On the Possibilistic Approach to Linear Regression with Rounded or Interval-Censored Data

Consider a linear regression model where some or all of the observations of the dependent variable have been either rounded or interval-censored and only the resulting interval is available. Given a linear estimator β of the vector of regression parameters, we consider its possibilistic generalization for the model with rounded/censored data, which is called the OLS-set in the special case β = Ordinary Least Squares. We derive a geometric characterization of the set: we show that it is a zonotope in the parameter space. We show that even for models with a small number of regression parameters and a small number of observations, the combinatorial complexity of the polyhedron can be high. We therefore derive simple bounds on the OLS-set. These bounds allow to quantify the worst-case impact of rounding/censoring on the estimator β. This approach is illustrated by an example. We also observe that the method can be used for quantification of the rounding/censoring effect in advance, before the experiment is made, and hence can provide information on the choice of measurement precision when the experiment is being planned.