Formation matrix explained

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) .^[1]

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 ^{ij} 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 ^{ij} following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by g _{ij} so that using Einstein notation we have g _{ik} g ^{kj} = \delta ^{j} _{i} .

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.
• 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.

Notes and References

1. Edwards (1984) p104