Stability of a characterization of the von Mises distribution (Q916228)

Let \(\vec x=(x_ 1,x_ 2,...,x_ n)\) be a repeated sample drawn from the population \(P_ g(A)=P(A-g)\), \(g\in T=[-\pi,\pi]\), where P(A) is the distribution on the unit circle T. \textit{A. L. Ruhin} [Some statistical and probabilistic problems on groups. Proc. Steklov Inst. Math. 111 (1970), 59-129 (1972); translation from Trudy Mat. Inst. Steklov 111, 52- 109 (1970; Zbl 0283.62007)] proved that the property of the location parameter estimator \[ d(\vec x)=\arg (\sum^{n}_{j=1}e^{ix_ j}) \] to be an optimal equivariant estimator in the sense of the loss function \[ r(d(\vec x),\theta)=| e^{id(\vec x)}- e^{i\theta}|^ 2 \] characterizes the von Mises distribution with density \[ P(x)=D(\gamma)\exp \{2\gamma^{-1} \cos x\},\quad x\in T, \] where \(\gamma >0\) and \(D(\gamma)>0\) are constant. The stability of this phenomenon is investigated. The formulation of the main result requires a long list of definitions, so we omit it. Note that the main result is based on the following estimator \[ \sup_{x\in T}| F(x)-G(x)| \leq 2\pi^{-1}\sum^{k}_{j=-k}| h_ j/j| +3\pi k^{- 1}\sup_{x\in T}| G'(x)| \] for the difference between the distribution function F(x) and the differentiable distribution function G(x), \(x\in T\), where k is any integer and \(h_ j\) are the Fourier coefficients of \(H(x)=F(x)-G(x)\).