Tailmeasurability in monotone latent variable models
 Jules L. Ellis,
 Brian W. Junker
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We consider latent variable models for an infinite sequence (or universe) of manifest (observable) variables that may be discrete, continuous or some combination of these. The main theorem is a general characterization by empirical conditions of when it is possible to construct latent variable models that satisfy unidimensionality, monotonicity, conditional independence, andtailmeasurability. Tailmeasurability means that the latent variable can be estimated consistently from the sequence of manifest variables even though an arbitrary finite subsequence has been removed. The characterizing,necessary and sufficient, conditions that the manifest variables must satisfy for these models are conditional association and vanishing conditional dependence (as one conditions upon successively more other manifest variables). Our main theorem considerably generalizes and sharpens earlier results of Ellis and van den Wollenberg (1993), Holland and Rosenbaum (1986), and Junker (1993). It is also related to the work of Stout (1990).
The main theorem is preceded by many results for latent variable modelsin general—not necessarily unidimensional and monotone. They pertain to the uniqueness of latent variables and are connected with the conditional independence theorem of Suppes and Zanotti (1981). We discuss new definitions of the concepts of “truescore” and “subpopulation,” which generalize these notions from the “stochastic subject,” “random sampling,” and “domain sampling” formulations of latent variable models (e.g., Holland, 1990; Lord & Novick, 1968). These definitions do not require the a priori specification of a latent variable model.
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 Title
 Tailmeasurability in monotone latent variable models
 Journal

Psychometrika
Volume 62, Issue 4 , pp 495523
 Cover Date
 19971201
 DOI
 10.1007/BF02294640
 Print ISSN
 00333123
 Online ISSN
 18600980
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 unidimensionality
 latent variable models
 tailmeasurable
 conditional independence
 conditional association
 monotonicity
 vanishing conditional dependence
 consistency
 truescore variable
 subpopulation
 subtailmeasurable
 Industry Sectors
 Authors

 Jules L. Ellis ^{(1)}
 Brian W. Junker ^{(2)}
 Author Affiliations

 1. Department of Mathematical Psychology, University of Nijmegen, P.O. Box 9104, 6500 HE, Nijmegen, The Netherlands
 2. Carnegie Mellon University, USA