The mean square error of a random fuzzy vector based on the support function and the Steiner point

Metrics between fuzzy values are a topic with interest for different purposes. Among them, statistics with fuzzy data is growing in modelling and techniques largely through the use of suitable distances between such data. This paper introduces a generalized (actually, parameterized) L2 metric between fuzzy vectors which is based on their representation in terms of their support function and Steiner points. Consequently, the metric takes into account the deviation in ‘central location’ (represented by the Steiner points) and the deviation in ‘shape’ (represented by a deviation defined in terms of the support function and Steiner points). Then, sufficient conditions can be given for this representation to characterize fuzzy vectors, which is valuable for different aims, like optimization studies. Properties of the metric are analyzed and its application to quantify the mean (square) error of a fuzzy value in estimating the value of a fuzzy vector-valued random element is examined. Some immediate implications from this mean error are finally described.