He developed the theory of the VII type Pearson distribution with the diagonal information matrix, and he proposed the method of getting of its parameters effective estimations, which is simpler, than the method proposed by Jeffreys.
32.
The family of all normal distributions, parametrized by the expected value " & mu; " and the variance " ? " 2 e " 0, with the Riemannian metric given by the Fisher information matrix, is a statistical manifold.
33.
When the Fisher information matrix is computed at the evaluates of the parameters ( " the observed Fisher information matrix " ) it is equivalent to the replacement of the true log likelihood surface by a Taylor's series approximation, taken as far as the quadratic terms.
34.
When the Fisher information matrix is computed at the evaluates of the parameters ( " the observed Fisher information matrix " ) it is equivalent to the replacement of the true log likelihood surface by a Taylor's series approximation, taken as far as the quadratic terms.
35.
The Fisher information matrix for the four parameter case does not depend on the individual values of the minimum " a " and the maximum " c ", but only on the total range ( " c " " " a " ).
36.
From Sylvester's criterion ( checking whether the diagonal elements are all positive ), it follows that the Fisher information matrix for the two parameter case is positive-definite ( under the standard condition that the shape parameters are positive ? > 0 and ? > 0 ).
37.
In the Fisher information matrix, and the curvature of the log likelihood function, the logarithm of the geometric variance of the reflected variable ( 1-" X " ) and the logarithm of the geometric covariance between " X " and ( 1-" X " ) appear:
38.
The Fisher information matrix component \ mathcal { I } _ { a, a } for the minimum " a " approaches infinity for exponent ? approaching 2 from above, and the Fisher information matrix component \ mathcal { I } _ { c, c } for the maximum " c " approaches infinity for exponent ? approaching 2 from above.
39.
The Fisher information matrix component \ mathcal { I } _ { a, a } for the minimum " a " approaches infinity for exponent ? approaching 2 from above, and the Fisher information matrix component \ mathcal { I } _ { c, c } for the maximum " c " approaches infinity for exponent ? approaching 2 from above.
40.
In general, the Fisher information matrix provides a Riemannian metric ( more precisely, the Fisher-Rao metric ) for the manifold of thermodynamic states, and can be used as an information-geometric complexity measure for a classification of phase transitions, e . g ., the scalar curvature of the thermodynamic metric tensor diverges at ( and only at ) a phase transition point.
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