Abstract
We consider 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 four in the model with one factor and six with two factors.
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References
Anderson, T. W., Rubin, H. (1956). Statistical inference in factor analysis. In Proceedings of the third Berkeley symposium on mathematical statistics and probability, 1954–1955 (Vol. V, pp. 111–150). Berkeley: University of California Press.
Draisma J. (2010) Finiteness for the k-factor model and chirality varieties. Advances in Mathematics 223(1): 243–256
Drton M., Sturmfels B., Sullivant S. (2007) Algebraic factor analysis: Tetrads, pentads and beyond. Probabability Theory and Related Fields 138(3–4): 463–493
Drton M., Sturmfels B., Sullivant S. (2009) Lectures on algebraic statistics. Birkhäuser, Basel
Harman H.H. (1976) Modern factor analysis (revised ed). University of Chicago Press, Chicago
Schott J.R. (2005) Testing for complete independence in high dimensions. Biometrika 92(4): 951–956
Sullivant S. (2009) A Gröbner basis for the secant ideal of the second hypersimplex. Journal of Commutative Algebra 1(2): 327–338
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Drton, M., Xiao, H. Finiteness of small factor analysis models. Ann Inst Stat Math 62, 775–783 (2010). https://doi.org/10.1007/s10463-010-0293-6
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DOI: https://doi.org/10.1007/s10463-010-0293-6