Let C be a convex set of symmetric distribution functions, H is the class of estimators of the location parameter that are linear combinations of order statistics, and $V(h,F)$ the asymptotic variance of such estimators with $F\in C$, $h\in H$. It is proved that under certain conditions the point $(h_0,F_0)$ is a saddle poind of the function $V(h,F)$, $F\in C$, $h\in H$, where $F_0$ is the distribution function for which a minimum Fisher information number is attained. and $h_0(t)=(f'_0/f_0)(F^{-1}_0(t))$.