  Motivated by the information bound for the asymptotic variance of M-estimates
for scale, we define Fisher information of scale of any distribution function F
on the real line as a suitable supremum. In addition, we enforce equivariance
by a scale factor.
  Fisher information of scale is weakly lower semicontinuous and convex. It is
finite iff the usual assumptions on densities hold, under which Fisher
information of scale is classically defined, and then both classical and our
notions agree.
  Fisher information of scale finite is also equivalent to
L_2-differentiability and local asymptotic normality, respectively, of the
scale model induced by F.
