Applications to Subset Selection for MVE Estimator and Multiclass Discrimination

The MVE estimator is an important estimator in robust statistics and characterized by an ellipsoid which contains inside 100β percent of given points in IR n and attains the minimal volume, where β ∈ [0.5, 1.0) is of usual interest. The associated minimal ellipsoid can be considered as a generalization of the minimum volume covering ellipsoid which covers all the given points since the latter ellipsoid corresponds to the former one with β = 1.0. Though the computation of the minimal covering ellipsoid is tractable, that of the MVE estimator or, equivalently, the associated minimal ellipsoid is cumbersome since it has a nonconvex structure in general. In this paper, we present a new formulation for constructing an ellipsoid which also generalizes the notion of the minimum volume covering ellipsoid on the basis of the CVaR minimization technique which is proposed by Rockafellar and Uryasev [13]. In contrast to computing the MVE estimator, the proposed ellipsoid construction is formulated as a convex optimization and an interior point algorithm for the solution can be developed. In addition, the optimization gives an upper bound of the volume of the ellipsoid associated with the MVE estimator, which fact can be exploited for approximate computations of the estimator. Also, potential applicability of the new ellipsoid construction is discussed through two statistical problems: 1) robust statistics computations including outlier detection and the computation of the MVE estimator; 2) a multiclass discrimination problem, where the maximization of the normal likelihood function is characterized in the context of the ellipsoid construction. Numerical results are given, showing the nice computational efficiency of the proposed interior point algorithm and the capability of the proposed generalization.