Von Mises Distribution - Distribution of The Mean

Distribution of The Mean

The distribution of the sample mean for the von Mises distribution is given by:

P(\bar{R},\bar{\theta})\,d\bar{R}\,d\bar{\theta}=\frac{1}{ (2\pi I_0(k))^N}\int_\Gamma \prod_{n=1}^N \left( e^{\kappa\cos(\theta_n-\mu)} d\theta_n\right) = \frac{e^{\kappa N\bar{R}\cos(\bar{\theta}-\mu)}}{I_0(\kappa)^N}\left(\frac{1}{(2\pi)^N}\int_\Gamma \prod_{n=1}^N d\theta_n\right)

where N is the number of measurements and consists of intervals of in the variables, subject to the constraint that and are constant, where is the mean resultant:

\bar{R}^2=|\bar{z}|^2= \left(\frac{1}{N}\sum_{n=1}^N \cos(\theta_n) \right)^2 + \left(\frac{1}{N}\sum_{n=1}^N \sin(\theta_n) \right)^2

and is the mean angle:

\overline{\theta}=\mathrm{Arg}(\overline{z}). \,

Note that product term in parentheses is just the distribution of the mean for a circular uniform distribution.

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