A.17 Circular normal distribution

X has a circular normal or von Mises’ distribution with mean μ and concentration parameter κ, denoted

if it has density

where X is any angle, so and is a constant called the modified Bessel function of the first kind and order zero (besselI(κ,0) in R). It turns out that

and that asymptotically for large κ

For large κ , we have approximately

while for small κ , we have approximately

which density is sometimes referred to as a cardioid distribution. The circular normal distribution is discussed by Mardia (1972), Mardia and Jupp (2001) and Batschelet (1981).

One point related to this distribution arises in a Bayesian context in connection with the reference prior

when we have observations such that

The only sensible estimator of μ on the basis of the posterior distribution is .

The mode of the posterior distribution of κ is approximately , where

(both of which are approximately 1.87 when ) according to Schmitt (1969, Section 10.2). Because of the skewness of the distribution of κ, its posterior mean is greater than its posterior mode.