  We propose the generalised Fisher information or the one-parameter extended
class of the Fisher information for the case of one random variable. This new
form of the Fisher information is obtained from the intriguing connection
between the standard Fisher information and the variational principle together
with the non-uniqueness property of the Lagrangian. The generalised
Cram\'er-Rao inequality is also derived. The interesting point is about the
fact that the whole Fisher information hierarchy, except for the standard
Fisher information, does not follow the additive rule. The whole Fisher
information hierarchy is also obtained from the two-parameter Kullback-Leibler
divergence. Furthermore, the idea can be directly extended to obtain the
one-parameter generalised Fisher information matrix for the case of one random
variable, but with multi-estimated parameters. The hierarchy of the Fisher
information matrix is obtained. The geometrical meaning of the first two
matrices in the hierarchy is studied through the normal distribution. An
interesting point is that these first two Fisher matrices give different nature
of curvature on the same statistical manifold of the normal distribution.
