Formation Matrix

In statistics and information theory, the expected formation matrix of a likelihood function $L(\theta)$ is the matrix inverse of the Fisher information matrix of $L(\theta)$, while the observed formation matrix of $L(\theta)$ is the inverse of the observed information matrix of $L(\theta)$.Edwards (1984) p104

Currently, no notation for dealing with formation matrices is widely used, but in books and articles by Ole E. Barndorff-Nielsen and Peter McCullagh, the symbol $j^$ is used to denote the element of the i-th line and j-th column of the observed formation matrix. The geometric interpretation of the Fisher information matrix (metric) leads to a notation of $g^$ following the notation of the ( contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by $g_$ so that using Einstein notation we have $g_g^ = \delta_i^j$.

These matrices appear naturally in the asymptotic expansion of the distribution of many statistics related to the likelihood ratio.

See also

* Fisher information
*Shannon entropy

Notes

References

*Barndorff-Nielsen, O.E., Cox, D.R. (1989), Asymptotic Techniques for Use in Statistics, Chapman and Hall, London.
*Barndorff-Nielsen, O.E., Cox, D.R., (1994). Inference and Asymptotics. Chapman & Hall, London.
*P. McCullagh, "Tensor Methods in Statistics", Monographs on Statistics and Applied Probability, Chapman and Hall, 1987.
*Edwards, A.W.F. (1984) ''Likelihood''. CUP.

Estimation theory
Information theory

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