Formation matrix

In statistics and information theory, the expected formation matrix of a likelihood function ${\displaystyle L(\theta )}$ is the matrix inverse of the Fisher information matrix of ${\displaystyle L(\theta )}$, while the observed formation matrix of ${\displaystyle L(\theta )}$ is the inverse of the observed information matrix of ${\displaystyle L(\theta )}$.[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 ${\displaystyle j^{ij}}$ 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 ${\displaystyle g^{ij}}$ following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by ${\displaystyle g_{ij}}$ so that using Einstein notation we have ${\displaystyle g_{ik}g^{kj}=\delta _{i}^{j}}$.

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