Research Letters in the Information and Mathematical Sciences

Permanent URI for this collectionhttps://mro.massey.ac.nz/handle/10179/4332

Research Letters welcomes papers from staff and graduate students at Massey University in the areas of: Computer Science, Information Science, Mathematics, Statistics and the Physical and Engineering Sciences. Research letters is a preprint series that accepts articles of completed research work, technical reports, or preliminary results from ongoing research. After editing, articles are published online and can be referenced, or handed out at conferences. Copyright remains with the authors and the articles can be used as preprints to academic journal publications or handed out at conferences. Editors Dr Elena Calude Dr Napoleon Reyes The guidelines for writing a manuscript can be accessed here.

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    On a matrix with integer eigenvalues and its relation to conditional Poisson sampling
    (Massey University, 2005) Bondesson, L.; Traat, I.
    A special non-symmetric N × N matrix with eigenvalues 0, 1, 2, . . . ,N − 1 is presented. The matrix appears in sampling theory. Its right eigenvectors, if properly normalized, give the inclusion probabilities of the Conditional Poisson design (for all different fixed sample sizes). The explicit expressions for the right eigenvectors become complicated for N large. Nevertheless, the left eigenvectors have a simple analytic form. An inversion of the left eigenvector matrix produces the right eigenvectors − the inclusion probabilities. Finally, a more general matrix with similar properties is defined and expressions for its left and right eigenvectors are derived.
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    Eigenvectors of block circulant and alternating circulant matrices
    (Massey University, 2005) Tee, Garry J.
    The eigenvectors and eigenvalues of block circulant matrices had been found for real symmetric matrices with symmetric submatrices, and for block circulant matrices with circulant submatrices. The eigenvectors are now found for general block circulant matrices, including the Jordan Canonical Form for defective eigenvectors. That analysis is applied to Stephen J. Watson’s alternating circulant matrices, which reduce to block circulant matrices with square submatrices of order 2.
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    Appoximation-assisted [sic] estimation of eigenvectors under quadratic loss
    (Massey University, 2005) Ahmed, S.E.
    Improved estimation of eigen vector of covariance matrix is considered under uncertain prior information (UPI) regarding the parameter vector. Like statistical models underlying the statistical inferences to be made, the prior information will be susceptible to uncertainty and the practitioners may be reluctant to impose the additional information regarding parameters in the estimation process. A very large gain in precision may be achieved by judiciously exploiting the information about the parameters which in practice will be available in any realistic problem. Several estimators based on preliminary test and the Stein-type shrinkage rules are constructed. The expressions for the bias and risk of the proposed estimators are derived and compared with the usual estimators. We demonstrate that how the classical large sample theory of the conventional estimator can be extended to shrinkage and preliminary test estimators for the eigenvector of a covariance matrix. It is established that shrinkage estimators are asymptotically superior to the usual sample estimators. For illustration purposes, the method is applied to three datasets.
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    Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics
    (Massey University, 2005) Rao, Radhakrishna
    Let A be p × p positive definite matrix. A p-vector x such that Ax = x is called an eigenvector with the associated with eigenvalue . Equivalent characterizations are: (i) cos = 1, where is the angle between x and Ax. (ii) (x0Ax)−1 = xA−1x. (iii) cos = 1, where is the angle between A1/2x and A−1/2x. We ask the question what is x such that cos as defined in (i) is a minimum or the angle of separation between x and Ax is a maximum. Such a vector is called an anti-eigenvector and cos an anti-eigenvalue of A. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as min[(x0Ax)(x0A−1x)]−1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle between A1/2x and A−1/2x as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an r(< p) dimensional subspace S and its transform AS. Similar problems are considered for a general matrix A and its singular values leading to anti-singular values. Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented.