The {2}-inverse with applications in statistics |
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Authors: | Francis Hsuan Patricia Langenberg Albert Getson |
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Institution: | Department of Statistics Temple University Philadelphia, Pennsylvania 19122 USA;Epidemiology and Biometry Program The University of Illinois at Chicago Chicago, Illinois USA |
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Abstract: | An inverse G of a given matrix A which satisfies the property GAG = G is known as a {2}-inverse. This paper presents a three-phase inversion procedure for which the {2}-inverse is a special case. We present the geometry of {2}-inverses and show that, starting from {2}-inverses, various types of generalized inverses can be derived. Two examples of the occurrence of {2}-inverses in statistics are given: one concerning the constrained least-squares estimator, the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi-square distribution. |
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