A linear regression model with LR fuzzy random variables

In standard regression analysis the relationship between one (response) variable and a set of (explanatory) variables is investigated. In a classical framework the response is affected by probabilistic uncertainty (randomness) and, thus, treated as a random variable. However, the data can be also subjected to other kinds of uncertainty, such as imprecision, vagueness, etc. A possible way to manage all of these uncertainties is represented by the concept of fuzzy random variable (FRV). The most common class of FRVs is the LR family, which allows us to express every FRV in terms of three random variables, namely, the center, the left and the right spread. In this work, limiting our attention to the LR FRVs, we address the linear regression problem in presence of one or more imprecise random elements. The procedure for estimating the model parameters is discussed, and the statistical properties of the estimates are analyzed. Furthermore, in order to illustrate how the proposed model works in practice, the results of some case-studies are given.