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Probability Spaces, Hilbert Spaces, and the Axioms of Test Theory

Published online by Cambridge University Press:  01 January 2025

Donald W. Zimmerman
Donald W. Zimmerman*
Affiliation:
Carleton University
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Abstract

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.

Type
Original Paper
Information
Psychometrika , Volume 40 , Issue 3 , September 1975 , pp. 395 - 412
Copyright
Copyright © 1975 The Psychometric Society

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References

Bachman, G., Narici, L.. Functional analysis, 1966, New York: Academic Press.Google Scholar
Burrill, C. W.. Measure, integration, and probability, 1972, New York: McGraw-Hill.Google Scholar
Feller, W.. An introduction to probability theory and its applications (Vol. II), 1966, New York: Wiley.Google Scholar
Gulliksen, H.. Theory of mental tests, 1950, New York: Wiley.CrossRefGoogle Scholar
Guttman, L.. A basis for analyzing test-retest reliability. Psychometrika, 1945, 10, 255–282.CrossRefGoogle ScholarPubMed
Halmos, P.. Introduction to Hilbert space, 1951, New York: Chelsea.Google Scholar
Loève, M.. Probability theory, 1963, Princeton, N. J.: Van Nostrand.Google Scholar
Lord, F. M., Novick, M.. Statistical theories of mental test scores, 1968, Reading, Mass.: Addison-Wesley.Google Scholar
Rényi, A.. Foundations of probability, 1970, San Francisco: Holden-Day.Google Scholar