Formation matrix

{{technical|date=May 2014}}

In statistics and information theory, the expected formation matrix of a likelihood function $L(\backslash theta)$ is the matrix inverse of the Fisher information matrix of $L(\backslash theta)$, while the observed formation matrix of $L(\backslash theta)$ is the inverse of the observed information matrix of $L(\backslash theta)$.Edwards (1984) p104

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 $g\_\{ik\}g^\{kj\}\; =\; \backslash delta\_i^j$.

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