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
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.