## Fisher Information in Single Factor Analysis

The single factor analysis model for data $\{ {\bf x}_1, {\bf x}_2, \ldots, {\bf x}_n \}$ is

${\bf x}_i = {\bf m} + v_i {\bf a} + \boldsymbol{\epsilon}_i \quad \; (i=1,2,\ldots,n),$

where ${\bf x}_i \in \mathbb{R}^k$, $\boldsymbol{\epsilon}_i$ is an i.i.d. zero-mean multivariate Gaussian random variable with a symmetric, positive definite variance-covariance matrix $\boldsymbol{\Sigma} \in \mathbb{R}^{k \times k}$; that is, $\boldsymbol{\epsilon}_i \sim {\rm N}_k({\bf 0}, \boldsymbol{\Sigma})$ for all $i=1,2,\ldots,n$. It is thus assumed that each data vector ${\bf x}_i$ follows a $k$-variate Gaussian distribution with moments

${\rm E} \{ {\bf x}_i \} = {\bf m} + v_i {\bf a},$

${\rm Var} \{ {\bf x} \} = \boldsymbol{\Sigma}.$

The aim of single factor analysis is to estimate the unknown factor scores ${\bf v} = (v_1, v_2, \ldots, v_n)^\prime \in \mathbb{R}^n$, the mean ${\bf m} = (m_1,m_2, \ldots, m_k)^\prime \in \mathbb{R}^k$, the variance-covariance matrix $\boldsymbol{\Sigma}$ and the factor loadings ${\bf a} = (a_1, a_2, \ldots, a_k)^\prime \in \mathbb{R}^k$.

I’ve written this note to show how one may derive the Fisher information for the single factor model. This turns out to be a fairly involved task requiring a lot of linear algebra and matrix differential calculus identities. I’ve uploaded a document describing all the necessary mathematical steps to the Publications section. Anyone interested in matrix differential calculus and linear algebra should find it useful.