Formation matrix

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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).[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_i^j.

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

See also[edit]


  1. ^ Edwards (1984) p104


  • 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