Moments of minors of Wishart matrices
Mathias Drton, Hélène Massam, and Ingram Olkin
Source: Ann. Statist. Volume 36, Number 5 (2008), 2261-2283.
Abstract
For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order m is populated by all m×m-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many classical multivariate problems.
Primary Subjects: 60E05, 62H10
Keywords: Compound matrix; graphical models; multivariate analysis; random determinant; random matrix; tetrad
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1223908092
Digital Object Identifier: doi:10.1214/07-AOS522
References
[1] Aitken, A. C. (1956). Determinants and Matrices, 9th ed. Oliver and Boyd, Edinburgh.
[2] Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
Mathematical Reviews (MathSciNet):
MR1990662
Zentralblatt MATH:
1039.62044
[3] Bollen, K. A. and Ting, K. (1993). Confirmatory tetrad analysis. In Sociological Methodology (P. M. Marsden, ed.) 147–75. American Sociological Association, Washington, DC.
[4] Casalis, M. and Letac, G. (1996). The Lukacs–Olkin–Rubin characterization of Wishart distributions on symmetric cones. Ann. Statist. 24 763–786.
Mathematical Reviews (MathSciNet):
MR1394987
Digital Object Identifier: doi:10.1214/aos/1032894464
Project Euclid: euclid.aos/1032894464
[5] Cox, D., Little, J. and O’Shea, D. (1997). Ideals, Varieties, and Algorithms, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1417938
[6] Drton, M., Sturmfels, B. and Sullivant, S. (2007). Algebraic factor analysis: Tetrads, pentads and beyond. Probab. Theory Related Fields 138 463–493.
Mathematical Reviews (MathSciNet):
MR2299716
Digital Object Identifier: doi:10.1007/s00440-006-0033-2
[7] Eaton, M. L. (1983). Multivariate Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR716321
Zentralblatt MATH:
0587.62097
[8] Hipp, J. R., Bauer, D. J. and Bollen, K. A. (2005). Conducting tetrad tests of model fit and contrasts of tetrad-nested models: A new SAS macro. Struct. Equ. Model. 12 76–93.
Mathematical Reviews (MathSciNet):
MR2086530
Digital Object Identifier: doi:10.1207/s15328007sem1201_4
[9] Holzinger, K. J. and Harman, H. H. (1941). Factor Analysis. A Synthesis of Factorial Methods. Univ. Chicago Press.
Mathematical Reviews (MathSciNet):
MR6637
[10] Lauritzen, S. L. (1996). Graphical Models. Oxford Univ. Press, New York.
Mathematical Reviews (MathSciNet):
MR1419991
[11] Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
Mathematical Reviews (MathSciNet):
MR552278
Zentralblatt MATH:
0437.26007
[12] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR652932
Zentralblatt MATH:
0556.62028
[13] Olkin, I. and Rubin, H. (1962). A characterization of the Wishart distribution. Ann. Math. Statist. 33 1272–1280.
Mathematical Reviews (MathSciNet):
MR141186
Digital Object Identifier: doi:10.1214/aoms/1177704359
Project Euclid: euclid.aoms/1177704359
[14] Spearman, C. (1927). The Abilities of Man, Their Nature and Measurement. Macmillan & Co., London.
[15] Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. MIT Press, Cambridge, MA.
Mathematical Reviews (MathSciNet):
MR1815675
[16] Wishart, J. (1928). Sampling errors in the theory of two factors. British J. Psychology 19 180–187.
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