Quadratic minimisation problems in statistics
We consider the problem minx(x¡t)0A(x¡t) subject to x0Bx+2b0x = k where A is positive deflnite or positive semi- deflnite. Commonly occurring statistical variants of this problem are discussed within the framework of a general unifying method- ology. These include non-trivial considerations that arise when (i) A and/or B are not of full rank and (ii) t takes special forms (es- pecially t = 0 which, under further conditions, reduces to the well-known two-sided eigenvalue solution). Special emphasis is placed on insights provided by geometrical interpretations. Algo- rithmic considerations are discussed and examples given.