Advanced Studies in Pure Mathematics

The multiple roots phenomenon in maximum likelihood estimation for factor analysis

Elizabeth Gross, Sonja Petrović, Donald Richards, and Despina Stasi

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Abstract

Multiple root estimation problems in statistical inference arise in many contexts in the literature. In maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inference.

In this paper, we study the multiple roots phenomenon in maximum likelihood estimation for factor analysis. We prove that the corresponding likelihood equations have uncountably many feasible solutions even in the simplest cases. For the case in which the observed data are two-dimensional and the unobserved factor scores are one-dimensional, we prove that the solutions to the likelihood equations form a one-dimensional real curve.

Article information

Source
The 50th Anniversary of Gröbner Bases, T. Hibi, ed. (Tokyo: Mathematical Society of Japan, 2018), 109-119

Dates
Received: 1 September 2016
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1537499599

Digital Object Identifier
doi:10.2969/aspm/07710109

Mathematical Reviews number (MathSciNet)
MR3839707

Zentralblatt MATH identifier
07034250

Subjects
Primary: 62F10: Point estimation 62N02: Estimation 62H25: Factor analysis and principal components; correspondence analysis 62H12: Estimation 62H20: Measures of association (correlation, canonical correlation, etc.) 62F30: Inference under constraints

Keywords
maximum likelihood estimation factor analysis multiple roots inference small sample

Citation

Gross, Elizabeth; Petrović, Sonja; Richards, Donald; Stasi, Despina. The multiple roots phenomenon in maximum likelihood estimation for factor analysis. The 50th Anniversary of Gröbner Bases, 109--119, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07710109. https://projecteuclid.org/euclid.aspm/1537499599


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