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Tail-Measurability in Monotone Latent Variable Models

Published online by Cambridge University Press:  01 January 2025

Jules L. Ellis  and
Brian W. Junker
Jules L. Ellis*
Affiliation:
University of Nijmegen
Brian W. Junker
Affiliation:
Carnegie Mellon University
*
Requests for reprints should be send to Jules L. Ellis, Department of Mathematical Psychology, University of Nijmegen, P.O. Box 9104, 6500 HE Nijmegen, THE NETHERLANDS.
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Abstract

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, and tail-measurability. Tail-measurability 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 models in 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 “true-score” 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.

Keywords

unidimensionalitylatent variable modelstail-measurableconditional independenceconditional associationmonotonicityvanishing conditional dependenceconsistencytrue-score variablesubpopulationsubtail-measurable

Information

Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 4 , December 1997 , pp. 495 - 523
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

The authors made equivalent contributions to the results of this article. Ellis' research was supported by the Dutch Interuniversitary Graduate School of Psychometrics and Sociometrics. Junker's research was supported by ONR Grant N00014-87-K-0277, NIMH Grant MH15758, and a Carnegie Mellon University Faculty Development Grant. In addition Junker would like to acknowledge the hospitality of the Nijmegen Institute for Cognition and Information during his visit to the University of Nijmegen in August 5–10, 1993.

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