Applications of quadratic minimisation problems in statistics

Albers, C. J.; Critchley, F. and Gower, J. C. (2011). Applications of quadratic minimisation problems in statistics. Journal of Multivariate Analysis, 102(3) pp. 714–722.

DOI: https://doi.org/10.1016/j.jmva.2010.11.009

Abstract

Albers et al. (2010) showed that the problem minx(x-t)'A(x-t) subject to x'Bx+2b'x=k where A is positive definite or positive semi-definite has a unique computable solution. Here, several statistical applications of this problem are shown to generate special cases of the general problem that may all be handled within a general unifying methodology. These include non-trivial considerations that arise when (i) A and/or B are not of full rank and (ii) where B is indefinite. General canonical forms for A and B that underpin the minimisation methodology give insight into structure that informs understanding.

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