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.

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