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 $$L(\theta)$$.

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.