Central moments of a fuzzy random variable using the signed distance: A look towards the variance

Abstract

The central moments of a random variable are extensively used to understand the characteristics of distributions in classical statistics. It is well known that the second central moment of a given random variable is simply its variance. When fuzziness in data occurs, the situation becomes much more complicated. The central moments of a fuzzy random variable are often very difficult to be calculated because of the analytical complexity associated with the product of two fuzzy numbers. An estimation is needed. Our research showed that the so-called signed distance is a great tool for this task. The main contribution of this paper is to present the central moments of a fuzzy random variable using this distance. Furthermore, since we are interested in the statistical measures of the distribution, particularly the variance, we put an attention on its estimation using the signed distance. Using this distance in approximating the square of a fuzzy difference, we can get an unbiased estimator of the variance. Finally, we prove that in some conditions our methodology related to the signed distance returns an exact crisp variance.