Applications of quadratic minimisation problems in statistics

• Published in: Journal of multivariate analysis Vol. 102; no. 3; pp. 714 - 722
• Main Authors: Albers, C.J., Critchley, F., Gower, J.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.03.2011; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Algebra; Canonical analysis; Canonical analysis Constraints Constrained regression Hardy-Weinberg Minimisation Optimal scaling Procrustes analysis Quadratic forms Ratios Reduced rank Splines; Constrained regression; Constraints; Exact sciences and technology; Hardy–Weinberg; Linear inference, regression; Mathematical problems; Mathematics; Minimisation; Multivariate analysis; Number theory; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Optimal scaling; Optimization; Probability and statistics; Procrustes analysis; Quadratic forms; Quadratic programming; Ratios; Reduced rank; Sciences and techniques of general use; Splines; Statistical methods; Statistics; Studies; Minimisation; 62P99; Splines; Reduced rank; Optimal scaling; Canonical analysis; Constrained regression; Ratios; Constraints; Quadratic forms; 12A63; 15A21; Hardy–Weinberg; Procrustes analysis; Solution uniqueness; Multivariate analysis; Spline approximation; Canonical form; Numerical approximation; Hardy-Weinberg; Quadratic form; Numerical linear algebra; Statistical method; Statistical regression; Numerical analysis; Condition number
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2010.11.009

Albers et al. (2010) [2] showed that the problem min x ( x − t ) ′ A ( x − t ) subject to x ′ B x + 2 b ′ 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.