Formation matrix

In statistics and information theory, the expected formation matrix of a likelihood function [math]\displaystyle{ L(\theta) }[/math] is the matrix inverse of the Fisher information matrix of [math]\displaystyle{ L(\theta) }[/math], while the observed formation matrix of [math]\displaystyle{ L(\theta) }[/math] is the inverse of the observed information matrix of [math]\displaystyle{ L(\theta) }[/math].

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 [math]\displaystyle{ j^{ij} }[/math] 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 [math]\displaystyle{ g^{ij} }[/math] following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by [math]\displaystyle{ g_{ij} }[/math] so that using Einstein notation we have [math]\displaystyle{ g_{ik}g^{kj} = \delta_i^j }[/math].

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

References

• Barndorff-Nielsen, O.E., Cox, D.R. (1989), Asymptotic Techniques for Use in Statistics, Chapman and Hall, London. ISBN:0-412-31400-2
• 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. ISBN:0-521-31871-8