The location of a minimum variance squared distance functional

Abstract

In this paper, we introduce a novel multivariate functional that represents a position where the intrinsic uncertainty of a system of mutually dependent risks is maximally reduced. The proposed multivariate functional defines the location of the minimum variance of squared distance (LVS) for some n-variate vector of risks X. We compute the analytical representation of LVS(X), which consists of the location of the minimum expected squared distance, LES(X), covariance matrix A, and a matrix B of the multivariate central moments of the third order of X. From this representation it follows that LVS(X) coincides with LES(X) when X has a multivariate symmetric distribution, but differs from it in the non-symmetric case. As LES(X) is often considered a neutral multivariate risk measure, we show that LVS(X) also possesses the important properties of multivariate risk measures: translation invariance, positive homogeneity, and partial monotonicity. We also study the mean-variance approach based on the balanced sum of an expectation and a variance of the square of the aforementioned Euclidean distance and control for the closeness of LES(X) and LVS(X). The proposed theory and the results are distribution free, meaning that we do not assume any particular distribution for the random vector X. The results are demonstrated with real data of Danish fire losses.