Applications of quadratic minimisation problems in statistics

Albers et al. (2010) [2] 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.