Upper expectation parametric regression

Every observation may follow a distribution that is randomly selected in a class of distributions. It is called the distribution uncertainty. This is a fact acknowledged in some research fields such as financial risk measure. Thus, the classical expectation is not identifiable in general.In this paper, a distribution uncertainty is defined, and then an upper expectation regression is proposed, which can describe the relationship between extreme events and relevant covariates under the framework of distribution uncertainty. As there are no classical methods available to estimate the parameters in the upper expectation regression, a two-step penalized maximum least squares procedure is proposed to estimate the mean function and the upper expectation of the error. The resulting estimators are consistent and asymptotically normal in a certain sense.Simulation studies and a real data example are conducted to show that the classical least squares estimation does not work and the penalized maximum least squares performs well.