Skewness and kurtosis of VonMises distribution

Question

I could not find a reference that has mentioned the skewness and kurtosis of a circular data that has Von Mises distribution. What are the values? and it is great if someone can introduce some references in this regard.

Answer 1

The distribution is symmetric, so its skewness is zero. I'll give you a partial answer for how to get the kurtosis. One reasonable way to obtain the kurtosis is from the cumulant generating function, which is:

$$K(t) = t \mu + \log I_{|t|} (\kappa) - \log I_0 (\kappa),$$

where $I_t$ is the modified Bessel function. Now, if you can find a power series expansion for the logarithm of the modified Bessel function (which should be around in some book on special functions somewhere) then you will be able to write the cumulant generating function as a power series, and get the kurtosis from that. If you can't find a power series, but can find a different type of series expansion, then you might still be able to get the kurtosis via differentiation of the cumulant generating function. Specifically, the kurtosis of the distribution is:

$$\mathbb{Kurt} = \frac{d^4 K}{dt^4}(t) + 3 \Bigg( \frac{d^2 K}{dt^2}(t) \Bigg)^2 = \frac{d^4}{dt^4} \log I_{|t|} (\kappa) + 3 \Bigg( \frac{d^2}{dt^2} \log I_{|t|} (\kappa) \Bigg)^2.$$

Now, since the Von Mises distribution converges to the normal distribution for $\kappa \rightarrow \infty$ the distribution is asymptotically mesokurtotic. Thus, as a check on your answer you should ensure that the kurtosis formula converges to three in this limit.