I'm working with the Multimodal Generalized von Mises (MGvM) distribution, and I am interested in finding the variance of this distribution. The density function is given by:

$$ f(\theta) = G_M(\mu_1, \ldots, \mu_S, \kappa_1, \ldots, \kappa_S) \exp\left(\sum_{s=1}^{S} \kappa_s \cos(s(\theta - \mu_s))\right), \quad -\pi \leq \theta < \pi, $$

where ( $\mu_1, \mu_2, \ldots, \mu_S$ ) are the locations of the modes, ( $\kappa_1, \kappa_2, \ldots, \kappa_S$ ) are the concentration parameters, and ( $G_M$ ) is the normalizing constant. The normalizing constant involves an infinite series of modified Bessel functions as indicated in the expansions:

$$ \exp(\kappa \cos(\theta)) = I_0(\kappa) + 2 \sum_{j=1}^{\infty} I_j(\kappa) \cos(j\theta), $$

$$ \exp(\kappa \sin(\theta)) = I_0(\kappa) + 2 \sum_{j=0}^{\infty} (-1)^j I_{2j+1}(\kappa) \sin((2j + 1)\theta) + 2 \sum_{j=1}^{\infty} (-1)^j I_{2j}(\kappa) \cos(2j\theta). $$

The distribution is symmetric about each ( $\mu_s$ ) and becomes bimodal in the case where all ( $\mu_s $ ) are equal. The modes and antimodes of the distribution are solutions to the equation ( $\sum_{s=1}^{S} \kappa_s \sin(s(\theta - \mu_s)) = 0$ ).

I am not sure how to approach the calculation of the variance for this distribution, especially due to the presence of multiple modes and the involvement of Bessel functions. I am looking for either an analytical approach or a numerical method that could be used to estimate the variance of the MGvM distribution.

Can someone provide guidance on how to calculate the variance of this distribution or point me toward relevant literature?

Any help or insight would be greatly appreciated!

The image depicting the problem: enter image description here



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