Title: The location of a minimum variance squared distance functional
Authors: Zinoviy Landsman - University of Haifa (Israel) [presenting]
Tomer Shushi - Ben Gurion University of the Negev (Israel)
Abstract: A novel multivariate functional is introduced 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 on Danish fire losses.