Applications of quadratic minimisation problems in statistics

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.