D-optimal design methods for robust estimation of multivariate location and scatter

Using ideas and techniques from related disciplines frequently proves productive and often yields new insights and methods. In this paper, a method from experimental design is applied to the robust estimation of multivariate location and scatter. In particular, the procedure of determining discrete D-optimal designs is applied to the problem of finding the robust estimator called the minimum volume ellipsoid (MVE). The objective of the D-optimal design problem is to select h points to include in the design from a set of n candidate points such that the determinant of the information matrix is maximized. To calculate the MVE, a subset of h points must be selected where the volume of the ellipsoid covering them is the minimum over all possible subsets of size h. We demonstrate the relationship of these optimization problems and propose a technique to select the subset of points for both applications. The subset selection method is applied to several regression data sets where the true MVE estimate is known.