What transformations preserve the von Mises distribution?

Question

The von Mises distribution is entirely defined on the circle with a density  given by
$$f(x) = (2,pi, I_0(kappa))^{-1} exp(kappa cos(x-mu)) ,$$
where $x$ is in an arbitrary real interval of length $2pi$, $I_0$ is the Bessel-I function of order 0, $mu$ is a location parameter, and $kappa>0$ is a concentration parameter.

My question is: What transformations of a von Mises random variable preserve the distribution family?

A slightly simplified version of this question (considering an important special case) can be restated as follows: Is there a bijective function (defined on the circle) $T$ and a $kappa>0$ solving
$$(I_0(1))^{-1}, left|T'(x)right|, exp(cos(T(x))) = (I_0(kappa))^{-1} exp(kappa cdot cos(x)) ,$$

The answer is easy for $kappa=1$, because $T$ might be chosen as the identity,  a simple shift, or a suitable reflection, Thus, I am interested if this can be solved when $kappaneq 1$. Unfortunately, I am not aware whether there are any results available on this issue.

This is a crosspost from Mathoverflow.

whuber · Answer

Obviously $mu$ is a location parameter, meaning that translations of the variable preserve the family.

Focus now on the shape parameter $kappa$.  Consider any family $Omega={F_theta|thetainTheta}$ of continuous distributions.  By virtue of this continuity, whenever $Xsim F_theta$ and $0le qle 1$,

$$Pr(F_theta(X)le q) = q.$$

The transformation

$$G_{theta^prime,theta}(X) = F_{theta^prime}^{-1}(F_theta(X))$$

maps any such random variable into $Y = G_{theta^prime,theta}(X)$ and

$$Pr(Y le y) = Pr(F_{theta^prime}^{-1}(F_theta(X)) le y) = Pr(F_theta(X) le F_{theta^prime}(y))=F_{theta^prime}(y)$$

shows that $Y sim F_{theta^prime}.$

The question, then, is whether the family ${G_{theta^prime,theta}| thetainTheta, theta^primeinTheta}$ is closed under composition.  Suspecting that it should not be for the shape family of the Von Mises distribution (with $mu=0$ and $theta=kappa$), I numerically searched for a solution $(alpha,beta)$ to the equation

$$ G_{alpha,beta} = G_{2,1} circ G_{1/2,1}$$

by minimizing the $L^2$ norm between the two sides.  The difference between the best solution (with $alpha=2.96234ldots$ and $beta = 2.48773ldots$) and the right hand side is small but so clear that I doubt there was an error in the calculation.

Consequently the answer to the question--as understood in the sense described here--appears to be that only the translations (modulo $2pi$) and, of course, the reflections $xto a-x mod 2pi$ preserve the entire family of Von Mises distributions.

What transformations preserve the von Mises distribution?

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