Fisher information of scale

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 the supremum of all (∫xϕ′(x)F(dx))2/∫ϕ2(x)F(dx), where ϕ ranges over the continuously differentiable functions with derivative of compact support and where, by convention, 0/0≔0. 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 finite scale is also equivalent to L2-differentiability and local asymptotic normality, respectively, of the scale model induced by F.