Finiteness of small factor analysis models

TL;DR

This paper proves a finiteness property for small factor analysis models, showing that for certain sizes, membership in the model can be determined by examining smaller principal submatrices.

Contribution

It establishes a finite characterization of covariance matrices in small factor analysis models with one or two factors, identifying specific matrix sizes for this property.

Findings

For one-factor models, the distinguished size is four.

For two-factor models, the distinguished size is six.

Membership can be checked via principal submatrices of these sizes.

Abstract

We consider small factor analysis models with one or two factors. Fixing the number of factors, we prove a finiteness result about the covariance matrix parameter space when the size of the covariance matrix increases. According to this result, there exists a distinguished matrix size starting at which one can determine whether a given covariance matrix belongs to the parameter space by determining whether all principal submatrices of the distinguished size belong to the corresponding parameter space. We show that the distinguished matrix size is equal to four in the one-factor model and six with two factors.