Formation matrix

In <a href="/facts/Statistics/DNPsKGYU">statistics</a> and <a href="/facts/Information_theory/qNLH3U5P">information theory</a>, the expected formation matrix of a <a href="/facts/Likelihood_function/1L8P4NBX">likelihood function</a> 
 
 
 
 L
 (
 θ
 )
 
 
 {\displaystyle L(\theta )}
 
 is the matrix inverse of the <a href="/facts/Fisher_information_matrix/Q2JLexN9">Fisher information matrix</a> of 
 
 
 
 L
 (
 θ
 )
 
 
 {\displaystyle L(\theta )}
 
, while the observed formation matrix of 
 
 
 
 L
 (
 θ
 )
 
 
 {\displaystyle L(\theta )}
 
 is the inverse of the <a href="/facts/Observed_information_matrix/S2gDCn8V">observed information matrix</a> of 
 
 
 
 L
 (
 θ
 )
 
 
 {\displaystyle L(\theta )}
 
.
Currently, no notation for dealing with formation matrices is widely used, but in books and articles by <a href="/facts/Ole_E._Barndorff-Nielsen/1Xr71bYs">Ole E. Barndorff-Nielsen</a> and <a href="/facts/Peter_McCullagh/GWv03UXR">Peter McCullagh</a>, the symbol 
 
 
 
 
 j
 
 i
 j
 
 
 
 
 {\displaystyle j^{ij}}
 
 is used to denote the element of the i-th line and j-th column of the observed formation matrix. The <a href="/facts/Information_Geometry/sFBShVbT">geometric interpretation</a> of the Fisher information matrix (metric) leads to a notation of 
 
 
 
 
 g
 
 i
 j
 
 
 
 
 {\displaystyle g^{ij}}
 
 following the notation of the (<a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">contravariant</a>) metric tensor in <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>. The Fisher information metric is denoted by 
 
 
 
 
 g
 
 i
 j
 
 
 
 
 {\displaystyle g_{ij}}
 
 so that using <a href="/facts/Einstein_notation/UXwS43wU">Einstein notation</a> we have 
 
 
 
 
 g
 
 i
 k
 
 
 
 g
 
 k
 j
 
 
 =
 
 δ
 
 i
 
 
 j
 
 
 
 
 {\displaystyle g_{ik}g^{kj}=\delta _{i}^{j}}
 
.
These matrices appear naturally in the <a href="/facts/Asymptotic_expansion/4W79BNve">asymptotic expansion</a> of the distribution of many statistics related to the <a href="/facts/Likelihood-ratio_test/oFqLh3R0">likelihood ratio</a>.

Formation matrix open-in-new

Formation matrix