $2 \pi F(\theta_1), \dots, 2 \pi F(\theta_n)$, which will be the p-value.


For von Mises, Jones-Pewsey, and inverse Batschelet distributions, this is implemented in the workspace belonging to 'Circular Statistics in R', by Pewsey, Neuhauser & Ruxton (2013), and can be obtained by 
load(url("http://circstatinr.st-andrews.ac.uk/resources/CircStatsInR.RData")) 

This is general in the sense that the bootstrapping can be applied for each circular data model for which we can find a CDF.
However, the previous applies only to the univariate case. The bivariate case would technically require a test statistic for bivariate isotropy (circular uniformity). However, if $\Theta_1$ and $\Theta_2$ are independent, it would seem for some univariate test statistic $t$, we could calculate $t_1$ and $t_2$ for $\Theta_1$ and $\Theta_2$ respectively, and use $t_b = t_1 + t_2$ as the test statistic in the bootstrapping procedure. Precise properties of such a method have not been discussed in the literature.
For dependent samples $\Theta_1$ and $\Theta_2$, I am not aware of any test statistics in the literature.
