Abstract: In regression analysis, some predictors might be unobservable, not observed, or ignored. These factors actually affect the response randomly. The observed data thus follows a conditional distribution when these factors are given. This phenomenon is called the distribution randomness. For such a working model, we propose an upper expectation regression and a two-step penalized maximum least squares procedure to estimate parameters in the mean function and the upper expectation of the error. The resulting estimators are consistent and asymptotically normal under certain conditions. Simulation studies and a data example are used to show that the classical least squares estimation fails but the proposed estimation performs well.

Key words and phrases: Distribution randomness, penalized least squares, upper expectation.