One-parameter generalised Fisher information matrix: One random variable

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.