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Weighted Minimum Trace Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Alexander Shapiro
Alexander Shapiro*
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev
*
Requests for reprints should be sent to: A. Shapiro, Department of Statistics & O.R., University of South Africa, P. O. Box 392, PRETORIA 0001, South Africa.
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Abstract

In the last decade several authors discussed the so-called minimum trace factor analysis (MTFA), which provides the greatest lower bound (g.l.b.) to reliability. However, the MTFA fails to be scale free. In this paper we propose to solve the scale problem by maximization of the g.l.b. as the function of weights. Closely related to the primal problem of the g.l.b. maximization is the dual problem. We investigate the primal and dual problems utilizing convex analysis techniques. The asymptotic distribution of the maximal g.l.b. is obtained provided the population covariance matrix satisfies sone uniqueness and regularity assumptions. Finally we outline computational algorithms and consider numerical examples.

Keywords

reliabilityreduced ranksample estimates
Type
Original Paper
Information
Psychometrika , Volume 47 , Issue 3 , September 1982 , pp. 243 - 264
Copyright
Copyright © 1982 The Psychometric Society

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Footnotes

I wish to express my gratitude to Dr. A. Melkman for the idea of theorem 3.3.

References

Reference Notes

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Della Riccia, G. & Shapiro, A. Minimum rank and minimum trace of covariance matrices. Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel, 1980.Google Scholar
Moss, A. G. Theoretical aspects of weight-vector maximization of composite test reliability, 1977, England: Institute of Educational Technology, The Open University, Walton Hall.Google Scholar

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