An L1-type estimator of multivariate location and shape

The proposed estimator is a location and shape estimator which generalizes the L1-idea to a multivariate context. Consider a sample x1, . . . , xn of p-variate observations. Then the estimator is defined as the solution (μ̂, V̂ ) that yields the minimum of the sum of the distances di(μ, V ) = √ (xi − μ)′V −1(xi − μ), minimized under the constraint that V has determinant 1. The constraint det(V ) = 1 implies that we will get an estimate for the shape of the data cloud. To compute these estimates we use an Iteratively Reweighted Least Squares algorithm. We can also consider the estimator (μ̂, Σ̂) which is obtained as the solution to the problem of minimizing detΣ subject to the constraint