Estimation of a simple linear regression model for random star-shaped sets

The estimation of a simple linear regression model is undertaken when both the independent and dependent variables are star-shaped set-valued random elements. The suggested regression model is defined by using the set arithmetic, assuming that the components representing location and imprecision of the random elements in the model are handled independently. Once the theoretical framework is established, the least squares estimation for the linear model is performed, taking into account an appropriate distance within the space of star-shaped sets. This approach results in a constrained minimization problem, which is analytically solved. Furthermore, the strong consistency of the obtained estimators is analyzed. Finally, the model is applied to a real-life situation and a simulation study is carried out.