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June 2019 The maximum likelihood threshold of a path diagram
Mathias Drton, Christopher Fox, Andreas Käufl, Guillaume Pouliot
Ann. Statist. 47(3): 1536-1553 (June 2019). DOI: 10.1214/18-AOS1724


Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops.


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Mathias Drton. Christopher Fox. Andreas Käufl. Guillaume Pouliot. "The maximum likelihood threshold of a path diagram." Ann. Statist. 47 (3) 1536 - 1553, June 2019.


Received: 1 February 2018; Published: June 2019
First available in Project Euclid: 13 February 2019

zbMATH: 07053517
MathSciNet: MR3911121
Digital Object Identifier: 10.1214/18-AOS1724

Primary: 62H12, 62J05

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 3 • June 2019
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