ONE-PARAMETER GENERALISED FISHER INFORMATION MATRIX: ONE RANDOM VARIABLE

We propose a generalised Fisher information or a 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 nonuniqueness property of the Lagrangian. A generalised Cramér--Rao inequality is also derived and a Fisher information hierarchy is also obtained from the two-parameter Kullback-Leibler divergence. An interesting point is that the whole Fisher information hierarchy, except for the standard Fisher information, does not follow the additive rule. Furthermore, the idea can be directly extended to obtain the one-parameter generalised Fisher information matrix for the case of one random variable with multi-estimated parameters. The hierarchy of the Fisher information matrices is obtained. The geometrical meaning of the first two matrices in the hierarchy is studied through the normal distribution. What we find is that these first two Fisher matrices give different nature of curvature on the same statistical manifold for the normal distribution.